Biological cells, tissues, organs and organisms exhibit a remarkable ability to grow by changing their mass and remodel by changing their internal structure. Growth and remodelling enable normal development and somatic growth, they drive adaptations to changes in external stimuli, and they mediate many responses to injury, disease and therapeutic interventions. In many cases, growth and remodelling processes depend strongly on mechanical factors and the associated mechanobiological response at the cellular level. Experience throughout the past three decades reveals that mathematical modelling of growth and remodelling processes can provide valuable insight into the basic biology and physiology, help guide the design and interpretation of appropriate experiments, and inform planning for therapeutic intervention. In this review, we discuss some of the historical and conceptual foundations upon which theories of growth and remodelling have emerged and provide illustrative examples of their use, focusing mainly on soft tissues. We conclude by identifying future challenges and opportunities. Show Scientific interest in using mechanics to understand fundamental aspects of biological systems dates back at least to the beginnings of modern science itself and some early examples are summarized in figure 1. While Galileo Galilei (1564–1642) was interested in the strength of bones, particularly in the optimal strength-to-weight relation in animals of different sizes, a true recognition of the role of mechanics in biological growth and remodelling only took place towards the end of the nineteenth century. Specifically, after some initial work, the period between 1867 and 1893 marks a golden age of growth mechanics with scientists from multiple disciplines observing, discussing and employing simple relationships between forces acting on growing organs and organisms and their overall response in terms of shape evolution or mass addition. In short order, scientists postulated new laws of physiology such as Davis’s law of soft tissue remodelling, Wolff’s law for bones and Woods’ law for the heart that paved the way to modern studies in biomechanics and mechanobiology [3–5]. These studies address different physiological systems, and include morphogenesis. The influential work of the Swiss anatomist Wilhelm His, in a series of essays on Unsere Körperform and Entwicklungsmechanik, suggested that developmental mechanics is a key driver for shaping organs and, in particular, is necessary to explain the characteristic folding pattern of our brain [7]. Following this early period, amply discussed in the monumental 1917 book On Growth and Form by Sir D’Arcy Thompson [8], interest of the biological community shifted to the biochemical and genetic components of growing organisms. It was thus not until the 1960s, with the rise of quantitative physiology and biomedical engineering, especially the modern field of biomechanics, that mechanics again became a main object of interest in the context of growth and remodelling.  Figure 1. A golden age of discovery and invention in growth and remodelling: Wilhelm His’s mechanical analogy between (a) the folding of a rubber tube and (b) the folding of a gut tube during morphogenesis [1]. (c) Wolff’s structural study of a bone [2]. (d) Traction methods by Davis to exploit mechanical homeostasis [3]. (e) Woods’ study of the heart (from Burton [4]). (f) Joseph Nutt’s innovative techniques such as a traction shoe to help elongate the gastrocnemius muscle [5] were based on the idea that stress influences growth and remodelling in soft tissues and bones. Adapted from [6].
An important modern contribution to the field of growth and remodelling of bone was the theory of adaptive elasticity in the 1970s [9–11]. This work distinguishes hard tissue growth via appositional, surface-based processes from soft tissue growth via interstitial, volume-based processes. Soon thereafter, an alternate approach for studying growth focused primarily on large strain kinematics to describe changes in size and shape [12,13], which motivated the theory of finite kinematic or volumetric growth. Several groups built upon these ideas, with an elegant conceptual approach for soft tissues put forward in a seminal publication in 1994 [14]. This theory has been adopted by many and is described in detail below. Alternatively, several other approaches evolved based on rate-dependent formulations for soft tissues using the concept of evolving stress-free natural configurations and evolving microstructural changes that affect macroscopic stiffness [15]. At about the same time, in the mid-1990s, it was recognized that growth and remodelling mechanics should include mass–stress relations that account for changes in the production or removal of material in response to changes in loads [16]. Motivated by these ideas and by an increased appreciation of the different mechanical properties and rates of turnover of different extracellular matrix components, a theory of constrained mixtures was proposed in 2002 [17], which is also described in more detail below. Others similarly adopted concepts from the continuum theory of mixtures, though employing other approaches [18–21]. Table 1 provides a brief summary of some of the key discoveries that connect growth and mechanics.
Biomechanics is the development, extension and application of the principles and methods of mechanics for studying problems of biology and medicine. Modern continuum biomechanics emerged in the mid-1960s following advances in nonlinear continuum mechanics and rapidly grew with advances in computational methods and computer technology, which enabled solution of complex initial-boundary value problems as well as the performance and interpretation of complex biomechanical experiments. Mechanobiology is the study of biological responses by cells to mechanical stimuli. Modern mechanobiology emerged in the mid-1970s following advances in mammalian cell culture and molecular and cell biology, noting that many biological responses to changes in the mechanical environment of the cell are mediated by changes in gene expression. While biomechanics includes diverse areas ranging from protein folding to gait analysis, continuum biomechanics focuses on cells, tissues and organs and naturally complements studies in mechanobiology: one exploits advances in mechanics and the other advances in biology, while both seek to understand questions of structure–function relationships and growth and remodelling throughout the cycle of life. Some of the first observations of mechanobiological responses in mammalian cells were in vascular cells: endothelial cells are very responsive to local blood flow-induced wall shear stress, which is typically of the order 1.5 Pa, while vascular smooth muscle cells are very responsive to blood pressure-induced intramural stress, which is typically of the order of 150 kPa. Clearly, therefore, different cell types can respond to very different magnitudes of imposed loads, in this case differing by five orders of magnitude. Indeed, this comparison reminds us that computed values of stress can be important mechanobiologically even if negligible biomechanically. Mechano-sensitive cells include other myocytes, chondrocytes, fibroblasts, macrophages and osteoblasts, to name a few [22]. Fibroblasts, for example, differentiate into myofibroblasts in response to increased stress and the cytokine transforming growth factor-beta. Like differentiated cells, stem cells also respond to their mechanical environment. Stem cell fate, that is, differentiation, can be driven in part by the stiffness of the matrix on which or in which the cells reside [23]. In general, progressively increasing stiffness tends to drive mesenchymal stem cells towards an adipocyte, myocyte, chondrocyte or osteoblast phenotype, respectively. Not surprisingly then, many cell types actively mechano-sense and mechano-regulate the extracellular matrix, which is facilitated by transmembrane proteins, notably integrins, that connect the extracellular matrix to the cytoskeleton that includes actin and myosin filaments that allow active force sensing or application [24]. There is, therefore, a pressing need to understand the mechanobiology—the transduction, transcription and translation of mechano-chemical information—and to mathematically model these pheonomena [25]. We now understand that many biological cells, tissues and organs exhibit a mechanical homeostasis, namely a tendency to maintain or restore a preferred mechanical state. When perturbed from this state, cells tend to engage mechanobiological processes to relax themselves or the associated extracellular matrix back to the preferred state. This behaviour is conceptually similar to stress relaxation in viscoelasticity or the tendency towards thermomechanical equilibrium though achieved via active cell-mediated rather than just innate physio-chemical processes. Similar to studies of stability in thermomechanics, much can be learned by studying the mechanobiological stability. The theory of finite growth was first formalized in the mid-1990s [14] and rapidly gained popularity with the use of computational methods to solve the underlying set of governing equations [26]. In contrast to the traditional theory of finite elasticity that consists of the classical set of kinematic, balance and constitutive relations, the theory of finite kinematic growth requires two additional sets of equations: kinematic and kinetic equations of growth [27]. Those two relations have to be prescribed constitutively to close the system of governing equations and thus are specific to the type of physiological system—the brain [28,29], the vasculature [30], the gut [31,32], the airways of the lung [33], the skin [34,35] or the heart [36]. The theory of finite growth is based on a particular multiplicative decomposition of the deformation gradient. Consider a motion φ : B0→Bt that maps an initial reference configuration B0 to a current configuration Bt via x(X,t)=φ(X,t), where x(X,t) is the position at time t of the material point originally located at X at time t = 0. The main idea is to decompose the deformation gradient, F=∇ φ=∂x/∂X, into an elastic part Fe and a growth part Fg [14,37] F=Fe⋅Fg.2.1 The growth tensor Fg effectively represents the addition or the subtraction of mass to a local volume element. We typically prescribe Fg constitutively, either directly or in rate form to characterize the evolution of growth. Typically, only the elastic contributions Fe generate mechanical stresses. Figure 2 illustrates that we can understand growth via a series of stress-free configurations. In the simplest case, we can assume a stress–strain behaviour of neo-Hookean type with Cauchy stressσ=[λln(det(Fe))−μ]I+μFe⋅(Fe)t,2.2 where λ and μ are the elastic Lamé constants and I is the second-order identity tensor. Similar to the classical theory of finite elasticity, this stress enters the linear momentum equation in equilibriumdiv(σ)+ρb=0,2.3 where ρ is the overall mass density and b is the body force. Provided we know the growth tensor Fg, we can solve this equation, with appropriate boundary conditions, either analytically for simple geometries or numerically using nonlinear finite-element solvers. A defining feature of the theory of finite growth is the series of incompatible growth configurations expressed mathematically by the growth tensor Fg. Importantly, the growth tensor is not necessarily a gradient of a vector field. Physically, this implies that, once grown, the initial pieces of a living system may become incompatible and may no longer fit together [38]. Figure 2 illustrates that these pieces must be deformed elastically to remain a continuum without openings or overlaps, which results in growth-induced residual stresses. These residual stresses can arise from differential growth and are a hallmark of living tissues that fulfil many functions. We can easily visualize the existence of residual stresses through the classical opening angle experiment by introducing a radial cut in an isolated arterial ring [39]. In the developing brain, differential growth is increasingly recognized as one of the major factors that modulates the complex surface morphology of the cerebral cortex [40]. Representing growth and remodelling via the multiplicative decomposition of the deformation gradient poses a number of interesting mathematical questions. For a hyperelastic material characterized by a strain energy functional W, we can parametrize this functional as W(Fe)=W(F⋅Fg−1), recalling that the growth tensor Fg has to be defined constitutively [41]. In the following, we revisit the mathematical properties of W(Fe) to understand the consequences of growth Fg.Figure 2. Schematic drawing of evolving configurations of importance in both a theory of finite kinematic growth (bottom portion) and a constrained mixture theory (top portion). In particular, note the common reference κo(0) and current κ(s) configurations. In the kinematic growth theory, one imagines that infinitesimal stress-free portions of the body grow independently via the transformation Fg, which need not result in compatible growth. An elastic ‘assembly’ transformation Fa ensures a contiguous traction-free body, which typically is residually stressed. Finally, an elastic load-dependent transformation FE yields the current configuration of interest, with Fe=FE⋅Fa that part of the deformation that is elastic and determines the stress field. Conversely, in the constrained mixture theory, it is the constituent-specific deformation Fn(τ)α from an individual stress-free configuration that dictates the elastic stress within that constituent. It is easy to show that Fn(τ)α(s)=F(s)⋅F−1(τ)⋅Gα(τ), where Gα(τ) is a so-called ‘deposition stretch’ tensor that accounts for cells depositing new extracellular matrix under stress when incorporating it within stressed extant matrix. Both approaches require multiplicative deformations, one in terms of the prescribed growth of stress-free elements and one in terms of a deformation that is built into individual constituents when they are incorporated within extant tissue.
This guiding principle requires that constitutive relations remain invariant under changes in the frame of reference, e.g. that material properties are independent of superimposed rigid-body motions. This implies that W(F¯e)=W(Fe)with F¯e=Q⋅Fe∀ Q∈ SO(3), 2.4 where Fe=F⋅(Fg)−1 and SO(3) is the group of all rotations about the origin of three-dimensional Euclidean space. To a priori satisfy this requirement, we can select strain energy functionals W with an explicit dependence on the elastic right Cauchy–Green deformation tensor CeW(C¯e)=W(Ce)with C¯e=(Q⋅Fe)t⋅(Q⋅Fe)∀ Q∈ SO(3), 2.5 where Ce=(Fe)t⋅Fe=(Fg)−t⋅Ft⋅F⋅(Fg)−1. We immediately realize that our constitutive choice for the Cauchy stress (2.2) satisfies the requirement of material frame indifference a priori.This consideration of symmetry is more subtle. Physically, for an isotropic material, unless growth itself is isotropic, the overall response of the material need not remain isotropic. The problem is then to find the transformation of a given material symmetry group G for W(Fe) in the presence of growth. To determine the material symmetry group, we notice that if Q∈G, a linear transformation belonging to the group of symmetry of the grown material [42], then W(F⋅(Fg)−1)=W(F⋅(Fg)−1⋅Q)=W(F⋅(Fg)−1⋅Q⋅Fg⋅(Fg)−1).2.6 This identity tells us that the material symmetry group after the growth-remodelling process is the conjugate of G through Fg, G¯={(Fg)−1⋅Q⋅Fg;Q∈G}. We should keep this in mind when considering the growth and remodelling of an anisotropic material. A remarkable example is a transverse isotropic material when Fg is a rotation itself: the symmetry axis rotates as it occurs in the reorientation of trabeculae in the bones.A crucial point of the multiplicative Fe⋅Fg decomposition is the form of the tensor Fg that gives rise to a biomechanically expected behaviour. A reinterpretation of the classical dissipation inequality of continuum mechanics in open systems [43–45] can help to determine a suitable growth law. An energetic characterization of the dissipative nature of the growth process can suggest thermodynamically admissible evolution laws. However, we must take great care when using thermodynamics. While the entropy inequality can provide useful guidelines in closed systems, it often does not provide valuable information in open systems that may contain entropy sinks. Early thermodynamic considerations identified the Mandel stress and the Eshelby stress as key quantities in formulating growth laws [43,46]. They also identify two types of growth processes [6]: first, passive growth processes during which the dissipation inequality is satisfied even in the absence of entropy sinks. This situation is typical for physical systems and arises, for instance, in plasticity, thermoelasticity or gel swelling, where a non-compliant entropy contribution is not required for the process to take place [47]. The observed macroscopic response is then slaved to the overall dissipation mechanism. This type of growth is not incompatible with physical processes. Second, active growth processes, during which case an additional sink of entropy must be included to satisfy the dissipation inequality. This situation arises in many biological systems where the cell, through its genetic information and internal energy contribution, can alter the entropy of a system by forcing a pre-programmed response to external stimuli, against the physical increase of entropy. Therefore, active processes at the microscopic level must be at work, and, indeed, are key to the organization of highly organized structures in a dissipative environment [48]. From a geometric perspective, the deformation gradient is a map from the tangent space of a point in a reference configuration to the tangent space of the same point in a current configuration. In classic nonlinear elasticity, a convenient measure of changes in distances and angles is the right Cauchy–Green tensor C=Ft⋅F.2.7 In the presence of growth, the tensor Fg maps the tangent space at each point of the initial configuration to the tangent space of a virtual stress-free state. The equivalent to the Cauchy–Green tensor, now associated with growth,Cg=(Fg)t⋅Fg,2.8 acts as a metric for the intermediate virtual configuration. The problem is now that the union of all the tangent spaces forms a tangent bundle that defines this intermediate configuration. But this configuration is not clearly defined. An alternative approach is to start with a reference configuration that is not Euclidean and to characterize inelastic effects such as growth by the intrinsic geometry. The natural structure to achieve equivalence between a theory of multiple configurations and a non-Euclidean theory is a Weitzenböck manifold [49,50]. In this framework, the reference configuration is a material manifold with a vanishing curvature tensor but with a torsion tensor T defined by the growth tensor FgT(Fg)=Fg−1 skw(∇Fg),2.9 where skw(°) defines the skew-symmetric part of a tensor (°), defined by skw(T)kij = Tkij − Tkji in a Cartesian basis. We can then formally define the intermediate configuration that is routinely used in finite inelasticity as a Weitzenböck manifold with torsion tensor T(Fg) and its tangent bundle is the natural space on which we can define all kinematic quantities of the theory with distorsions, e.g. morphoelasticity, thermoelasticity or elasto-plasticity. This a posteriori justification of the classical morphoelasticity approach provides a rigorous way to answer fundamental questions on the mathematical nature of growth processes. Borrowing ideas from the geometric theory of defects in solids, we can further generalize the theory to introduce new effects associated with growth- and remodelling-localized point or line growth. These effects require a generalization of the Weitzenböck manifold to include non-vanishing curvature and non-metricity [49,51,52]. Using differential geometry also opens the door to developing new numerical schemes that take advantage of the underlying geometric structure [53]. Another interesting difficulty arises when considering the dynamic evolution of growth. If we assume that the growth tensor Fg depends on the stress, the torsion is determined by a set of partial differential equations involving torsion itself. The evolution of the geometry and topology of manifolds through differential equations, such as Ricci flows [54], is an important topic of differential geometry that is of direct relevance to the mathematical description of kinematic growth. Taken together, the theory of differential geometry provides an elegant theoretical basis for growth and remodelling of a single constituent. However, in a theory of mixtures, multiple constituents with different reference configurations are mixed and the underlying structure of the material manifold is less obvious. Much work must be done in this regard to elucidate key foundational aspects of a mixture theory.In this review, we focus primarily on volume growth, which assumes an addition or subtraction of mass within regions of existing tissue. Alternatively, tissues may grow by surface growth, which assumes an addition of material at the tissue surface Γ. Typical examples are growing horns [55], tusks [30], shells [56] or bones [57]. Surface growth models also often adopt a multiplicative decomposition of the deformation gradient, F=Fe⋅Fg, into an elastic part Fe and a growth part Fg, or an additive decomposition of the material velocity V=VΓ+Vg into the surface velocity VΓ and the velocity of the grown material Vg [30]. Independently of the geometric nature of the governing equations, we can formulate the problem of growth either as a variational problem with respect to the modified strain-energy density function or as a set of nonlinear partial differential equations. Representing growth through a set of partial differential equations allows us to establish valuable results on well posedness and local existence of solution [58], which give hope that general global results will follow. A natural question to ask is how the classical problems of elasticity extend to morphoelasticity. For instance, there exist several classes of universal solutions for isotropic materials [59]. In the case of compressible, isotropic materials, a complete generalization of the classic problem of Ericksen is possible [60], but the case of incompressible isotropic materials is still open. Once the existence of such solutions is established, we can study general bifurcation and stability phenomena for problems that involve both mixed boundary conditions and varying growth parameters. The continuum theory of mixtures is a logical starting point for modelling many aspects of cell, tissue and organ growth and remodelling since processes and properties differ by cell type and extracellular matrix constituent. For example, actin and intermediate filaments polymerize/depolymerize at different rates within an adapting cell; elastic and collagen fibres have different half-lives in the extracellular matrix [17,24]. Yet, a full mixture theory is complicated by numerous factors, including the difficulty of quantifying momentum exchanges between constituents within nonlinear solids and specifying how traction conditions partition on boundaries, especially in the presence of evolving mass or microstructure. Motivated by Fung’s call for mass–stress relations, a theory of constrained mixtures was proposed to exploit advantages of a mixture theory while avoiding inherent complexities [17]. This approach allows us to model different mechanical properties, rates of turnover and natural configurations of the different constituents. In a continuum theory of mixtures, we first consider mass balance for a mixture of α = 1, 2, …, N structurally significant constituents ∂ρα∂s+ div(ραvα)=m¯α,3.1 where ρα is the spatial mass density, vα is the velocity and m¯α is the net rate of mass density production or removal, which we must prescribe constitutively. It can depend on various chemical or mechanical factors, including stress and the growth and remodelling time s. Three assumptions for a constrained mixture theory of growth and remodelling are: first, that individual constituents can have separate natural, stress-free configurations, but they are constrained to move with the mixture as a whole, xα≡x; second, that growth and remodelling are typically slow relative to rates of mechanical loading, x˙α=vα≡v≅0; third, that the net rate of mass production of removal can be modelled via a multiplicative decomposition m¯α=mαqα, where mα(τ) > 0 is the true rate of mass production and qα(s, τ) ∈ [0, 1] is a survival function that tracks that part of the constituent produced at growth and remodelling time τ ∈ [0, s] that remains at current time s. These assumptions render mass balance integrableρRα(s)=ρRα(0)Qα(s)+∫0smRα(τ)qα(s,τ) dτ,ρR(s)=∑α=1NρRα(s), 3.2 where the subscript R refers to quantities defined per unit reference volume, for example ρRα=(det(F))ρα, and Qα(s)∈[0,1] is similar to qα(s,τ) but for constituents produced at or before time 0 and having the property that Qα(0)=1 similar to qα(τ,τ)=1.To use the classical equation of linear momentum balance (2.3), rather than a full mixture relation that necessarily includes momentum exchanges, we further assume that the Cauchy stress can be determined from a rule-of-mixtures relation for the stored energy per unit reference volume, WR=∑WRα, consistent with a standard constitutive relation of finite elasticity σ=2det(F)F⋅∂WR∂C⋅Ft−pI,3.3 where F=∇φ=∂x/∂X is the deformation gradient, C=Ft⋅F is the right Cauchy–Green tensor and p is a Lagrange multiplier that enforces incompressibility during transient motions. The key requirement is a constitutive form for the energy stored in each constituent due to its deposition within extant tissue and its individual deformation. With (3.2), we can positρWRα(s)=ρRα(0)Qα(s)W^α(Cn(0)α(s))+∫0smRα(τ)qα(s,τ)W^α(Cn(τ)α(s)) dτ,3.4 where ρ is the mass density of the mixture, W^α are stored energy functions for individual structurally significant constituents that depend on constituent-specific right Cauchy–Green tensors Cn(τ)α(s)=(Fn(τ)α(s))t⋅Fn(τ)α(s), with Fn(τ)α(s) the deformation gradient experienced by an individual constituent α relative to its own natural configuration κnα(τ), which can evolve and is denoted simply by a subscript n(τ); recall figure 2. Two simple special cases illustrate the motivation for this particular general form. For case 1, consider the special case of no growth and remodelling, namely a normal mechanical behaviour defined within finite elasticity. In this case, s = 0 and equation (3.4) reduces toWRα(s=0)=ρRα(0)W^α(Cn(0)α(0))ρ=ϕRα(0)W^α(0),3.5 which is a simple rule-of-mixtures relation, as desired, with ϕRα a mass fraction. For case 2, consider the special case of tissue maintenance wherein tissue turns over continually, with production balancing removal within an unchanging mechanical configuration with unchanging material properties. In this case, the natural configuration is κnα(τ)≡κnα(0), whereby the constituent-specific strain energy function within the integral does not change with growth and remodelling time τ ∈ [0, s]ρWRα(s)=(ρRα(0)Qα(s)+∫0smRα(τ)qα(s,τ) dτ)W^α(Cn(0)α(s)),3.6 whereby, from equation (3.4) and the definition of the mass fraction, we again recover a simple rule of mixtures.Equation (3.4) reveals the need to determine the constituent-specific deformation gradient Fn(τ)α(s). From figure 2, we can imagine evolving configurations of the mixture from an in vivo reference configuration κ(0) to an intermediate κ(τ) or final κ(s) configuration. We can define deformations associated with these configurations similar to standard finite elasticity, for example F(τ) and F(s). Importantly, there can be a natural, stress-free configuration at any of these same times for each constituent, from which we can consider the constituent to be deformed when deposited within extant matrix at the time of its synthesis, say τ ∈ [0, s]. From figure 2, we conclude that Fn(τ)α(s)=F(s)⋅F−1(τ)⋅Gα(τ),3.7 where the linear transformation Gα is the deposition stretch; it accounts for cells depositing new constituents with an intrinsic pre-stress, an important aspect of mechanical homeostasis. Interestingly, both the theory of finite growth and the constrained mixture theory require multiplicative decompositions of the motion to account for growth and remodelling according to figure 2. The former requires a growth tensor that depends constitutively on chemo-mechanical stimuli and accounts for rates of change; the latter requires a deposition stretch that plays the role of an internal variable while the rates of change are accounted for by mass production and removal functions.Equation (3.4) also reveals that the constrained mixture theory requires three constitutive functions, one for mass production mRα(τ), one for mass removal qα(s,τ) and one for the mechanical properties of the existing mass W^α(Cn(τ)α(s)). We can define the constituent-specific stored energy functions within the context of finite elasticity and can include neo-Hookean or Fung exponential forms, which are common in tissue biomechanics. Consider, therefore, the relations related to mass turnover. Importantly, many matrix constituents are constantly being produced, elastin being a counter-example, and most cell types are constantly dividing, cardiomyocytes being a counter-example, hence it seems reasonable to posit production in terms of a basal rate as well as changes therein due to chemo-mechanical or other stimuli. It is also known that many processes such as degradation of matrix and death of populations of cells follow first-order-type kinetics, also at basal rates that can be modulated by chemo-mechanical stimuli. We assume that mRα(τ)= moαg(stimuli)andqα(s,τ)=exp(−∫τskoαf(stimuli) dτ), where moα is an original basal or homeostatic production rate and koα is an original basal or homeostatic rate parameter, with g and f scalar functions that need to be determined constitutively for individual cells and tissues, typically as functions of deviations in mechanical stress or chemical factors from normal values, with g = 1 and f = 1 at a homeostatic state. We can show that basal homeostatic processes, such as tissue maintenance, further require that basal rates of production and removal balance, for example moα/koα=ρoα.It is important to emphasize that growth and remodelling often proceed hand in hand. Changes in internal microstructure can arise when old constituents are replaced with new constituents, via degradation and deposition, that have a different orientation, diameter, cross-linking or other characteristic. Modelling this change in cytoskeleton or matrix, which need not involve a change in mass, is thus critical. Indeed, there can also be cases wherein cells remodel tissue microstructure independent of mass production and removal, that is, by simply refashioning extant matrix, which requires actomyosin activity. Just as the fibroblast is the prototypical synthetic cell, collagen is the prototypical matrix constituent, frequently central to remodelling in adaptations, disease and injury, especially wound healing, wherein the fibroblast differentiates into a myofibroblast. Our commentary here is brief for excellent reviews on cell-mediated collagen remodelling can be found elsewhere [61,62]. Note, however, that matrix orientation defines tissue anisotropy, a key characteristic of matrix mechanics. If the deformation is affine, filament or fibre orientation can be calculated simply from λm=F⋅M, where m and M define the filament or fibre direction in current and reference configurations, respectively, and λ=C : M⊗M is the filament or fibre stretch. In remodelling, we are interested in changes in orientation separate from those induced simply by deformations. Cell-mediated fibre alignment often results from mechanical stimuli, with cells orienting the fibres along the directions of principal strain, principal stress or some angle between [62]. Different cell types appear to use different rules to dictate such alignment, and it has been shown that testing various hypotheses against known alignments under normal conditions can often be used to identify cell-specific alignment rules as well as relations describing the rate of change of the orientation during remodelling. The interested reader is referred to the aforementioned reviews or some key papers [63–65]. Various direct uses of this constrained mixture theory range from modelling aneurysmal enlargement [66] and cerebral vasospasm [67] to the in vivo development of tissue-engineered constructs [68]. The associated computational implementation is expensive owing to the heredity integrals and the need to store past histories, thus multiple methods have been developed to render the method more computationally tractable. One is a simple reduction of the modelling to consider only an initial and a single perturbed state, thus eliminating the need for the heredity integrals [69,70]. Other methods have included a temporal homogenization to yield rate equations [71] and a concept of mechanobiological equilibration [72]. In addition, there are many other mixture-based models of biological growth and remodelling, some of which include methods of finite kinematic growth coupled with the concept of a constrained mixture. The interested reader is referred to the many other examples of mixture-based theories, including a study of cartilage growth [18], a model of tumour growth [19], a general framework for growth [73], growth in tissue engineering [74], a study of residual stress [21], a new multi-generational theory of growth [75], a focus on mass transfer within growth mechanics [76], hypertensive remodelling of arteries [77], development of the aorta [78], a study of diverse applications including cervical remodelling in pregnancy [79], a coupling of haemodynamics and arterial wall growth and remodelling [80], additional studies of tissue engineering [81,82], anisotropic volumetric remodelling [83] and asthmatic airway remodelling [84], to name a few. An interesting mechanism through which biological shape can develop is a mechanical instability resulting from prior growth [37,85]. For example, if different parts of a tissue grow at different rates, they can build sufficient stresses to create a mechanical instability similar to the well-known Euler buckling instability [86]. For instance, the shapes of multiple organs arise as a direct result of growth and remodelling during early development. This period of embryonic or fetal life, dominated by cell proliferation and tissue formation, gives multiple examples of pattern formation via buckling and post-buckling induced by growth as illustrated in figure 3. Figure 3. Mechanical instability. Various patterns emerge from multi-layered systems during embryogenesis. Zigzag patterning in pre-villus ridges in the jejunum of turkey embryos (a) and simulations using the theory of finite kinematic growth (b) show the importance of mechanical instabilities in governing shape in morphogenesis. Adapted from [31]. (Online version in colour.)
Each part of our body is structured in multiple adjacent layers that had experienced growth and remodelling. With very simple modelling of volumetric growth, we can explain the development of various structures in our body, from circumvolutions of the basal membrane that separates the dermis from the epidermis [34] to the undulations of our small intestine [31] or the convolutions that define our brain. These patterns are not only amazing structures, they also play a role in physiology: dips are the location of adult stem cell niches in our skin or small intestine; defects in brain circumvolutions, localized or not, induce severe pathologies of newborns [87]. Typically, the time scale associated with the growth process is very long compared with the visco-elasticity of the sample. This implies that, from the biomechanical point of view, the structure always remains in equilibrium and can be modelled using the minimization of a free energy. Variational methods in finite elasticity with volumetric growth are powerful tools to analytically investigate pattern formation; they establish not only the instability threshold, the so-called control parameter in the theory of bifurcations, but also the first nonlinearities. Figure 4 shows that, in the simple geometry of a slab subjected to growth in one direction, the Biot instability [88] predicts a periodic pattern without specification of the geometry: stripes, squares or two families of hexagons depending on their centre elevation. Even for neo-Hookean elasticity, a weakly nonlinear analysis establishes that the selected pattern is indeed the pattern that is observed experimentally [89]. Figure 4. Mechanical stability. Only small changes can induce large variations in pattern formation and the generation of shape. Typical undulations obtained after buckling instability include stripes, chessboards, hexagon+ and hexagon− patterns. Adapted from [90]. (Online version in colour.)
One approach is based on defining an energy density in terms of invariants I1, I2, I3 that in turn depend on the deformation gradient F=∂x/∂X. Again using Fe=F⋅Fg−1, the basic strategy consists of minimizing the free energy of the system under constraints. These constraints include incompressibility, as commonly applied in biomechanics. Surface energies, as, for example, from capillarity, compete with the boundary conditions deduced from the variational process E=∫V[xX2+yY2+τ2(xX2+yY2)−2−2p(J−1)]τ dX dY, where p is a Lagrange parameter enforcing two-dimensional incompressibility, and τ=ϑ1/ϑ2 where ϑ1 and ϑ2 are eigenvalues of the two-dimensional growth tensor Fg= diag{ϑ1,ϑ2}, which are monotonically increasing functions in time. Growth implies that ϑ1,ϑ2>1. Stresses originate at borders due to connections to other tissues, as well as due to anisotropic or heterogeneous growth. As growth proceeds, stresses increase in the sample and, being compressive, can induce bifurcations that are revealed with this method. In complex geometrical systems that are incompressible and involve several layers, directly eliminating the Lagrange parameter p helps to render the system of nonlinear partial differential equations tractable. This is also why the introduction of a nonlinear stream function can be helpful [34]. As in viscous two-dimensional hydrodynamics, this function allows us to solve the Euler–Lagrange equations derived from the free energy, though it is restricted to systems that can be reduced to two dimensions and do not develop fully three-dimensional instability patterns. Variational methods may also include viscoelasticity. Motivated by cyclic tensional tests, which are commonly performed to study living tissues [91,92], and inspired by the difficulty of generalizing the classical one-dimensional Maxwell and Kelvin–Voigt models of linear viscoelasticity into the nonlinear regime, the Rayleighian method turns out to be especially powerful [93,94]. Without difficulty, this method provides the correct formulation of dissipative tensors, which have been a matter of extensive debates in the acoustic domain [95].Whereas the growth-induced instabilities discussed in the prior section refer to possible instabilities of mechanical equilibria that arose from prior growth and remodelling, there is also a need to study possible instabilities of the actual growth and remodelling process itself over long periods. When growth and remodelling arises from mechanobiological responses by cells, we can refer to this analysis as one of mechanobiological stability [96], which we can, of course, extend to include responses to biochemical or other stimuli as well. We recall that most biological cells continually reorganize, produce or degrade their surrounding extracellular matrix, often resulting in different geometries and properties. There exist special configurations wherein the net effects of these cellular processes cancel so that these configurations persist over long periods—such configurations are called homeostatic and thus represent mechanobiological equilibria. A key question is whether such equilibria are stable, that is, if they can be maintained despite particular perturbations. It appears that, in healthy adults, most tissues typically maintain their geometry and mechanical properties for many years despite myriad perturbations in diet, exercise and external stimuli, including brief infections, hence suggesting mechanobiological stability under normal cases. Natural ageing, disease progression and responses to injuries, however, can be very different; they may reflect a compromised homeostasis and in some cases mechanobiological instability. Theories of finite growth and mixture-based theories can both be used to examine growth and remodelling related stabilities. For example, consistent with ideas from §2, consider a growth law of the form [97] Fg−1⋅F˙g=K : (σ−σh),5.1 where the dot represents the material time derivative, Fg is a growth tensor that effectively characterizes geometric consequences of local changes in mass in evolving stress-free configurations, and K is a fourth-order tensor characterizing the growth response rate to differences in Cauchy stress σ from homeostatic values σh. In this case, mechanobiological equilibrium is defined by Fg˙=0, whereas mechanical equilibrium is enforced implicitly by using values of stress from equilibrated mechanical solutions. In a simple two-dimensional case, for example, we obtain two first-order differential equations of the form γ˙i=fij(K,σ)γ j. We can conduct a usual stability analysis by considering γi=γieq+ϵγi¯, where the superscript eq denotes equilibrium values and the overbar denotes perturbations, with ε ≪ 1. Eigenvalues of the associated Jacobian matrix dictate the mechanobiological stability as indicated in figure 5.Figure 5. Mechanobiological stability. Phase diagram (a) showing distinct regions with different equilibrium states and stability behaviours for a growing tubular structure as a function of its homeostatic stress and anisotropy [97]. Phase-plane-type plots (b,c) showing both unstable, for a low value of a parameter governing the rate of matrix synthesis (b), and asymptotically stable, for a higher value of this parameter (c), growth and remodelling responses of an artery (normalized wall thickness h versus luminal radius a) following a transient perturbation in blood pressure [98]. (Online version in colour.)
Studies using the constrained mixture theory of §3 similarly use methods from dynamical systems, for example stability analyses, but in terms of different characteristic equations [96,98–100]. In particular, consistent with equations (3.2)–(3.4), the focus turns towards rates of change in mass (equivalently, referential mass density ρR=(det F)ρ=(det F)∑ρα) and stress, with mechanobiological equilibrium requiring both mass densities and associated mechanically equilibrated stresses to remain unchanged during the period of interest [72,96]. Dynamic stability analyses around these equilibria thus generally require us to consider equivalent nonlinear evolution equations for mass and stress in rate form. For example, in a simple two-dimensional case, we can derive a non-autonomous system of first-order differential equations for mass density and in-plane stresses for a mixture of the form [98] y˙(s)=[f](y(s),s) andy=[ρR,σ1,σ2]t.5.2 The growth and remodelling time s ≥ 0, with s = 0 representing a homeostatic state denoted by superscript h, which requires σ(0)=σh. Importantly, we can examine both mechanobiologically static and dynamic stability analogous to static (e.g. limit point bifurcations) and dynamic (e.g. self-excited) mechanical stability. Similar to the volumetric growth approach, we seek stable and unstable solutions given small perturbations of the form y(0)=yh+δyh, which can be examined from an eigenvalue analysis of the associated linearized, autonomous system of differential equations. Depending on the formulation and type of perturbation, one can identify neutral [96,100] or asymptotically [98] stable mechanobiological solutions, the latter consistent with clinical experience that many normal growth and remodelling processes appear to preserve homeostatic states over long periods. Studies have shown that many parameters affect the mechanobiological stability, including but not limited to the level of homeostatic intramural and wall shear stresses, the intrinsic material stiffness and mass density of the tissue or its constituent parts, the associated rates of matrix synthesis and degradation, the presence or not of muscle contractility, and so forth (figure 5). Such insight could be useful in better understanding disease progression or healing following injury, particularly with regard to possible palliative clinical treatments. For example, antagonizing a particular microRNA can hasten collagen production by mesenchymal cells, and simulations show that such increases could slow the rate of enlargement of certain arterial aneurysms [96]. Given the recent recognition that many disease processes involve biological instabilities associated with positive feed-back loops [101], there is continuing motivation to study mechanobiological and related instabilities.Mathematical models and computational simulations of growth and remodelling have been widely used to study myriad problems ranging from cellular mechano-sensing to developmental biology, understanding disease progression and engineering tissue replacements. In this section, we highlight a few representative examples to show the depth and breadth of these efforts. For each case, we identify problem-specific constitutive relations. In constrained mixture theories, we must identify problem- and constituent-specific constitutive relations for rates of mass production mRα>0 and survival qα(s,τ)∈[0,1] and stored energy W^α(Cn(τ)α(s)). In the finite growth theories, we can define the growth kinematics to be isotropic, e.g. in tumours or in the brain, with Fg=ϑI, transversely isotropic along a specific direction n, e.g. in skeletal or cardiac muscle, with Fg=I+[ϑ−1]n⊗n, or orthogonal to a specific direction n, e.g. in skin, with Fg=ϑI+[1−ϑ]n⊗n, or completely arbitrary [30]. In these examples, ϑ is a growth parameter that follows specific kinetic definitions. In the simplest case, growth is purely morphogenetic, which implies that the evolution of ϑ is independent of physical factors [102]. In tumours or tissue engineering, however, growth depends primarily on the availability of nutrients, and we have to account for nutrient supply [103]. In many other living systems, growth is controlled by the mechanical environment, through shear stresses or pressure in the vascular system, through the area stretch in skin [35] or through the fibre stretch in skeletal muscle [104]. A classical everyday example is the chronic shortening of the gastrocnemius muscle in women who frequently wear high-heeled footwear, as highlighted in figure 6. In the following, we consider a variety of illustrative examples of tissue growth and remodelling. Of course, given the vast number of possible examples and methods, this list is necessarily limited. As noted above, we do not consider growth and remodelling of bone, plants, sea life, etc. Moreover, we focus on continuum theories, noting that other approaches exist as well, including agent-based models [106,107] and stochastic lattice-based models [108,109]. Figure 6. Application to skeletal muscle. Skeletal muscle can lengthen and shorten in response to sustained stretch. A classical everyday example is the chronic shortening of the gastrocnemius muscle in women who frequently wear high heels. The muscle shortens by a chronic loss of sarcomeres, which results in chronic muscle fatigue and an increased injury risk. Adapted from [105]. (Online version in colour.)
Motility refers to the ability to move spontaneously and actively. Animal locomotion, which is powered by the contractility of skeletal muscle, is part of the story while, at the level of individual cells and biological tissues, motility is related to cell migration, the immune system response, the establishment of neuron synapses and even wound healing. More broadly, motility is fundamental to both the generation of life and the propagation of diseases, for example through unicellular swimming of sperm, bacteria, parasites and invasion of metastatic tumour cells, as well as to many other biological phenomena of great relevance for life. Muscle contraction is at the root of animal locomotion. Active remodelling of muscle tissue is, thus, central in all motility phenomena involving higher organisms. Quoting from the 1924 Linacre lecture of the English physiologist and Nobel laureate C. S. Sherrington: ‘To move things is all that mankind can do … for such the sole executant is muscle, whether in whispering a syllable or in felling a forest’ [110]. Study of the mechanics of muscle is a large field, and it would be impossible to even scratch its surface in a few lines. Yet, there are a few classical mathematical references that discuss the basic mechanisms [111,112]. The impact of muscle activity and coordination in the study of animal locomotion is also a vast subject. Classical textbooks [113,114] and reviews [115] can provide a starting point. In recent years, the pioneering work ‘Studies in animal locomotion’ [116] has inspired a renewed interest in the motility of limbless organisms which, in many respects, provide simpler model systems for studying coordination and locomotion. Locomotion arises from the mechanical interactions of an active elastic body with its surroundings, driven by the action–reaction principle. Muscle activity selects a preferred state of deformation, the configuration that the body would acquire in the absence of external forces. Modulating this state of spontaneous deformation in time in the presence of a surrounding medium, e.g. a fish waving a fin, generates reactive forces from the environment. These can be frictional ground forces in the undulatory locomotion of snakes [117] and in the peristaltic locomotion of worms [118], or viscous and inertial forces from a surrounding fluid in the cases of swimming and flying [119–121], which can be exploited as propulsive forces. Growth is the engine of some distinctive forms of single cell motility. Indeed, while most forms of biological motility are powered by molecular motors, which could be viewed as instances of motility by remodelling, since the steps executed by molecular motors are nothing but chemically activated conformational changes, actin polymerization is a distinctive engine for some specific forms of cell locomotion. Examples include the protrusion of lamellipodia of spreading and migrating embryonic cells, and the bacterium Listeria monocytogenes that propels itself through its host’s cytoplasm by constructing behind it a polymerized tail of cross-linked actin filaments [122–124]. Remarkably, neuronal growth works in a similar manner, with the protrusion of neurites from the main axonal body, which is powered by the polymerization of actin filaments at the neurite tips known as growth cones [125]. To understand how growth of actin fibres can act as the propulsive engine for a crawling cell, it is instructive to look at the propulsion mechanism of non-adherent cancer cells migrating inside a capillary tube [126]. Actin filaments polymerizing at the leading edge protrude the plasma membrane forward at a velocity V in the laboratory frame, positive if forward. At the same time, they are advected backward by a retrograde actin flow at local velocity v in the cell body frame, negative if backward, powered by myosin molecular motors. Friction from the tube walls is described by a force per unit area, τ = −μ(v + V), where μ > 0 is the viscous friction coefficient. Balance of force along the tube axis yields 0=−πR2αV−2πRμ∫0L(v(x)+V) dx, where α is a coefficient for the hydrodynamic resistance due to flow induced in the tube, and R and L are the radius and length of the cell–tube contact, respectively. We denote the average velocity of the actomyosin retrograde flow by ⟨v⟩:=∫0Lv(x) dx/L, and solve for the velocity of the leading edge of the plasma membrane VV=−11+(αR)/(μ2R)⟨v⟩. The above equation shows that the velocity is generally intermediate between the value −〈v〉, for perfect cell-wall grip, μ → +∞, and the value zero, for perfect cell-wall slip, α = 0. Motility is relevant also at the sub-cellular level, where it becomes complicated but also fascinating and relatively unexplored in the details of its mechanics. Again, the basic mechanism is remodelling, in the sense of chemically activated conformational changes of complex molecular machines. This includes, for example, the ribosome translating a protein, a molecular process based on the highly coordinated motile behaviour of a nano-scale machine [127]. Understanding the molecular machinery for DNA duplication, editing and transcription, which works in similar ways, is an equally significant problem to explore with the tools of modern mathematics and mechanics.One of the most exciting new applications of the theory of growth and remodelling over the last decade has been the systematic study of the geometric features found in the brain. This is not, however, a new topic as the characteristic convolutions of the human brain were first reported in an Egyptian manuscript dated 1700 BC that compares brain convolutions to the corrugations or wrinkles found in molten metal [128]. The description, development and function of these convolutions have also been major topics of research for the last two centuries [87]. The upper part of the convolutions are the gyri and the bottom groves are the sulci. This folded shape increases the surface area of the brain for a given volume. Functionally, convolutions have the strategic function of increasing the number of neuronal bodies located in the cortex and facilitating the connections between neurons, hence reducing the travelling time of the electric signals between different regions. The mechanism responsible for gyrification, the morphogenetic creation of these shapes, is not yet fully understood [129]. However, it is now accepted that intrinsic mechanical forces, rather than external constraints, are responsible for folding in the human brain [130] and recent observational studies [131,132] further support the role of the rapid tangential expansion of the cortex during development as the primary driver for folding [133–136]. Mechanically, the onset of folding can be understood as a build-up of elastic energy in the compressed upper cortex and its partial release by a wrinkling deformation of the film, the grey matter cortex, and substrate, the white matter core. Experimentally, this instability can be reproduced by the constrained polymeric swelling of a circular shell bounded to an elastic disc, which triggers the same type of wrinkling pattern [85,137,138]. Similar experiments performed on a two-layered brain prototype made of polymeric gels with differential swelling properties reproduce folds similar to the gyri and sulci of a real brain [139]. In this simple two-layer system, it is well appreciated that the pattern adopted by the system depends on a number of important factors such as the relative stiffnesses of the two layers [140], the thickness of the thin layer, the growth of the top layer [141], the curvature of the foundation [142], the adhesion energy between the layers [143,144], the imperfection of the substrate [145], the anisotropic response [146,147], the surface tension and pressure [148] and the nonlinear elastic response of the materials [149]. For small ratios of layer μl to foundation μs stiffnesses, μl/μs <∼ 10, as the wrinkling patterns develop, the system localizes this initial deformation and a fold or crease appears as observed in many biological systems. The deep folding patterns that are formed during the growth of brains are believed to be partially caused by this mechanical instability [87,150]. The analysis of this instability is particularly difficult owing to the existence of multiple unstable linear modes and possible contact. Surprisingly, a complete theoretical description is still lacking and is one of the great challenges of morphoelasticity. For larger ratios of μl/μs >∼ 10 and sufficient growth, a period-doubling instability occurs due to nonlinearities in the substrate response [151,152], as shown in figure 7. Whereas period doubling is well understood in dynamical systems, understanding the development of a spatial period-doubling pattern is more challenging even in the absence of growth [153]. The theory of growth and remodelling can now be used to explore some of the fine features of this pattern-forming system such as the variation of cortical thickness [154] or the role of initial curvature in aligning primary fissures [155,156]. Figure 7. Application to the brain. Differential growth during development creates the characteristic convoluted surface morphology of our brain. By varying the radius-to-thickness ratio r : t or the degree of ellipticity rz : r, mechanical concepts can help explain the varying degrees of complexity of the mammalian brain, for example the brachycephalic, rounded brain of the wombat and the dolichocephalic, elongated brain of the hyrax. Adapted from [40]. (Online version in colour.)
Chronic heart failure is a medical condition that involves structural and functional changes of the heart and a progressive reduction in cardiac output. Heart failure is classified into two categories: diastolic heart failure, a thickening of the ventricular wall associated with impaired filling; and systolic heart failure, a dilation of the ventricles associated with reduced pump function. We can model both conditions through the multiplicative decomposition of the deformation gradient into an elastic part and a growth part, F=Fe⋅Fg. To model diastolic heart failure through chronic cardiomyocyte thickening, we can introduce a growth multiplier ϑ⊥ that represents the parallel deposition of sarcomeres on the molecular level [36]. The growth tensor for transverse fibre growth follows as the rank-one update of the growth-weighted unity tensor in the plane perpendicular to the fibre direction f0 as Fg=ϑ⊥I+[1−ϑ⊥]f0⊗f0. The growth multiplier, ϑ⊥=det(Fg)=Jg, represents the thickening of the individual muscle cells through the parallel deposition of new myofibrils. To model longitudinal fibre growth through chronic cardiomyocyte lengthening, we can introduce a scalar-valued growth multiplier ϑ‖ that reflects the serial deposition of sarcomeres on the molecular level [36]. The growth tensor for longitudinal fibre growth follows as the rank-one update of the unity tensor along the fibre direction f0 as Fg=I+[ϑ‖−1]f0⊗f0. The growth multiplier, ϑ‖= det(Fg)=Jg, now takes the physiological interpretation of the longitudinal growth of the individual cardiac muscle cells through the serial deposition of new sarcomere units. This model naturally connects molecular events of parallel and serial sarcomere deposition with cellular phenomena of myofibrillogenesis and sarcomerogenesis to whole organ function. Figure 8 illustrates that our simulation predicts chronic alterations in wall thickness, chamber size and cardiac geometry, which agree favourably with the clinical observations in patients with diastolic and systolic heart failure. A recent longitudinal heart failure study in pigs has shown that changes in sarcomere number alone can explain 88% of myocyte lengthening, which, in turn, can explain 54% of cardiac dilation [158]. This whole heart model can also predict characteristic secondary effects including papillary muscle dislocation, annular dilation, regurgitant flow and outflow obstruction. Computational modelling provides a patient-specific window into the progression of heart failure with a view towards personalized treatment planning. Figure 8. Application to the heart. Our heart responds to a chronic increase in blood pressure by gradual wall thickening and to chronic volume overload by ventricular dilation. Personalized multi-scale models of cardiac growth can predict the time line of cardiac wall thickening in response to local muscle fibre thickening triggered by stresses and of cardiac dilation in response to local muscle fibre lengthening triggered by elevated strains [157]. (Online version in colour.)
Arteries enlarge in response to sustained increases in blood flow, they thicken in response to hypertension, they stiffen in ageing, they change size and shape in aneurysmal dilatation and they assume a dramatically different composition and properties in atherosclerosis. These few cases, and many more, reveal that blood vessels experience significant growth and remodelling in health and disease. Soon after its introduction, the theory of kinematic growth was employed to describe flow- and pressure-mediated growth and remodelling in arteries [159,160]. Combined with a computational solution and patient-specific modelling based on medical images, this approach provides insight into important clinical interventions such as angioplasty and stenting [161]. Yet, like all biological soft tissues, arteries consist of myriad proteins, glycoproteins and glycosaminoglycans. Each has different natural configurations as revealed by early experiments using elastase, collagenase and chondroitinase to selectively degrade individual constituents; each also has different material properties and rates of turnover in the form of synthesis and degradation. In parallel, therefore, different constrained mixture models arose to study arterial growth and remodelling, including predictions of aneurysmal enlargement [67,162] (figure 9). Indeed, such models helped explain the effects of different levels of elastolytic insults versus different rates of collagen deposition on overall rates of lesion enlargement, the latter of which predicted a subsequent experimental finding that suggests the therapeutic potential of antagonizing certain microRNAs to control rates of matrix production. Mixture models similarly allow modelling of separate biomechanical effects of active and passive smooth muscle [163], as well as stimuli other than mechanical, including biochemical effects of thrombus in cerebro-vasospasm [67] or infiltrating inflammatory cells in fibrosis [164]. The former accurately predicted the time course of vasospasm and its resolution over a one-month period, revealing that it is rapid collagen turnover, stimulated by extensive growth factors, cytokines and vasoactive substances, in progressively narrowed configurations, not smooth muscle contraction per se, that dominates this deadly condition, hence explaining why vasodilators are ineffective therapeutics for vasospasm. The ability to model separately the contributions of individual constituents and their separate growth and remodelling can thus provide important mechanistic insight. Figure 9. A circular–cylindrical model blood vessel (left) is perturbed by minor damage to its elastin layer. A healthy blood vessel is mechanobiologically stable, and its growth and remodelling will compensate for the damage over time to ensure just a minor permanent change of geometry (top). A diseased blood vessel may be mechanobiologically unstable, and its growth and remodelling can result in an uncontrolled dilatation over time, possibly resulting in aneurysm formation (bottom), depending on many factors, including matrix turnover rates, values of the deposition stretch and so forth [96,98]. (Online version in colour.)
Skin is our interface with the outside world. In its natural environment, it displays unique mechanical characteristics including prestrain and growth. While there is general agreement on the physiological importance of these features, they remain poorly characterized mainly because they are difficult to access with standard laboratory techniques. By combining recent developments in multi-view stereo and isogeometric analysis, it is now possible to analyse living skin in vivo at virtually no experimental cost [165]. Based on easy-to-create hand-held camera images, we can quantify prestretch, deformation and growth by longitudinally following characteristic anatomic landmarks throughout a chronic skin expansion experiment. Figure 10 shows the gradual inflation of a subcutaneously implanted balloon. By taking weekly photographs of the experimental scene, we can reconstruct the geometry from a tattooed surface grid, and create parametric representations of the grown skin surface. We have analysed these representations using the theory of finite area growth based on the multiplicative decomposition of the deformation gradient F=Fe⋅Fg into an elastic tensor Fe and a growth tensor Fg=ϑI+[1−ϑ]n⊗n, where ϑ defines the area growth. This model assumes that changes in thickness are purely elastic and no growth takes place in the thickness direction n. Surface growth modelling of skin allows us to quantify both the amount of average area prestretch and area growth ϑ as a function of time. We can use these data to calibrate skin growth models and simulate clinical cases of skin expansion; for example, in pediatric forehead reconstruction [35]. These simulations can accurately predict the clinically observed mechanical and structural response of living skin both acutely and chronically. This living skin model can easily be generalized to arbitrary biological membranes and serve as a valuable tool to virtually manipulate living systems, simply by means of changes in their mechanical environment. Figure 10. Skin responds to a chronic overstretch by generating new skin, a concept that is frequently used to repair birth defects or burn injuries. Using concepts of multi-view stereo analysis, we can characterize the amount of skin growth from a series of three-dimensional hand-held camera images. Cutting the grown skin into individual pieces in the spirit of figure 2 reveals the effects of differential growth and incompatibility [165]. (Online version in colour.)
Fibrosis around soft implants, e.g. mammary implants, results from a complex inflammatory response [166] characterized by myriad chemical signalling pathways and involves a variety of cells, among them fibroblasts and myofibroblasts. Fibroblasts are the cells that synthesize collagenous fibres of tissues while myofibroblasts are muscle-like cells with active contractile properties [61]. Although fibroblasts exist in any type of tissue, in case of inflammation, new cells are primarily directed towards the allergen source. These cells demonstrate a wounding response to the presence of an implant. Fibrosis always exists around implants, but can be exacerbated after mastectomy and radiotherapy treatment. This can ultimately lead to withdrawal after only a few months because of unbearable pain. In situ rupture of the breast prosthesis is another potential complication, which, due to the implant’s composition, usually silicon gels, is not good for a patient. A recent study revealed new insights into the biomechanics of the human capsule [167]: fibrosis manifests itself in the appearance of a thin fibrous layer called a capsule [168]. The thickness of the capsule increases with the grade of the pathology, from 0.5 mm thick in grade II up to 3 mm thick in the more severe cases of grade IV. We can estimate the level of compressive stress within the capsule using the theory of finite kinematic growth. Indeed we can compare the formation of the capsule to the growth of a thin layer attached to a semi-spherical surface. Evaluating the stiffness of fibrotic tissue in a tensile test revealed compressive stresses of the order of 10 MPa [167]. This represents a significant stiffening compared with healthy breast tissue with stiffnesses of the order of 1 kPa [169]. This difference may well explain the pain, although the link between pain and compressive stresses has not been quantified. In addition, this estimation discards the possibility of having active compressive cells as myofibroblasts, which is probable for patients at grade III or IV of the pathology. Because of compressive stresses, we expect a wrinkling instability at the implant–capsule interface. We have modelled the deformation at the interface using a two-dimensional neo-Hookean model for both implant and capsule. Figure 11 shows the implant–capsule interface for low values of growth. Strikingly, although the capsule is in compression in the ortho-radial direction, the implant is mainly in tension, which could potentially explain its failure. In conclusion, after more than 50 years of plastic surgery, no real explanation has been found for the existence of fibrosis around soft implants that affects approximately 30% of patients post radiotherapy. Figure 11. Finite-element simulations of the fibrous capsule in a two-dimensional neo-Hookean model for both implant and capsule. The aspect ratio reflects an implant of radius R and a capsule of order millimetres. In (a–c), the stiffness ratio ρ between capsule and implant is 10; in (d) it is 100. From (a) to (c) relative growth per unit length varies from g = 1.28 for (a), to g = 1.48 for (b), and g = 1.60 for (c). For this choice of ρ and g, the outer boundary does not buckle but flattens. The colour code indicates the maximum in-plane stress and demonstrates that the implant is in tension while the capsule is in compression. In (d), ρ = 100, g = 1.18 and we observe buckling. Magnification on the right shows the resulting deformation more clearly and reveals that stress inhomogeneities occurs only at the interface. (Online version in colour.)
Cancer is increasingly responsible for death and disability; it is characterized by accelerated cellular proliferation and local changes in matrix and vascular networks. Studies of in vivo growth and remodelling are vital, but so too in vitro models of disease. The multi-cellular spheroid is a standard in vitro mechanobiological system for studying the uncontrolled duplication rate of a tumour cell aggregate [170]. A tumour spheroid is a cluster of cells floating in culture medium, proliferating freely in an environment with abundant nutrients. Malignant cells have lost the ability to self-regulate their number through normal apoptosis, regulated by homeostasis with the environment: they duplicate isotropically in an uncontrolled manner, producing a nearly spherical shape. In the case of free, unconstrained growth, the diameter of the tumour typically exhibits an early exponential growth, followed by linear growth. The transition from one regime to the other is mainly regulated by the availability of nutrients, which, in turn, is driven by diffusion through the intercellular space. In fact, when the size of the tumour R(t) is smaller than the typical diffusion length, nutrient is available everywhere in the spheroid and the growth is purely volumetric, dR3/dt ≃ R3, and the radius increases exponentially in time, R ≃ et. Conversely, when the diameter of the spheroid is much larger than the penetration length of the nutrient, growth occurs primarily at the surface, dR3/dt ≃ R2, and radius increases linearly in time, R ≃ t. In the intermediate regime, the concentration of nutrients decays exponentially with the radius [171], favouring external over internal proliferation. A mechanical stress, produced by external loads or geometrical constraints, makes the simple scenario illustrated above more complex. The mechanical influence of external loads on tumour growth was first demonstrated in the late 1990s [172]. Growing cell spheroids in agarose gels revealed that the surrounding material is compressed by the expansion of the inner volume and—by the action–reaction principle—the traction exerted by the gel affects the growth rate of the multi-cell spheroid. Gels can be produced at a tunable stiffness by varying the concentration of the solid phase. An a priori mechanical characterization of the gel allows one to calculate the pressure exerted by the gel on the growing spheroid, as a function of the spheroid radius. The major finding of the gel experiments is that the generated stress reduces the final size of the spheroid, with decreased apoptosis and non-significant changes in proliferation [172]. It is therefore clear that a precise determination of the constitutive laws that characterize the mechanical behaviour of a tumour spheroid is a pre-requisite to reliably quantify the stress–growth relationship. Early attempts in this respect assumed that a cell conglomerate behaves like a viscoelastic fluid, able to bear a static load because of its surface tension. At equilibrium, measurements of the curvature of a loaded sample provide the surface tension of the fluid. Alternative studies support the idea that tumours behave like solids [173]. In stress–relaxation experiments of various tumour types in confined compression, all tumours equilibrated at a constant, non-zero stress at the end of the experiment, typical of viscoelastic solids. A second argument supporting the assumption of solid-like constitutive equations is the spatial correlation between stress and apoptosis–mitosis in loaded ellipsoidal spheroids [174]. A non-homogeneous proliferation pattern can be produced only by a solid-like material: a hydrostatic behaviour would generate a state of hydrostatic pressure, independent of position within the sample, whereas a solid behaviour generates high stress concentrates around the tips. Finally, there is increasing evidence of residual stresses in murine and human tumours [175], similar to the aforementioned observation in arteries. Cutting the tumour azimuthally results in a non-zero opening angle, which is the signature of a solid-like behaviour. Residual stresses are likely to be produced by an inhomogeneous duplication rate of cells and by mechanical interactions between the cells and their extracellular matrix, particularly collagen and hyaluronic acid, that strain the tumour microenvironment. Only solids can sustain residual stress, which can be modelled in multiple ways, including differences in deposition stretches or by the evolution of relaxed configurations produced by incompatible growth [103]. This implies that energy can be stored elastically in the unloaded body only if it behaves like a solid. Strikingly, experiments reveal compressive residual stress with negative opening angles in the centre and a tensile residual stress with positive opening angles in the outer shell of the tumour [175]. This observation seems paradox in terms of availability of nutrient concentrations being larger near the tumour boundary, which would favour proliferation and eventually generate compressive stress. In a later series of experiments, the compression of the spheroid was determined by the concentration of Dextran, a large molecule soluted in the bath [176]. As Dextran molecules can enter neither the cell membrane nor the interstitial space, an imbalance of osmotic pressure at the boundary loads the cellular aggregate. For larger concentrations of Dextran, the diameter of the spheroid grew slower and reached a plateau at smaller radius, in agreement with the initial predictions [172]. While a single cell is almost incompressible in response to the pressure generated by Dextran, the volume of the cell aggregate strongly depends on its hydration, that is, the osmotic pressure [177]. This technology allows a precise control of the mechanical stress: the osmotic pressure at the spheroid interface, the force per unit current surface, is constant in time. Considerable data suggest that a cell aggregate behaves as a binary mixture, or poroelastic material: a fluid phase and a solid phase composed of cells and extracellular matrix. Mathematical modelling of solid tumours as porous deformable media has been addressed in a number of publications [19,178,179]; it is a suitable mechanical framework to account for the coupled dynamics of cells, extracellular matrix and interstitial fluid. The interstitial flow is typically represented by a Darcy-type equation, and the mass exchange among individual phases allows a prediction of tumour growth. In the controlled osmotic pressure experiments with Dextran [176], the theory of porous media offers a transparent explanation for the control of the solid stress in the spheroid. Observing that the diameter of the macromolecules is typically larger than the size of the intracellular pores, we assume continuity of total stress and chemical potential at the spheroid boundary [180] (σ−pinI)⋅n=−poutnwithpin=pout−π, where pin is the pressure of the interstitial fluid, π > 0 is the osmotic pressure caused by Dextran, σ is the Cauchy stress in the cellular aggregate, I is the identity tensor and n is the radially outward pointing normal [181]. According to the above equation, the osmotic pressure only loads the solid phase, in agreement with the observation that the solid stress is not affected by the interstitial fluid pressure [175].Despite the apparent advancements, a number of open questions remain to be addressed: measurements of pressure as a function of the radial position within a spheroid exhibit a trend that remains to be theoretically explained [177]. Similarly, it is not yet clear whether the reduction in volume in the cellular aggregate occurs because of a reduction of the intercellular space only, or because of self-regulation of the osmotic pressure at a cellular level. Experiments report a sodium efflux from the loaded cells, which could explain a fluid outflow to re-establish the equilibrium of chemical potential [182]. This mechanism of regulation at the cellular level contributes to a macroscopic decrease in the diameter of the spheroid that should, in principle, be distinguished from variations in apoptotic and mitotic rates. This points towards the need for a more precise cell-level interpretation of the observed macroscopic tumour size versus time behaviour. The presence of many diverse confounding factors significantly complicates the identification of key factors that govern biological growth and remodelling in vivo. As in the case of studying tumour growth, an interesting way to overcome this problem is to use in vitro experiments with so-called tissue equivalents [183]. Tissue equivalents are highly simplified model systems of living tissues. They are typically studied in bioreactors where many relevant factors can be easily controlled. One of the simplest and most widely used experimental set-ups is a free-floating disc of collagen fibres seeded with living fibroblasts and immersed in a culture medium. Within just 24 h, the cells can compact the discs to a fraction of their original diameter [184], a remodelling process that introduces residual stresses [185]. Indeed, similar to the case of spheroids, the stress field is compressive in the interior and tensile in the outer region. To identify the governing principle behind this behaviour, a slightly modified study found that fibroblasts have a natural tendency to remodel the initially relaxed collagen gels towards a state of homeostatic stress [186]. Perturbations of a once established homeostatic state of stress stimulate cells to remodel the matrix until the homeostatic state is restored (figure 12). The notion of a homeostatic stress is well understood in a one-dimensional setting; yet, its three-dimensional generalization remains an open question. Recently developed biaxial experiments with tissue equivalents may help generalize this concept to higher dimensions, which could provide important information for the development and validation of accurate mathematical and computational models of soft tissue mechanobiology [187]. Figure 12. Application to tissue equivalents. Schema of a tissue equivalent being remodelled (compacted) by the embedded cells over a short period in a traction-free environment (left) versus a model-based prediction of remodelling-induced stresses in a cell-seeded uniaxial tissue equivalent, with the homeostatic target stress indicated by the horizontal dashed line and the actual response indicated by the solid curve (right). Note the initial build-up of stress as the cells attempt to compact the gel against fixed end constraints and the subsequent ‘relaxation’ of stress back towards homeostatic values following either an abrupt release of (first) or increase in (second) the imposed stress. (Online version in colour.)
Tissue equivalents are simple model systems of living tissues, mainly suited for academic research. Tissue engineering, by contrast, goes one step further and aims to provide human-made tissue substitutes for clinical applications, for example vascular grafts [188] or heart valves [189]. For example, simulations can quantify the amount of growth required to prevent leakage when tissue engineering valves. Figure 13 shows the effects of somatic growth averaged over n = 6 infants, n = 8 adolescents and n = 10 adults [191] on the aortic and pulmonary valves. Simulations without and with leaflet growth during the early and late stages of development reveal that growth is a natural and mandatory mechanism to prevent regurgitation [190]. Mathematical and computational modelling can help accelerate the expensive experimental and clinical studies that are necessary to push the boundaries in the field of tissue engineering. Figure 13. Application to tissue engineering. Characterization of growth during the early and late stages of human heart valve development reveals the amount of leaflet growth that is a mandatory mechanism to prevent regurgitation. Adapted from [190]. (Online version in colour.)
Mixture models are particularly useful in this regard since many engineering tissue strategies often use synthetic, biodegradable polymers as scaffolds to promote cell and tissue growth. For example, mixture-based models can be used to examine conditions that affect cell motility and aggregation within scaffolds used in tissue engineering. When focusing on the earliest cellular responses, it can be convenient to assume that the scaffold is rigid and unchanging so as to focus on the interstitial fluid movement, nutrient exchange, and cellular migration or proliferation [74]. Alternatively, scaffold properties are important when considering neotissue production and overall functionality [81]. Because cell-produced neotissue and scaffold necessarily differ in mechanical properties, often with the scaffold consisting of a polymer that degrades, mixture-based models are useful for delineating the separate effects of neotissue turnover and scaffold degradation both in vitro and in vivo [82,188]. Finally, whereas many successes within tissue engineering have been achieved via extensive trial-and-error empirical studies, mixture-based models promise to aid in scaffold design for one can perform time- and cost-efficient parametric studies and formal optimization using computational models of tissue development [68]. Growth and remodelling are fundamental to most biological processes and considerable progress has been made towards their mathematical understanding. Nevertheless, much remains to be accomplished. Among the many opportunities, we believe that there is a pressing need for the following. (i) To understand and model the relative influences of genetic, epigenetic and environmental factors on tissue and organ development. (ii) To meld increasing information from systems biology models of cell signalling within continuum models of growth and remodelling, e.g. a single mechanical stimulus can elicit myriad changes in cell signalling, simultaneously affecting cytoskeletal proteins, cytokine and protease production, and overall matrix integrity. (iii) To account for pathophysiological constraints imposed by genetic mutations, e.g. mutations to the gene FBN1 that codes fibrillin-1, an essential glycoprotein that associates with elastin to increase the long-term biological stability of elastic fibres and compromises the function of many soft tissues. (iv) To expand modelling of mechanobiology to include other effectors, such as immunobiology and pathobiology. (v) To enable modelling of the effects of specific pharmacological treatments on growth and remodelling, e.g. the differential effects of beta-blockers versus angiotensin receptor blockers in the treatment of hypertension. (vi) To increase the computational efficiency of the various growth and remodelling frameworks to enable advanced multi-scale and personalized modelling. (vii) To collaborate closely with data scientists to reduce the incredible complexity of biological processes to fundamental laws and rules of growth and remodelling that are amenable to simulation, for example using machine learning and artificial intelligence. Other opportunities exist in the spirit of biomimicry, for example elucidating basic mechanisms of constrained shape optimization and realizing self-healing engineering materials [192,193]. Applications of growth and remodelling thus seem limitless and, with the rapid advancement of medical imaging technologies, personalized simulations seem within reach in the near future. Towards this goal, one of the major challenges in the coming years will be to work hand in hand with biological scientists and clinical researchers to design experiments to rigorously calibrate and validate the requisite evolution equations under physiological and pathological conditions. We hope that this brief review stimulates such advances and many more not yet envisioned. This article has no additional data. We declare we have no competing interests. We received no funding for this study. This perspective article stemmed from the Advanced School on ‘Growth and Remodelling in Soft Biological Tissue’ held 12–16 June 2017 at the International Centre for Mechanical Sciences (CISM) in Udine, Italy, with co-sponsorship provided by the International Union for Theoretical and Applied Mechanics (IUTAM). We acknowledge the tremendous support of CISM in advancing the mechanical sciences through its manifold lectures, textbooks and archival papers, and we dedicate this paper to its many directors and staff. FootnotesReferences
Page 2Antimicrobial resistance (AMR) is undeniably one of the greatest global public health challenges we are currently facing [1]. The recent discoveries on the spread of resistance genes for key antimicrobials such as NDM-1 for carbapenem resistance [2–4] suggest that to tackle this challenge, instead of only studying the spread of resistant bacteria, we must understand the processes by which individual resistance genes spread. The first is ‘vertical gene transfer’, where genes are passed from parent to progeny during bacterial replication. The second, which is our focus here, is ‘horizontal gene transfer’ (HGT). This allows bacteria to acquire genetic material, including AMR genes, from their environment or other bacteria [5–7]. There are three mechanisms of HGT. First, ‘transformation’ is the capacity of bacteria to intake genetic material from their environment. Second, ‘conjugation’ occurs when two bacteria come into contact with each other and form a conjugative bridge, enabling direct exchange of genetic material. Finally, ‘transduction’ occurs when a bacteriophage (a virus that can infect bacteria) replicates and packages a bacterial gene instead of its own genetic material and then acts as a vector and transfers this gene into another bacterium. The consequences of HGT of AMR in a bacterial population are varied and can change depending on the setting where this process occurs. First, HGT can often be at the origin of new combinations of resistances to multiple antimicrobials in single bacteria strains [8]. This is amplified by the fact that HGT can occur both intraspecies and interspecies [9], therefore allowing for mixing between many different gene pools. Fortunately, these resistance mechanisms often impose a fitness cost that reduces the competitiveness of bacteria with AMR genes in settings where antibiotics are absent [10], thereby limiting the increase in the prevalence of these bacteria in the environment. Studying HGT of AMR can be further complicated by differences in transfer rates and importance of transfer mechanisms between bacterial species [11], with transformation, for example, being rare for Staphylococcus aureus [12] but common for Neisseria gonorrhoea [13], and by differences between rates estimated in vitro and in vivo, as was seen with transduction in Staphylococcus aureus [14] and conjugation in Klebsiella pneumoniae and Escherichia coli [15]. Finally, HGT dynamics appear to vary depending on the presence or absence of antibiotics in the surrounding environment [16–20], therefore requiring studies to be conducted in multiple settings to fully capture this process. It is essential to fully understand HGT of AMR since it can impact the overall transmission of AMR, and therefore the predicted effect of interventions against bacterial infections, to varying degrees depending on the setting. A most striking example of this is phage therapy, where bacteriophages are proposed as antimicrobials. A risk is that therapeutic phages could perform transduction and increase the proportion of bacteria in the patient which carry a resistance gene. In that case, if the phage therapy treatment fails to clear all the bacteria, this could leave the patient at a higher risk of antimicrobial-resistant bacteria infection [21,22]. In addition to the aforementioned differences between bacterial species, HGT mechanisms themselves are biologically complex. For example, the capacity to form a conjugative bridge generally requires the presence of a specific set of ‘tra’ genes [23]. These can themselves be transferred, leading to an increase through time in the prevalence of bacteria that can perform conjugation. Transformation gene expression is extremely variable depending on the environmental conditions that bacteria are exposed to [6], and therefore we cannot realistically assume that bacteria are able to perform transformation at all times. Finally, some phages can undergo either a ‘lytic cycle’, where they immediately replicate upon infecting a bacterium, or a ‘lysogenic cycle’, where they first integrate into the bacterial genome for a variable duration [12]. Consequently, transduction dynamics can be further complicated by the characteristics of the phage life cycle. Therefore, HGT is complex in its dynamics, and studying these requires appropriate tools. Mathematical modelling is often used to study infectious disease processes [24]. It provides a simulation environment that can be informed by real-life data, in which dynamics can be disentangled and easily studied. Mathematical models can be split into ‘deterministic models', which always generate the same results for a given set of parameter values [24], and ‘stochastic models’, which generate variability in their results using random events [24]. Mathematical modelling is already being used to study AMR dynamics and their public health implications [25,26]. For example, it has been employed to study within-host bacterial dynamics (i.e. the bacterial processes that occur during colonization or infection of a host) and derive conclusions on patterns of AMR seen in the host population [27]. Consequently, it can provide novel insight into optimal strategies to combat AMR spread by analysing the effect that these have on the transmission dynamics [28]. However, existing models may not always capture the relevant and complex microbiological dynamics of HGT. In this systematic review, we aimed to find modelling studies that focus on HGT of AMR to record their methods and research questions and, hence, to identify potential research gaps and areas for improvement in this field. The methodology of our systematic review follows the recommended PRISMA guidelines [29]. In order to be included in this review, studies had to fulfil all of the following criteria:
The entire screening process is summarized in figure 1. We searched two databases on 26 April 2019 using the following terms:
Figure 1. PRISMA flow diagram of the search and exclusion process.
After removal of duplicates, these combined searches yielded a list of 272 studies. Both Q.J.L. and G.M.K. independently screened the titles and abstracts of all 272 studies. Fifty-four studies were retained by both authors, and two more were discussed and retained after an additional screen of the methods due to uncertainty, leading to a total of 56 studies retained after the first screening step. The full texts of these 56 studies were then screened by Q.J.L., leading to 34 studies being retained as relevant for this review. Finally, by screening the reference lists in these 34 studies, 9 more were included, leading to a total of 43 studies to discuss in this review. To maximize comparability between studies, we devised a list of 11 elements to extract from every study. These are summarized and explained in table 1.
Note that in our analysis, ‘Type of parameter values' and ‘Sensitivity analysis performed’ are two independent criteria. Therefore, we can report that a study only uses ‘Constant’ parameter values, yet still performs a sensitivity analysis. If a study is reported to have ‘Sampled’ parameters, this means that the values of the parameters vary for each model run and that this is represented in the main results, with figures showing the model output with ranges instead of single lines for example. If a sensitivity analysis was performed, this means that the authors report conducting such a procedure to support their findings (e.g. to argue that their choice of ‘Constant’ parameter values is a reasonable assumption and does not significantly affect their results). The table showing all of the recorded elements from the 43 included studies can be found in the electronic supplementary material of this paper. First, when looking at the transfer mechanism modelled by these studies, we observe that almost all exclusively focused on conjugation (40 of 43) [30–69] (figure 2). Of the remaining three, two focused on transformation [70,71] and one on transduction [72]. Additionally, more than a third of the studies (16/43) chose exclusively Escherichia coli as the bacteria in which to model the transfer processes [30,34,36,41–46,52,53,59,64,66,68,72] (figure 2). It is also worth noting that another one-third of the studies (15/43) did not model a specific organism and instead indicate that they are looking at bacteria in general [31,32,37,38,48,51,54,56–58,61,62,65,67,69]. Finally, while eight studies applied their model to more than one bacterial species [33,35,39,40,47,49,60,63], only four of these modelled two strains of bacteria simultaneously and captured interspecies transfer of resistance genes [39,49,60,63]. Figure 2. Transfer mechanisms and bacterial species modelled in the 43 studies included in our review. (Online version in colour.)
In terms of the aims of these studies, all except eight studies [32,55,58,60,63–65,69] used modelling approaches exclusively to improve the understanding of bacterial evolutionary dynamics (figure 3). This covered questions such as how the prevalence of resistance genes in the bacterial population changes over time (as in [34], for example), or how the rise of multidrug-resistant bacteria varied under different environmental conditions (as in [30], for example). Inversely, the remaining eight studies [32,55,58,60,63–65,69] attempted to place at least some of their results in a public health setting by, for example, quantifying the impact of transfer on the incidence of multidrug-resistant bacteria infection in humans [32,69]. In accordance with this previous point, almost half of the studies (20/43) modelled bacteria exclusively in culture [33–42,47,49,50,52,53,58,59,66,70,71], and only seven modelled bacteria in humans [30,32,55,60,63,65,69] (figure 3). In the remaining studies, seven did not specify an environment for their bacteria [31,48,56,57,61,62,67]. Figure 3. Aims and environments modelled in the 43 studies included in our review. (Online version in colour.)
Almost all of the studies included a bacterial fitness cost for the carriage of a resistance gene in their models (table 2), except for six [32,42,48,63,66,71]. On the other hand, despite the fact that in reality bacteria can acquire multiple AMR genes independently (i.e. the acquisition of each gene is a separate HGT event), only four studies included this feature [30,32,60,69] (table 2). Finally, it is important to note that almost half of the studies did not model the presence of antibiotics and therefore did not consider the effect of antibiotics on transfer rates [33–36,39–42,47,52,53,59,63,66,68,71,72] (table 2).
Almost half of these modelling studies (19/43) included their own experimental work to generate data and estimate at least some parameter values for their models [33–36,39–42,47,49,51–54,59,66,68,70,71] (figure 4). On the other hand, more than half (23/43) chose to assume the values of at least some of their parameters, without explicitly citing any sources to support their choices, and a quarter (12/43) assumed the values of all of their parameters [31,32,37,38,65,67]. Finally, a third (15/43) used previous studies to obtain at least some of their parameter values. For these, except for three studies (two of which were each the direct follow-up of another one on the same topic [44,50], and one an analysis of data collected during an outbreak [63]), more than one previous study was taken to estimate the value of parameters, with a median number of studies of 8 and a maximum of 42. Figure 4. Sources of parameter values in the 43 studies included in our review. ‘Assume’ ((a), green): no clear reference is given to support the choice of parameter value; ‘Experimental’ ((b), orange): the study generated its own experimental data to support the choice of parameter value; ‘External’ ((c) brown): the study references a previous study to support the choice of parameter value. Studies in an overlap region used each of the corresponding methods at least once to estimate the value of their parameters. (Online version in colour.)
Finally, more than three quarters of the studies (33/43) exclusively relied on deterministic models to obtain their results [30,32,34,36–40,42,43,45–51,53–56,58,59,61,63–69,71,72]. All of these deterministic models were composed of a set of ordinary differential equations to track the different subpopulations (susceptible bacteria, resistant bacteria, etc.) through time. As for the ten studies that relied on stochastic models [31,33,35,41,44,52,57,60,62,70], most of these were agent-based models, where the bacteria were tracked individually [31,33,41,52,57,60], while the remaining ones either used stochastic differential equations [44,62,70] or difference equations [35]. Of the studies that exclusively used deterministic models, only eight acknowledge variability in the parameter values by running their model multiple times and sampling parameters from distributions instead of assuming them to be constant [32,38,43,46,56,64,65,72]. Nevertheless, most studies performed sensitivity analyses of the effect of their parameter values on their model results (table 2). Overall, nine studies still relied solely on a deterministic model without either sampling their parameter values or performing sensitivity analyses [30,36,40,42,48,54,55,58,68]. We also noted that except for the one study on transduction [72], all the studies modelled transfer as a mass action process. This assumes that the number of transfer events is determined by multiplying the number of bacteria that can receive the gene, the number of bacteria that can transfer the gene and the rate at which transfer occurs. Therefore, this is generally written as some form of β × S × R/N, where β is a rate of transfer, S is the number of bacteria that can receive the resistance gene, R is the number of bacteria that can provide the resistance gene and N is the total bacterial population in the system. We used a systematic literature review of mathematical models of HGT to determine our current understanding of the dynamics of HGT of AMR. The first main observation from our results is that the majority of studies assessed only focus on HGT by conjugation (40 of 43). The likely reason for this is the simplicity of conjugation dynamics. Effectively, these are comparable to infections transmitted upon contact, such as influenza, where established modelling exists using mass action dynamics [24]. Consequently, modelling conjugation does not require much complexity to be added to these models. However, we know that transformation and transduction also contribute to HGT [7,14], and the lack of studies on these mechanisms is worrying. Conjugation, transformation and transduction fundamentally differ in their biology, making it essential to study each of them in their own modelling framework; it is unknown whether models of conjugation could be directly applied to transformation and transduction. When looking at the studies that attempted to model these two processes, we first see that the one that focused on transduction [72] attempted to place it in a complex setting, with the phage able to undergo both lytic and lysogenic cycles, and the possibility for some bacteria to be resistant to phage infection. Transduction is represented as a multistep process in this model, as opposed to relying on a single rate. The phage must first successfully infect a bacterium and then pick up a resistance gene, before successfully transferring this gene to a different bacterium. This model aims to accurately represent most of the biological complexity of transduction, which necessarily requires many assumptions regarding parameter values. Further study of this trade-off would be greatly beneficial; it is currently unclear whether this complexity is required, at the cost of more assumptions, or if the process of transduction could be simplified and modelled using fewer parameters, which could be estimated from the experimental data. The two studies that focused on transformation [70,71] applied similar mass action dynamics to this process to that which can be seen in models of conjugation. However, this approach assumes that the number of resistance genes available in the environment is equivalent to the number of bacteria carrying these genes. This is questionable, as we would only expect these genes to be available in the environment after the bacteria die and release their genetic material; although it is possible for bacteria to actively release their genetic material while still alive, the extent of this phenomenon is unclear [6]. Further exploration of this assumption and perhaps redesigns of model structures for transformation would be of value. E. coli is the most commonly studied model organism for bacteria in general [73]. Its rapid growth and consistent behaviour in in vitro settings make it amenable to experimental work, including transfer studies, and therefore its overwhelming presence as the organism of choice for the studies modelling HGT of AMR genes is not a surprise. However, HGT is known to occur with varying rates in multiple bacterial species, and consequently it is unlikely that the rates of transfer estimated by looking at E. coli are equally applicable to other bacterial species [7]. In addition, HGT of AMR is a process that can also occur between bacterial species [9,11], while most models here exclusively focused on E. coli alone. Some resistances in bacterial species are in fact thought to have been originally acquired following a gene transfer event with another species, such as the mecA resistance gene in Staphylococcus aureus acquired from Staphylococcus fleurettii [74]. Despite the fact that the carriage of an AMR gene often imposes a reduction in the growth rate of the bacteria [10], a few studies did not model this (6/43), but only one argued that this element could be ignored after fitting their model to the experimental data [66]. However, this was once more only based on observations in vitro, which are likely to differ from the in vivo reality. Including a fitness cost, while requiring the estimation of an additional parameter, does not add any particular complexity to the model structure itself, effectively only requiring a reduced growth rate value for the bacteria carrying AMR genes as opposed to bacteria susceptible to the modelled antibiotic (as can be seen in [68], for example), and should therefore be included at least for sensitivity analyses. In addition, although it is understandable that the first models of HGT of AMR should focus on tracking single genes to understand the basic dynamics of this process, in reality, many bacteria carry multiple AMR genes that can be transferred independently [8]. However, we only identified four studies in our review which included more than one independent AMR gene in their model [30,32,60,69]. Thirteen studies did model the transfer of multiple linked genes [34,35,40–42,47,49,53,55,59,66,68,70]; however, in these cases, a single HGT event causes the transfer of all of these genes, and therefore, there is little difference between the model structures of these 13 studies and those of other studies that modelled the transfer of single genes. Many studies did not allow for the presence of an antibiotic in their model. However, antibiotics are likely to modify HGT dynamics by directly affecting transfer rates, as well as the survival of bacteria not carrying the AMR gene [16–20]. The former has been shown to occur for transduction in S. aureus, where the addition of antibiotics induced a higher proportion of transducing phage compared with lytic phage [75]. On the other hand, some studies correctly argue that it is equally important to understand the dynamics of HGT in the absence of antibiotics. Effectively, it is common for bacterial populations to rapidly transition between being exposed to antibiotics or not, with the most obvious example being individuals transiently consuming antibiotics. Consequently, understanding the dynamics of HGT of AMR both in the presence and in the absence of antibiotics is essential. HGT of AMR has been studied in laboratory settings; consequently data around which models can be built have been generated and are available [7,76]. However, we note that, to the best of our knowledge, most data appear to focus on conjugation in in vitro settings. The availability of the experimental data on HGT of AMR by transformation or transduction, and on any of the three HGT mechanisms in more complex settings (such as in vivo), is unclear. This should be investigated in future work to further refine the recommendations we make here and identify where more data are needed to support the development of mathematical models. This is essential to understand which of the gaps we identify are due to theory outpacing data collection and which are due to underutilization of the available data. In any case, using these external data sources for purposes they were not originally designed for can require assumptions to be made in the model structure and parameters. In addition, it is essential to bear in mind how these data were originally collected since, for example, combining sources that look at bacteria in multiple environments to derive parameters in a single environment-specific model is far from ideal. On the other hand, the fact that a quarter of the studies we reviewed (12/43) assumed all of their parameter values is worrying. While the purpose of some of these studies was to exclusively test a range of parameter values to identify conditions for a specific event to occur (e.g. AMR prevalence increases), the absence of any clear sources for the limits of these ranges is questionable. Looking at studies that determined their parameter values experimentally, we see that some of these also assume values such as the initial proportion of bacteria capable of performing transformation and the rate at which they can gain this ability [70], the bacterial growth rate and the conjugation rate [40] or the fitness cost of carrying an AMR gene and the rate at which such genes are lost by the bacteria [34]. Informing models with data is essential to ensure that they are accurate representations of reality; therefore, as stated above, we believe that further work is required to review the availability of data on HGT of AMR and the methods that could be used to generate them when they are currently missing. Regarding model structures, the majority of studies relied on deterministic models. To allow variability in the dynamics and therefore increased realism, studies more often chose to sample their parameter values, run their deterministic model and repeat this process a number of times (as can be seen in [32,38,43,46,56,64,65,72]), a simpler alternative to developing new stochastic models. Acknowledging stochasticity when looking at HGT is essential; HGT rates are typically low (estimates from studies in our review include for example 5.1 × 10−15 (cells/ml)−1 h−1 for conjugation [49] and 10−16 (cells/ml)−1 h−1 for transformation [70]). These are therefore models of rare events which, by chance, might not always occur as expected, a feature that deterministic models fail to capture [24]. Sensitivity analysis is extremely important in any case since a small change in parameter value can lead to a greater change in the results. Despite this, nine studies exclusively relied on a deterministic model without sampling parameters or performing sensitivity analyses [30,36,40,42,48,54,55,58,68]. Interestingly, five of these nine studies also generated their own parameter values experimentally [36,40,42,54,68]. Although they capture variation when measuring the parameters experimentally, often providing distributions for their values, they then only retain fixed-point estimates for their corresponding model parameter values instead of sampling them from these distributions and only use these fixed estimates to derive their conclusions. Acknowledging variability in microbiological observations by specifying distributions rather than point estimates is essential, and this must be represented in the corresponding mathematical models. This also raises the question of how to best represent these microbiological events in mathematical models. Effectively, almost all of the models here describe transfer as a mass action process (42/43). However, as stated above, this approach is acceptable for conjugation, but might not fully apply to transformation, where transfer depends on the density of DNA in the surrounding environment rather than the number of bacteria, and transduction, which follows vector-like dynamics with the phage acting as carriers of resistance genes between bacteria. Therefore, transformation dynamics might be better represented by models of environmental transmission of infections (such as [77]) and transduction by models of vector-borne diseases (such as [78]), as opposed to mass action models. The degree of modelling complexity required to accurately represent HGT is therefore unclear. This is also true for models designed to understand the public health implications of HGT of AMR, for which the level of detail required to represent within-host dynamics must be clarified. In addition, since transfer dynamics have thus far been mostly studied in bacterial culture, mostly ‘short’ time frames have been explored (hours or days), with long-term dynamics remaining unclear despite our knowledge that even resistant bacteria can colonize us for weeks or months [79–81]. To best guide our public health policies with mathematical modelling, we must first clarify the complexity of the process we are actually attempting to model and the time required to fully capture its in vivo dynamics. This is the first attempt at providing an overview of existing mathematical modelling work on HGT of AMR. Our systematic review methods, with two individuals separately screening the titles and abstracts of candidate studies, allowed us to identify and bring together key studies on this topic. By using our list of comparison elements, we extracted and contrasted essential information between studies, overall allowing us to obtain a broad overview of the field and identify research gaps. However, our approach also has some limitations. First, it was necessary for us to specify ‘(math* OR dynamic*) model*’ rather than just ‘model*’ in the search, since otherwise it would have returned results on experimental models (e.g. mice) as opposed to mathematical models. Effectively, repeating our search with ‘model*’ instead of ‘(math* OR dynamic*) model*’ yields 2360 and 1560 results on PubMed and Web of Science, respectively, as opposed to our 171 and 185 results. However, the consequence of our choice was that nine relevant studies were missed in the search and were only identified by screening the references of already included studies. These nine studies were missed in the original literature search due to the absence of at least one of the search terms, with some studies for example referring to their models as ‘mass action models’ instead of ‘mathematical models’. In addition, we only searched for studies that modelled transfer of AMR genes, as opposed to HGT of any gene. This is first due to our specific research interest; horizontal transfer of AMR genes is an especially strong evolutionary driver for bacteria populations, compared with transfer of other genes. This is because AMR genes can be strongly selected for by environmental factors, such as the presence of antibiotics, while many other genes are often not subject to such selection pressures. In addition, AMR genes can be selected in more settings compared with other genes; for example, genes involved in immune evasion will be selected only during infection of the host, while AMR genes can also be selected for during asymptomatic colonization. The consequences of HGT of AMR in the bacterial population can therefore be greater than for other genes, which is why we believe that it is important to study this process. Second, repeating the search without ‘(antimicrobial OR antibacterial OR antibiotic) resist*’ yields 12 236 and 38 148 results on PubMed and Web of Science, respectively, which would be too many to cover in a single systematic review. Nevertheless, this suggests that there are other studies that model HGT more broadly. These could be a source of methodologies that could be applied to further develop the specific field of HGT of AMR modelling. In terms of the elements gathered from the studies to compare them, we were unable to extract any meaningful quantitative data (e.g. estimated gene transfer rates) common to all studies due to the high variability of study designs. This variability also prevented us from identifying common measures of study quality we could report aside from the presence or absence of sensitivity analysis. Studying the effect of HGT of AMR on bacterial evolutionary dynamics is a necessary first step to understand the overall importance of this process. This has been the focus of the majority of the studies identified in this review; however, the public health implications remain vastly unknown. This is related to the observation that the majority of studies model bacteria in an in vitro setting; to understand the public health impact of HGT of AMR, it is essential to expand this to include other bacterial environments such as within humans and animals. In addition, important differences have been identified between transfer rates estimated in vitro and in vivo, with in vivo transduction rates in S. aureus and conjugation rates in K. pneumoniae and E. coli for example being much higher than expected [14,15]. This difference in dynamics is attributable to the fact that in vitro conditions fail to capture essential biological mechanisms influencing bacteria and therefore HGT [6,10]. Studying HGT in vitro allows for a controlled environment to understand the basic dynamics of this process and the factors that might influence them (e.g. antibiotic exposure) and consequently offers a starting point to inform in vivo models. Therefore, we recommend that future modelling studies should build upon the work of existing in vitro studies to evaluate HGT of AMR in more complex scenarios, using parameter estimates from in vitro studies as a baseline and refining them using the data generated with in vivo model organisms such as mice [68]. Owing to the added complexity (e.g. immune system, simultaneous within-host and between-hosts dynamics, rapidly varying host exposure to antibiotics and therefore selection pressure on the bacteria), this will require major extensions to existing models. However, we believe that this is necessary to truly assess the potential consequences of HGT of AMR on human well-being. This systematic review allowed us to identify key research gaps on the dynamics of HGT of AMR. First, we recommend that future studies should focus on developing models of transformation and transduction to determine the required complexity to represent these dynamics. Since these mechanisms fundamentally differ in their biological characteristics, this will likely require substantial, novel modelling work as opposed to the extension of existing models of conjugation. In parallel, since the basic dynamics of conjugation are already reasonably well understood, future studies on this mechanism should focus on other bacterial species than E. coli, preferably in a setting where interspecific HGT and the movement of multiple, separate AMR genes can be observed. This should be achievable simply by re-parametrization or minor extension of existing models; the greatest challenge would be to generate new data on HGT in these currently unexplored settings. The optimal solution to address these research questions would be to design frameworks to study HGT of AMR that encompass both laboratory and modelling work; this would ensure that the data collected are appropriate for the modelling needs and that the actual model is a good representation of the situation measured in the laboratory. Therefore, we believe that to fully understand the complexity of both the biology and the dynamics of HGT, collaboration of both microbiologists and mathematical modellers would be the best strategy for future research on this topic and that studies should attempt to generate both their own data and models to reduce the assumptions they require. While exclusively microbiological approaches have successfully been used to identify when HGT occurs, combining these with modelling has allowed us to estimate rates at which these events occur and to disentangle the finer temporal dynamics of this process. For example, some studies we identified in our review, which combined microbiology and modelling work, answered questions such as how changing the exposure of bacteria to antibiotics influences the HGT rates [49], how a bacterium interacts in space with its neighbours to perform HGT [31] or how to adjust shaking speed to maximize contact between bacteria, and thus the rate of HGT, in a liquid culture [66]. Modelling also allows faster exploration of situations that could be harder to test using only microbiological methods, since an experiment where the bacteria need to grow for 24 h in the laboratory could be completed in a few seconds using a mathematical model. Crucially, this requires the model to be an accurate representation of reality, which in turn requires it to be informed by the microbiological data to begin with. Therefore, our conclusion here is not that either one of modelling or microbiology is superior to the other, but that both approaches complement each other. Consequently, we believe that close cooperation between these two fields would allow us to greatly improve our understanding of complex microbiological processes, such as HGT of AMR. In this systematic review, we aimed to assess the current state of mathematical modelling as a tool to improve our understanding of HGT of AMR. From the 43 studies identified, we found that the majority focused on conjugation in E. coli, exploring evolutionary dynamics of HGT in culture. While this provides a solid base for a key method of HGT, future work must also consider HGT by transformation and transduction, which are also essential drivers of HGT in bacteria. Importantly for public health implications, only one bacterial species was considered in most models when we know that interspecies transfer is responsible for many of our epidemic AMR clones, and much of the work was fitted to data in the absence of antibiotic exposure. Crucially, to answer these questions, we must first clarify the level of modelling complexity required to accurately represent HGT dynamics, as well as the availability and capacity to generate the experimental data on these processes. This complex topic requires close collaboration between mathematical modellers and microbiologists in order to determine the full impact of these processes on our ability to control the public health threat posed by AMR. This article has no additional data. All authors jointly developed the search strategy. Q.J.L. and G.M.K. independently screened the titles and abstracts of the identified studies. Q.J.L. then evaluated the full texts of the included studies and wrote the first draft of the manuscript. All authors subsequently edited the manuscript. We declare we have no competing interests. This work was supported by the Medical Research Council (grant no. MR/P014658/1) and by a Medical Research Council London Intercollegiate Doctoral Training Program studentship (grant no. MR/N013638/1). The authors would like to thank Dr Katherine Atkins for helpful discussions on the search strategy for this systematic review. FootnotesElectronic supplementary material is available online at http://dx.doi.org/10.6084/m9.figshare.c.4570967. References
Page 3Scientists have long tried to decipher the principles underlying bipedal locomotion with the aim of improving human gait performance and treatment of neuro-musculoskeletal disorders. A powerful approach to this problem is the use of physics-based predictive simulations that generate de novo movements based on a mathematical description of the neuro-musculoskeletal system without relying on experimental data. Such simulations can explore diverse hypotheses about mechanisms underlying locomotion that are difficult to study through experiments. The high computational time of predictive simulations has favoured the use of conceptual models that only describe the most prominent features of the musculoskeletal system. Predictive simulations based on conceptual models have contributed to our understanding of the mechanics [1,2] and energetics [3–6] of bipedal locomotion. However, such models provide limited support for personalized clinical decision-making, since they do not sufficiently describe the musculoskeletal structures and motor control processes underlying gait that may be affected by treatment. An orthopaedic surgeon considering a rectus femoris transfer in a patient with cerebral palsy cannot predict the effect of the surgery on the walking pattern of the patient using conceptual models. By contrast, complex musculoskeletal models that include the many degrees of freedom of the skeleton and the many muscles actuating the lower limbs have the potential to make such predictions. Yet these complex models are computationally expensive in predictive simulations [7–10] and, therefore, the field has not explored their ability to predict the broad range of gaits encountered under different environments, pathologies and augmentations. Such generalizability is a prerequisite for using predictive simulations to design optimal treatments. Predictive simulations typically optimize a performance criterion that describes the high-level goal of the motor task without relying on experimental motion data. Yet it remains unclear what such a criterion would be for human gait. Experimental studies suggest that humans select gait features, such as step frequency and length, that optimize the cost of transport (COT, defined as metabolic energy consumed per unit distance travelled) [11] and that they continuously optimize the COT during walking [12]. Similarly, energy considerations have been suggested to drive the walk-to-run transition as gait speed increases [13]. Following these experimental observations, numerous simulation studies have used energy-based performance criteria to predict human walking or running [5–10]. Predictive simulations based on conceptual models also showed that the same energy-based criterion produced walking at low speeds and running at high speeds [5,6]. However, criteria centred on muscle activity, used as surrogates for muscle effort and fatigue, have also been suggested to underlie gait [14–16]. Simulation studies based on two-dimensional musculoskeletal models reported that using a performance criterion based on muscle activity better predicted the preferred walking speed in elderly [16] and resulted in more accurate kinematics during running [15] as compared to using an energy-based performance criterion. Yet it is unclear whether these observations hold for simulations based on three-dimensional models. Further, gait might be governed by multiple performance criteria [15,17] but the effect of combining different criteria on the predicted gait pattern has not been widely explored with complex three-dimensional models, likely due to the associated computational costs. Finally, it remains unclear whether a single task-level performance criterion can explain the range of gaits adopted by humans in different contexts. No simulation study has yet explored whether different gaits, such as healthy walking and running or pathological gaits, can emerge from the same underlying control strategy when using complex three-dimensional models. The purpose of our study was threefold. First, we developed a computationally efficient optimal control framework to predict human gaits based on complex musculoskeletal models. To this aim, we combined direct collocation, implicit differential equations and algorithmic differentiation. Second, we sought a performance criterion that could accurately predict human walking. To this aim, we explored a wide range of walking-related performance criteria and selected the criteria that best described walking at a self-selected speed. Third, we tested whether our framework could predict healthy and pathological gaits when altering gait speed and musculoskeletal properties but without altering the control strategy (i.e. using the same performance criterion). To this aim, we simulated gait (i) at different speeds, (ii) with muscle strength deficits and (iii) with a lower leg prosthesis and compared our simulation results to those from experiments. The computational efficiency of our framework (goal 1) allowed us to explore potential cost functions (goal 2) and to test the ability to predict the mechanics and energetics of a range of human gaits based on complex musculoskeletal models (goal 3). We used an OpenSim musculoskeletal model with 29 d.f. (6 between pelvis and ground; 3, 1, and 2 at each hip, knee and ankle, respectively; 3 at the lumbar joint between trunk and pelvis; and 4 per arm), 92 muscles actuating the lower limbs and trunk (43 per leg and six actuating the lumbar joint), eight ideal torque actuators at the arms and six contact spheres per foot [18,19]. To increase computational speed, we fixed the moving knee flexion axis to its anatomical reference position; moving and fixed knee flexion axes give similar results for gait [20]. We added passive stiffness (exponential) and damping (linear) to the joints of the lower limbs and trunk to model ligaments and other passive structures [7]. We used Raasch's model [21,22] to describe muscle excitation–activation coupling and a Hill-type muscle model [23,24] to describe muscle–tendon interaction and the dependence of muscle force on fibre length and velocity. We modelled skeletal motion with Newtonian rigid body dynamics and compliant Hunt–Crossley foot-ground contacts [19,25]. We used smooth approximations of the Hunt–Crossley model that were twice continuously differentiable as required with gradient-based optimization [26]. Conditional if statements were smoothed using hyperbolic tangent functions (example in electronic supplementary material). To increase computational speed, we defined muscle–tendon lengths, velocities and moment arms as a polynomial function of joint positions and velocities [27]. We optimized the polynomial coefficients to fit muscle–tendon lengths and moment arms (maximal root mean square deviation: 3 mm; maximal order: ninth) obtained from OpenSim using a wide range of joint positions. We used experimental data for comparison with simulation outcomes as well as to provide some of the bounds and initial guesses of the predictive simulations. Not all initial guesses were based on experimental data. We collected data (marker coordinates, ground reaction forces and electromyography; recording details in the electronic supplementary material) from one healthy adult. The subject was instructed to walk over the ground at a self-selected speed and to run on a treadmill at 10 km h−1. The average walking speed, henceforth referred to as the preferred walking speed, was 1.33 ± 0.06 m s−1. We processed the experimental data with OpenSim 3.3 [19]. The musculoskeletal model was scaled to the subject's anthropometry based on marker information from a standing calibration trial. Joint kinematics were calculated based on marker coordinates by applying a Kalman smoothing algorithm [28]. Joint kinetics were calculated based on joint kinematics and ground reaction forces. We formulated predictive simulations of gait as optimal control problems. We identified muscle excitations and gait cycle duration that minimized a cost function subject to constraints describing muscle and skeleton dynamics, imposing left–right symmetry and prescribing gait speed (defined as the distance travelled by the pelvis divided by the gait cycle duration). This optimal control problem is challenging to solve because of the stiffness of the equations describing muscle and skeleton dynamics. Owing to these stiff differential equations, a small change in muscle excitations can have a large impact on the simulated movement pattern and the cost function because of, for example, the high sensitivity of the ground reaction forces to the kinematics. To overcome this challenge, we used an optimal control method called direct collocation [14,22]. Compared to other methods such as direct shooting [7], direct collocation reduces the sensitivity of the cost function to the optimization variables by reducing the time horizon of the integration. Applying direct collocation results in large sparse nonlinear programming problems (NLP) that readily available NLP solvers can solve efficiently. We formulated muscle and skeleton dynamics with implicit rather than explicit differential equations, which are more common [24,29]. Using implicit formulations improves the numerical conditioning of the NLP by, for example, removing the need to divide by small muscle activations [24] or invert the mass matrix that is near-singular due to the large range of masses and moments of inertia of the body segments [29]. For the muscle contraction and skeleton dynamics, we introduced additional controls udFt and udv that equal (dynamic constraints) the time derivatives of tendon forces Ft and joint velocities v, respectively, and we imposed the nonlinear dynamic equations describing muscle contraction and skeleton dynamics as algebraic constraints in their implicit rather than explicit form [24]. We used a slightly different approach for muscle activation dynamics [22]. We introduced additional controls uda that equal the time derivatives of activations a and imposed activation dynamics by linear constraints on a and uda. Hence, muscle excitations were eliminated from the problem but can be computed post-processing [22]. Activation dynamics of the ideal actuators driving the arms were described by a linear first-order approximation of a time delay relating excitations earms to activations aarms. This equation is linear and continuously differentiable and there was thus no computational rationale for using implicit formulations. Details of the problem formulation are in the electronic supplementary material. We formulated our problems in MATLAB (The Mathworks Inc., USA) using CasADi [30], applied direct collocation using a third-order Radau quadrature collocation scheme with 50 mesh intervals per half gait cycle and solved the resulting NLP with the solver IPOPT [31]. We increased computational efficiency by applying algorithmic differentiation [30], which is an alternative to finite differences for computing function derivatives required by the NLP solver. In contrast with finite differences, algorithmic differentiation is free of truncation errors. Further, it permits the evaluation of derivatives through both forward and reverse algorithms (the forward algorithm is comparable to finite differences). Typically, the reverse algorithm requires fewer function evaluations than the forward algorithm when the function has many more inputs than outputs, whereas the opposite holds when the function has many more outputs than inputs. Hence, the reverse algorithm is more efficient for computing, for example, the cost function gradient, since the cost function is a single value (one output) that depends on many variables (many inputs). We created custom versions of OpenSim and its dynamics engine Simbody [25] to enable the use of algorithmic differentiation. We first sought a performance criterion that could predict healthy human walking by generating simulations, at the subject's preferred walking speed (1.33 m s−1), using multi-objective cost functions describing trade-offs between physiologically relevant walking-related performance criteria. Our cost functions included metabolic energy rate, muscle activity, joint accelerations, passive joint torques and arm excitations: J=1d∫0tf(w1∥E˙∥22⏟Metabolic energy rate+w2∥a∥22⏟Muscle activity+w3∥udv,lt∥22⏟Joint accelerations+w4∥Tp∥22⏟Passive torques+w5∥earms∥22⏟Arm excitations) dt,2.1 where d is the distance travelled by the pelvis in the forward direction, tf is half gait cycle duration, udv,lt are joint accelerations of the lower limbs and trunk, Tp are passive joint torques, t is time and w1−5 are weight factors. We modelled the metabolic energy rate E˙ using a smooth approximation of the phenomenological model described by Bhargava et al. [32]. We obtained the parameters for fibre type composition and muscle-specific tension from the literature [33]. We did not include the length dependence of the model's maintenance heat rate. This function is very unsmooth, which is physiologically unlikely and numerically problematic. Further, muscles are working close to their optimal fibre lengths during gait and including length dependency is thus expected to have a minor effect. We smoothed the metabolic energy model in a similar way as the contact model. To avoid singular arcs, situations for which controls are not uniquely defined by the optimality conditions [34], we appended a penalty function Jp with the remaining controls to the cost function:Jp=1dwu∫0tf(∥uda∥22+∥udFt∥22+∥udv,arms∥22) dt,2.2 where wu=0.001 and udv,arms are joint accelerations of the arms. We explored many sets of weight factors, by manually tuning them, until we found a cost function that predicted human-like walking, henceforth referred to as nominal cost function. We started each optimization from two initial guesses (electronic supplementary material, table S1) and selected the result with the lowest optimal cost. Only one initial guess was based on experimental data.We investigated the effect of different terms in the cost function by consecutively replacing the metabolic energy rate term by 2w1∑m=1ME˙m where M is the number of muscles (i.e. not squaring metabolic energy rate), removing the metabolic energy rate term, removing the muscle activity term, lowering the weight on joint accelerations and removing the passive torque term. We then tested whether the nominal cost function could predict healthy and pathological gaits when altering gait speed and musculoskeletal properties. First, we generated predictive simulations at different gait speeds (from 0.73 to 2.73 m s−1 by increments of 0.1 m s−1). For each speed (except for the preferred walking speed), we used five initial guesses (electronic supplementary material, table S1). Our criterion to evaluate whether the model adopted a walking or running gait was potential and kinetic energy being out-of-phase or in-phase [35]. Second, we investigated the influence of weak hip muscles and ankle plantarflexors during walking. We generated predictive simulations, at the preferred walking speed, while successively decreasing the maximal isometric force of muscles in the corresponding muscle group by 50, 75 and 90%. Third, we explored the influence of a transtibial passive prosthesis during walking. To model the prosthesis, we removed the ankle and subtalar muscles (including the gastrocnemii) of the right leg and modelled a passive prosthesis by describing ankle and subtalar torques as linear functions of joint angles q: T=−kq,2.3 where k = 800 N m rad−1 is torsional stiffness [36]. We reduced the mass of the lower leg and foot segments by 35% and the moment of inertia by 60% compared to the biological leg [36]. We did not alter the foot–ground contact model. To allow for gait asymmetry, we imposed periodicity of the states over a complete gait cycle (except for the pelvis forward position) instead of symmetry over half a gait cycle. We used 100 rather than 50 mesh intervals to account for the longer motion.We evaluated the sensitivity of our simulations to different parameters. If not explicitly mentioned, these simulations minimized the nominal cost function at the preferred walking speed. First, we evaluated how using different metabolic energy models, namely the models proposed by Umberger et al. [37], Umberger [38] and Uchida et al. [33], influenced walking simulations. These models treat negative mechanical work, muscle lengthening heat rate and motor unit recruitment differently (electronic supplementary material, table S2). Second, we tested the influence of increasing the lower bound on muscle activations to simulate co-contraction (using 0.1, 0.15 or 0.2 instead of 0.05). Third, we evaluated the sensitivity of the simulations to the foot–ground contact model parameters. We first calibrated a subset of the contact model parameters (transverse plane locations and radii of the contact spheres) by minimizing tracking errors with respect to the subject's walking data (details in electronic supplementary material). We then used the optimized contact models in predictive simulations. Fourth, we evaluated the sensitivity of the results to the number of mesh intervals by using 100 rather than 50 intervals. Finer meshes increase accuracy but also problem size and likely computational time. Finally, we evaluated the sensitivity of the walk-to-run transition speed to the model's peak mechanical power. We increased muscle power by doubling the maximal muscle contraction velocities (from 10 s−1 to 20 s−1) and generated predictive simulations at increasing gait speeds (from 1.33 to 2.23 m s−1 by an increment of 0.1 m s−1). Our framework generated three-dimensional muscle-driven simulations that converged in an average of 36 min of computational time (over 197 simulations; electronic supplementary material, table S3) on a single core of a standard laptop computer (2.9 GHz Intel Core i7 processor). We found that metabolic energy rate, muscle activity, joint accelerations and, to a lesser extent, passive joint torques—all terms squared—were important criteria to capture key features of human walking. We identified a set of weight factors that predicted joint kinematics, kinetics, ground reaction forces and muscle activations resembling experimental data of the subject at the preferred walking speed (w1=5 ×102/92/body mass,w2=2×103/92,w3=5×104/21,w4=1 ×103/15, w5=1×106/8; each weight factor is scaled by the number of elements in the vector from which we take the norm). Minimizing this nominal cost function required 23 min of computational time (electronic supplementary material, table S3) and resulted in a human-like walking gait (figure 1; electronic supplementary material, movie S1). The COT from this simulation, 3.55 J kg−1 m−1, was in the range of experimental measurements (3.35 ± 0.25 J kg−1 m−1 [8]) (figure 1). As opposed to previous studies [7–10], we squared the metabolic energy rate term; minimizing energy rate without squaring resulted in exaggerated trunk sway (figure 1; electronic supplementary material, movie S2). Figure 1. Simulated walking gaits with nominal and alternative cost functions. (a) Joint angles (add, adduction). The nominal cost function predicted an extended knee during mid-stance and limited ankle plantarflexion at push-off. Not squaring or removing the metabolic energy rate term from the cost function increased knee flexion but also trunk sway (e) and step width (g). (b) Joint torques. An extended knee resulted in small knee torques but limited ankle plantarflexion did not result in reduced ankle torques. (c) Joint powers. Limited ankle plantarflexion resulted in reduced ankle powers. (d) Ground reaction forces (BW, body weight; GC, gait cycle). (e) Trunk sway (i.e. trunk rotation in frontal plane). (f) Muscle activations (gluteus med, gluteus medius; min, minimus; semiten, semitendinosus; bic, biceps; fem, femoris; sh, short head; lat, lateralis; gastroc med, gastrocnemius medialis; ant, anterior). Removing the muscle activity term from the cost function resulted in unrealistically high muscle activations. The experimental electromyography data (grey curves) were normalized to peak nominal activations (black curves). (g) Metabolic cost of transport (COT), step width and stride length. The nominal COT matched experimental data [8]. (h) Resultant walking pattern with nominal cost function (electronic supplementary material, movie S1). Experimental data are shown as mean ± 2 s.d. (Online version in colour.)
Each term in the proposed nominal cost function was necessary for predicting the prominent features of walking. Removing the metabolic energy rate term resulted in increased trunk sway and step width, whereas removing the muscle activity term resulted in unrealistically high muscle activations for several muscles (figure 1; electronic supplementary material, movie S2). The joint acceleration term was important for convergence of the optimization algorithm and smoothness of the motion, whereas the passive joint torque term limited knee overextension (electronic supplementary material, movie S3). Our framework predicted a continuum of walking and running gaits as we varied the prescribed gait speed (electronic supplementary material, movie S4). Further, the predicted COT, stride frequency and vertical ground reaction forces changed as a function of speed in agreement with reported data (figure 2). A transition from walking to running occurred at 2.23 m s−1 (electronic supplementary material, figure S1). In agreement with the literature, we found quadratic (coefficient of determination R2=0.98) and linear (R2=0.66) relations between COT and speed for walking and running, respectively (figure 2a) [35]; a linear relation (R2=0.99) between stride frequency and walking speed (figure 2b) [39]; and vertical ground reaction forces whose first peak increased and mid-stance magnitude decreased as walking speed increased (figure 2c) [40]. Figure 2. Alterations in gait features with speed. (a) Quadratic and linear regressions (black curves) based on simulation results (coloured markers) between metabolic cost of transport (COT) and speed for walking (0.73–2.23 m s−1; R2=0.98) and running (2.23–2.73 m s−1; R2=0.66), respectively. (b) Linear regression (black curve) based on simulation results from walking (coloured markers) between stride frequency and speed (R2=0.99). The regression line is compared with the one obtained from experimental data [39]. (c) Vertical ground reaction forces (BW, body weight) at walking speeds less than the preferred walking speed of 1.33 m s−1 (i), at walking speeds greater than the preferred walking speed (ii) and at running speeds (iii). Each coloured curve represents a simulation result for a different gait speed. (Online version in colour.)
Altering musculoskeletal properties led to gaits that accurately exhibited clinical gait deficiencies. Reduced hip muscle strength resulted in greater hip circumduction (i.e. conical movement of the legs) to reduce hip torques (figure 3a; electronic supplementary material, movie S5). This strategy, known as compensated Trendelenburg gait, may be observed in patients with neural injuries or myopathies affecting hip muscles [41]. Reduced ankle plantarflexor strength resulted in calcaneal gaits that reduced ankle torques (figure 3b; electronic supplementary material, movie S6). Such gaits may be observed in children with spastic diplegia who have a weak triceps surae, possibly due to an Achilles tendon lengthening surgery [42]. Figure 3. Effect of muscle weakness on walking pattern. (a) Hip muscle weakness. Reducing hip muscle strength by 50, 75 and 90% resulted in increased trunk sway and step width and decreased hip torques. (b) Ankle plantarflexor weakness. Reducing ankle plantarflexor strength by 50, 75 and 90% resulted in increased knee flexion and ankle dorsiflexion and decreased stride lengths that reduced ankle torques. Experimental data of the healthy subject are shown as mean ± 2 s.d. The simulations minimized the nominal cost function at the subject's preferred walking speed (1.33 m s−1). (Online version in colour.)
Our simulations produced ankle torques and COT that are typical of amputees with a transtibial passive prosthesis. In agreement with experiments [43], ankle plantarflexion torques of the affected leg were larger during early- and mid-stance than torques of the unaffected leg (figure 4; electronic supplementary material, movie S7). The COT was similar to the nominal COT, as expected for physically fit amputees [44]. Figure 4. Simulated walking gait of an amputee with a transtibial passive prosthesis. Simulated ankle torques (red curves) matched the average ankle torques of six transtibial amputees [43]. The metabolic cost of transport (COT) for healthy and amputee walking was similar. Experimental data (grey envelopes) are shown as mean ± 2 s.d. The simulations minimized the nominal cost function at an imposed speed of 1.33 m s−1. The prosthesis geometry is for visualization only.
The sensitivity analyses showed that different metabolic energy models resulted in qualitatively similar walking patterns, although we observed some differences in hip, knee and ankle angles caused by differences in the muscle lengthening heat rate component of the metabolic cost (electronic supplementary material, figure S2, table S2 and movie S8). Increasing the lower bound on muscle activations from 0.05 to 0.1 resulted in larger knee flexion angles, knee torques and vasti activity during stance but also in a larger COT (5.10 J kg−1 m−1). Lower bounds larger than 0.1 resulted in non-human-like gaits (electronic supplementary material, figure S3). The geometry of foot–ground contact and the mesh density had little influence on the simulations (electronic supplementary material, figures S4–S6). Finally, increasing the model's peak mechanical power reduced the transition speed to 2.13 m s−1 (electronic supplementary material, figure S1). We developed a computationally efficient framework for predictive simulations of three-dimensional human gaits that allowed us to explore a broad range of control strategies and conditions. Our simulations showed that healthy and pathological gaits can emerge from the same underlying control strategy, thereby providing further support for observations based on experimental protocols [45] and conceptual models that did not account for the large redundancy in the musculoskeletal system [5]. In addition, we showed that predictive simulations based on complex three-dimensional musculoskeletal models can capture healthy and pathological human gait mechanics and energetics, a first prerequisite for the use of our models and simulation framework for clinical outcome predictions. Our framework generated three-dimensional muscle-driven simulations in only 36 min on average, which is more than 20 times faster than existing simulations with similarly complex or simpler models (i.e. between 13 and 60 h) [8–10]. Note that a fair comparison of computational efficiency with published simulations is difficult as none of these studies solved the exact same problems. Further, prior studies might not have prioritized computational efficiency. Our use of direct collocation, implicit differential equations and algorithmic differentiation likely explains the superior computational efficiency of our framework but a comparison with alternative methods is required to further our insight into how each of these components contributed. We also made several assumptions that may have contributed to the rapid convergence of our simulations but that may not always hold. First, imposing symmetry allows simulating only half a gait cycle but is not valid, for example, in impaired gait. Optimizing for a complete gait cycle results in a higher computational time (e.g. prosthesis simulation in electronic supplementary material, table S3). Second, fixing the moving knee flexion axis increases computational speed but might not be valid when studying knee pathologies. Third, using an implicit formulation of activation dynamics eliminates muscle excitations from the optimization problems and, therefore, we used activations instead of excitations to evaluate motor unit recruitment in the metabolic energy model. Similarly, our implicit approach to impose activation dynamics does not allow us to directly formulate constraints on excitations (e.g. imposing that excitations are driven by muscle reflexes). Our cost function (i.e. control strategy) included both metabolic energy rate and muscle activity, which was important to predict physiological walking. Although human walking is often assumed to result from minimizing only energy consumption [5–8,10], including squared muscle activity might be important for minimizing signal-dependent motor noise since muscle activity is believed to directly affect motor noise [46,47]. Our cost function also included a joint acceleration term that was important to obtain convergence. Minimizing joint accelerations or jerks (i.e. rate of change of joint accelerations) to obtain smooth movements also resulted in good predictions of planar reaching movements [48,49]. However, observed reaching movements were predicted equally well when minimizing uncertainty due to sensorimotor noise [46]. Therefore, it remains unclear whether smoothness of the gait patterns is part of the control strategy or emerges from non-modelled neuro-musculoskeletal features (e.g. robustness against perturbations or soft tissue damping). Minimizing only squared muscle activations led to exaggerated trunk sway and step width (figure 1). Hence, the observation based on two-dimensional models that minimizing muscle activity rather than COT better predicts running kinematics [15] and preferred walking speed in elderly [16] might not hold for simulations based on three-dimensional models. Our simulations produced walking gaits at low speeds and running gaits at high speeds. Nevertheless, the simulated walk-to-run transition speed (2.23 m s−1) was slightly greater than reported values (1.89–2.16 m s−1 [50]) and most running gaits presented a longer stance phase than expected (close to 50% of the gait cycle), suggesting that our cost function may not capture all goals during running. Other strategies such as reducing maximum dorsiflexor moment [50] or locomotor variability and avoiding instabilities [51] have been suggested to trigger the walk-to-run transition. Increasing the model's muscle mechanical power reduced the transition speed to 2.13 m s−1. This is in agreement with the delayed walk-to-run transition observed in young children that has been attributed to reduced peak mechanical power compared to adults [52]. We might have obtained a similar decrease by reducing the cost of peak mechanical power over mechanical work [52] or by increasing muscle power through increased muscle volumes. Our simulations produced realistic gaits at different gait speeds, with muscle strength deficits, and with lower leg prosthesis use based on the same control strategy. Hence, a range of healthy and pathological human gaits emerged from the multi-objective cost function that was identified by judging the realism of a simulated gait pattern at a self-selected speed only. Yet this observation does not imply that the proposed cost function represents the underlying physiological control mechanisms that drive locomotion. Although omitting any term in our cost function resulted in less realistic patterns, it is possible that a different set of weight factors or alternative cost functions containing criteria proposed in the literature but not considered in this study (e.g. head stability [53] and angular momentum regulation [54]) will result in equally or more realistic gaits. Especially since many of the proposed criteria are related through system dynamics. Despite the absence of reflexes and other motor control pathways, our simulations captured the prominent features of human gait. This may be because preferred gait patterns are dictated in part by musculoskeletal mechanics and not only by a control strategy. This would explain why we obtained human-like walking patterns with different cost functions and is in line with observations based on passive walkers that natural dynamics may largely govern locomotion [55]. Second, in a healthy nervous system, different control pathways might interact in a way that optimizes a task-level goal. Our cost function might therefore capture the result of distributed pathways within the central nervous system without explicitly describing these pathways. Such optimization is less likely in the presence of pathologies of the central nervous system, such as spasticity, in which reflex loops are dysregulated. Third, reflexes and other feedback pathways might be especially important to move in a noisy world (i.e. to reject disturbances) but we did not model any noise. This omission might explain the lack of hamstrings activity at terminal swing (semiten in figure 1), which has been reported to be reflex-driven [56]. We expect our predictions to be more accurate if we model known control loops [57], especially in the presence of pathologies or in noisy environments. Alternatively, our framework could be used to test the effect of hypothesized control pathways on human locomotion. Combining our simulation workflow with experimental studies will advance our understanding of criteria driving gait selection and improve the accuracy of our predictions. The hypothesis that the COT is optimized during gait has been extensively tested both by observing natural behaviour and by manipulating the relation between COT and gait pattern (e.g. [12]). Our simulations suggested that minimizing metabolic rate alone does not result in realistic gaits and pointed to other performance criteria that can be experimentally tested. For example, passive joint torques could be manipulated through braces, whereas manipulating muscle activations or joint accelerations will be more challenging. Our simulations elicited the potential role of the musculoskeletal mechanics, which could also be manipulated experimentally, for example, by locking degrees of freedom, reducing the base of support (cfr., [58]), or adding mass to certain segments. These experiments could be combined with approaches based on inverse optimal control to automate the search for a walking control strategy. To realize the potential of our framework for optimal treatment design, experimental work is also needed to investigate how the control strategy changes in the case of motor control impairments. We have demonstrated the ability of our simulations to reproduce key features of healthy and pathological human locomotion. Nevertheless, our simulations deviated from measured data in two notable ways. First, our predicted knee flexion during mid-stance was limited, resulting in small knee torques (figure 1). Other predictive studies have reported limited knee flexion during mid-stance and argued that the cause is the lack of stability requirements [14,26]. Similar to these studies, we did not model any stability requirements and included both muscle activity and metabolic rate in the cost function, which might explain why reducing knee torques and, therefore, knee extensor activity was optimal. Co-contraction has been suggested to play a stabilizing role during walking. We found that imposing co-contraction resulted in a more flexed knee during stance at the cost of a higher COT. Future work should investigate more physiologically inspired approaches to account for stability, such as feedback control through spinal reflexes [57]. Alternatively, we could explicitly model and minimize the uncertainty on the simulated movement due to perturbations using approaches from the domain of robust optimal control [59] although such approaches induce significant computational costs. Second, our simulations produced less ankle plantarflexion at push-off (figure 1). The absence of a metatarsophalangeal (MTP) joint might explain reduced plantarflexion, as similar ankle kinematics have been observed experimentally when limiting the range of motion of the MTP joint [60]. The simplistic trunk model might have contributed further to the aforementioned differences between simulated and measured walking patterns. By contrast, the foot–ground contact geometry had little influence on the simulated walking pattern. Our evaluation of the simulated patterns was qualitative rather than quantitative as we found it hard to capture the realism of a gait pattern in a few numbers. Although depending on subjective interpretation, a comprehensive comparison of simulated and measured trajectories along with the animated movements allowed us to judge the realism of our simulations. Different gaits locally optimize the nominal cost function. We used a discretization scheme with a predefined number of mesh intervals and a gradient-based method to solve the optimization problems. Gradient-based methods find a local optimum, as opposed to the global optimum, and might hence be sensitive to the initial guess. Our simulations based on the nominal cost function at the preferred walking speed converged to similar results whether using an initial guess derived from walking data or a quasi-random initial guess (electronic supplementary material, figure S6), whereas previous studies required initial guesses derived from computationally expensive data-tracking simulations [8,10]. In addition, using a finer mesh led to similar results (electronic supplementary material, figure S6). However, we obtained a different gait pattern when using an initial guess derived from running data (electronic supplementary material, figure S6). The optimal cost and COT of this gait were much larger than those resulting from the two other initial guesses, suggesting different local optima. This is not surprising, since humans adopt a range of different gait patterns at a given speed depending on the context. For example, it has been shown that humans do not always adopt a walking pattern that minimizes energy consumption but prefer this option when instructed to self-explore different patterns [12]. Local optima might hence characterize a model describing human locomotion. How to model the context-dependent selection of a local optimum remains, however, an open question. Overall, our physics-based computational framework holds the potential to greatly expedite advances in understanding human locomotion. In particular, we expect our efficient simulations, when combined with patient-specific neuro-musculoskeletal models, to enable optimal design of treatments aiming to restore gait function by allowing in silico assessment of the effect of changes in the neuro-musculoskeletal system on the gait pattern. Currently, treatment of gait impairments resulting from interactions between motor control and musculoskeletal deficits, such as in cerebral palsy, is often unsuccessful [61]. Optimal treatment design might hence have a large impact on patients' quality of life. Yet the availability of models and methods characterizing these patient-specific deficits is still limited and should be the focus of future research. Further, the design of experimental protocols to collect data required to personalize these models will be particularly important as such protocols should be comprehensive enough to allow for accurate modelling while accounting for practical limitations in clinical contexts. We also envision numerous applications beyond personalized medicine. Our framework can be used to design assistive devices, to simulate gaits of extant and extinct species, to optimize performance in sports by designing equipment and training programmes or to synthesize realistic movements in animations. The participant gave informed consent to participate in the study that was approved by the Ethics Committee at UZ/KU Leuven (Belgium). All data, code and materials used in this study are available at https://simtk.org/projects/3dpredictsim. A.F., F.D.G. and I.J. conceptualized the methods; A.F. processed the data; A.F. performed the formal analysis; A.F., F.D.G. and I.J. acquired funding; A.F. and F.D.G. conducted the investigation; A.F., F.D.G., G.S., C.L.D. and J.G. developed the methodology; F.D.G. and I.J. administrated the project; A.F., F.D.G. and I.J. provided resources; A.F., F.D.G., G.S., C.L.D. and J.G. developed the software; F.D.G. and I.J. supervised the project; A.F. and F.D.G. validated the research outputs; A.F. prepared the data visualization; A.F. and F.D.G. drafted the manuscript; and all authors edited the manuscript. We declare we have no competing interests. This work was supported by the Research Foundation Flanders (FWO) under PhD grant no. 1S35416N and travel grant no. V441717N to A.F., research project grant no. G079216N to F.D.G., and by the NIH under grant no. P2C HD065690 to the National Center for Simulation in Rehabilitation Research (NCSRR). C.L.D. received a Stanford Bio-X Graduate Fellowship. J.G. has benefitted from KU Leuven-BOF PFV/10/002 Centre of Excellence: Optimization in Engineering (OPTEC) and from Flanders Make (Flemish strategic research centre for the manufacturing industry) ICON (DriveTrainCodesign). We thank H. Geyer and S. Song for insightful discussions on predictive simulations; M. Afschrift and T. Van Wouwe for data collection; L. Ting, M. Daley, P. Bishop, A. Lai, H. Kainz, M. Afschrift for editorial suggestions; and W. Aerts for providing the prosthesis model. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4606853. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 4Human walking is known to be energy efficient [1–4] and stable [5–7], but we do not know how these potentially conflicting properties are realized simultaneously. Specifically, we do not yet have a complete characterization of how humans control walking. One way to examine how humans control walking, or indeed any movement task, is to apply unforeseen perturbations and examine the transients back to steady state. While a number of such walking perturbation experiments have been performed [5–15], none of them have been used to derive a controller sufficient for a complete, even if simplified, simulation of walking. Instead, the perturbation experiments have only been used to obtain insights into limited aspects of walking control such as foot placement or centre of pressure modulation [5,7,15–18], ankle impedance [12,19], impulse response functions for some muscles [14,15] or some aspect of body motion [9]. Other authors have considered natural step-to-step variability in unperturbed steady-state locomotion [10,20–24] to obtain information about stability and control of walking and running, but again, we do not know of a complete synthesis of a walking controller from such measurements. Similarly, there have been many stable bipedal walking simulations [25–28], stable two-legged walking robots [29,30], and robotic prostheses and exoskeletons [31,32], but none of these simulations, robots or assistive devices have controllers that are quantitatively derived from human walking response to perturbations. Our study could be considered a partial analogue of [33] in which a running controller was obtained for a simple biped model by fitting model responses to human responses to natural step-to-step variability. Here, we derive a complete walking controller for a minimal and classical model of walking, based on human responses to perturbation experiments. Specifically, we perform human subject experiments in which we apply unforeseen perturbations on humans as they walked on a treadmill, and then use data from these experiments to derive a controller that describes how foot-placement and push-off impulse are controlled for the inverted pendulum model of walking. This is a model used extensively in both biomechanics (to understand human movement [34]) and robotics (as a simple model for control and planning [35,36]). We show that the human-derived controller stabilizes inverted pendulum walking, is robust to large changes in body parameters, and prevents falls from large perturbations to the upper body. The experimental protocol was approved by the Ohio State University Institutional Review Board. Twelve subjects (N = 12, eight male, four female, age = 25.5 ± 4 years, leg length = 0.97 ± 0.06; mean ± s.d.) participated with informed consent. Subjects walked on a treadmill at 1.2 m s−1, while discrete perturbations were applied to them by the experimenters, that is, pulls through a light stiff cable tied at the waist (figure 1). One subject walked at 1.1 m s−1. Figure 1. Experimental set-up. Subjects walked with a loose safety harness on an instrumented treadmill, with randomly spaced unforeseen perturbations being applied by pulling on the perturbing cables. The subject wore passive noise-cancelling earmuffs so they could not hear the perturbation coming (though the perturbation was indeed very quiet due to human pulling). The subject also wore cardboard blinders that prevented them from seeing the perturbation through peripheral vision. The figure shows the set-up for sideways perturbation experiment; the backwards perturbation experiment is identical, except the two sideways perturbing cables are replaced by one perturbing cable able to pull backwards. The perturbing cables were attached to the person via magnets so that the cable disengages when a force threshold of roughly 100 N is crossed. The notation for the coordinate directions (X, Y, Z) is shown, but the origin of the system is the stance foot at mid-stance for each step. (Online version in colour.)
We performed two different types of perturbation experiments: (i) anterior–posterior (AP) perturbations, involving only backward pulls, and (ii) sideways or medio-lateral (ML) perturbations, involving either leftward or rightward pulls in random order. Nine subjects participated in AP trials and seven subjects participated in ML trials, and four subjects participated in both. For each type of perturbation experiment, subjects walked on the treadmill for 10 trials of 4 min each. Each such trial had a total of 10 perturbations. The pulls were exerted manually by the experimenters, rather than by using a robotic device (say, as in [7]). The perturbing cables were usually slack (negligible forces) and became taut to apply forces only during the brief perturbations. About 20% of these perturbations were ‘fake’, so that the experimenter goes through the motion of pulling on the subject without making the cables taut and applying substantial forces. Such fake pulls were designed to ensure that participants could not reliably predict the onset of perturbations and to rule out the possibility that subjects were responding to the perceived motion of the experimenter rather than the effect of the perturbing forces. Subjects wore earmuffs and blinders to block any visual or audio cues (figure 1) and walked with arms crossed because the sideways perturbing cables interfere with arm swing. The timing of the pulls was randomized, with 15–45 s between consecutive pulls. We performed this randomization by repeatedly generating a uniformly distributed random number s between 15 and 45, and alerting the experimenter to manually exert a pull when exactly s seconds have passed after the previous pull. The phase of the perturbations were not controlled, and were randomly distributed over the whole gait cycle (see electronic supplementary material, figure S1 for perturbation phase histograms). For ML pulls, in addition to randomizing the pull timing, we also randomized the sequence of left versus right pulls by having one experimenter on each side and alerting either the left or the right experimenter to apply the pull. At the beginning and end of each subject’s session, a 30 s unperturbed walking bout was performed to serve as a baseline. Kinematic data for the feet and the pelvis were collected using a marker-based motion capture system (Vicon T20, eight cameras, 100 Hz, five pelvis markers and four markers on each foot). Ground reaction forces were captured through a pair of force plates within the treadmill (Bertec FIT split-belt instrumented treadmill), but not used in this study. Perturbing forces were also measured using light load cells (Phidgets 100 Kg S-type) in series with the perturbing cables. The data will be available without restrictions through an open database, Dryad. Figure 2 shows perturbing forces and impulses from two example trials, one each from the backward and sideways perturbation experiments. While the forces applied were not carefully controlled, typical maximum forces in each pull were in the 45 N to 75 N range. The pull directions were approximately along the treadmill (AP) or perpendicular to the treadmill (ML), enforced by having the experimenter pull from sufficiently far away from the subject (about 3 m) and appropriately positioned behind or on either side of the treadmill. Our analysis does not rely on the exact control of these perturbing force magnitudes or directions. Figure 2. Forces applied during the perturbation experiment. (a) Example of applied perturbation forces from one trial for one subject in the backwards perturbations experiment. (b) Histogram of applied impulses (normalized for each subject, pooled over all subjects) from all backwards perturbation trials for ‘highly perturbed’ steps. (c) Example of applied perturbation forces from one trial for one subject in the sideways perturbations experiment. (d) Histogram of applied impulses (normalized for each subject, pooled over all subjects) from all sideways perturbation trials for ‘highly perturbed’ steps. (Online version in colour.)
First, we simplify the human data by representing the walking motion with three salient points (as in [21]), one marker for each foot (heel marker) and one for the pelvis (approximated by averaging the pelvis markers as in [21,23]), as a proxy for the centre of mass (CoM) [37]. These three representative points are hereafter referred to as the pelvis and feet. See figure 1 for the coordinate system used: Z is vertical (upward positive), Y is forward and X is rightward, so that the pelvis position is denoted (Xpelvis, Ypelvis, Zpelvis). Mid-stance is defined as when the forward position of the averaged pelvis marker (Ypelvis) equals the forward position of the representative foot marker currently in stance. The position of the stance foot at mid-stance is taken to be the origin (0, 0, 0) of coordinates for each step, so that mid-stance happens when Ypelvis = 0, with the pelvis moving forward relative to the foot. Each walking bout was divided into two sequences of steps, steps starting at a left mid-stance and ending at a right mid-stance, and steps starting at a right mid-stance and ending at a left mid-stance. For each such step starting and ending at a mid-stance, the stance foot position at the initial mid-stance is the origin (0, 0, 0) and the foot position for the immediately subsequent contralateral stance foot in the same coordinate frame is (Xfoot, Yfoot, 0). The steps in which the applied external impulse was very small were labelled as unperturbed (impulse less than a small threshold, 2 Ns, designed to ignore noise and fake pulls). All subject data were then normalized using body mass m, leg length ℓ and acceleration due to gravity g; all analyses and results are presented in this non-dimensional form, for instance, distances normalized by ℓ, speeds normalized by gℓ. In the analysis below, we only use steps that were highly perturbed, corresponding to a state deviation greater than a threshold. To determine the set of all such perturbed steps, we used the sideways velocity and forward velocity (X˙pelvis,Y˙pelvis) of the CoM in the treadmill-belt frame. This velocity was compared to the mean mid-stance CoM velocity for the unperturbed steps. Steps where the difference between these two values exceeded twice the standard deviation of the unperturbed steps (characterizing the natural unperturbed variability) were labelled as ‘highly perturbed.’ This criterion implies that we ignore about 95% of state deviations that occur due to natural variability in the absence of perturbations, ensuring that we primarily capture state deviations due to externally applied perturbations. The normalized perturbed step data from all subjects were pooled together, producing a total of about 4900 perturbed steps of data across the two experiments: 2442 steps for backward perturbations and 2445 steps for sideways perturbations. Figure 2 also shows a histogram of normalized applied impulses (normalized using individual subject weights, subject leg lengths and the acceleration due to gravity) from all trials in both experiments, looking only at impulses from these highly perturbed steps. We use the measured human recovery responses subsequent to these highly perturbed steps to derive a stabilizing controller for inverted pendulum walking, perhaps the simplest model of human walking [34]. Here, a biped with a point-mass body (figure 3a) vaults over a constant-length leg during each step, describing, in general, a three-dimensional (3D) inverted pendulum motion (figure 3b,c). The transition from one step to the next is accomplished by two collisional impulses, pushing off impulsively with the trailing leg, followed by an impulsive heel strike with the leading leg. The heel strike is modelled as a passive plastic collision with a nominally rigid leg. By changing the push-off impulse and where the foot is placed (Xfoot, Yfoot), the inverted pendulum walker can achieve complex walking trajectories and also recover from perturbations [2,28]. See electronic supplementary material, S1 for mathematical details and the equations of motion. Figure 3. Inverted pendulum walking with a simple biped. (a) This biped consists of a point-mass and two mass-less legs. The inverted pendulum walking gait consists of single-stance inverted-pendulum motions separated by an instantaneous step-to-step transition consisting of a push-off and a heel-strike impulse along the trailing and leading legs. (b) The 3D inverted pendulum walking motion consists of 3D inverted pendulum phases, and the foot placements not in a line. A two-step periodic 3D inverted pendulum walking motion is shown. (c) The 3D inverted pendulum walking motion can be controlled by modulating the next target foot-position and the applied push-off impulse, allowing the biped to recover from perturbations. (Online version in colour.)
For this inverted pendulum model, first, we determine the unique two-step periodic motion that matches the mean step length, mean step width and mean walking speed of our subjects (figure 3b). See electronic supplementary material, S1 and S2 for the computational procedure for obtaining this best-fit periodic motion. We consider this to be ‘nominal motion’ for the walker in the absence of any perturbation. Deviations from this nominal motion are to be corrected by the feedback controller, and such deviations in state and control from their nominal values are represented by the prefix Δ. Specifically, we assume that the foot placement and the push-off impulse are changed in response to deviations in mid-stance state S of the biped, characterized by the three variables (XCoM,X˙CoM,Y˙CoM), in the belt-fixed frame with origin at the current stance foot. These three scalar variables form the complete state of the inverted pendulum walker: due to the choice of origin, we have YCoM = 0 at mid-stance; further, ZCoM and Z˙CoM are simple functions of (XCoM,X˙CoM,Y˙CoM) due to the constancy of the leg length ℓ. Next, we describe how the feedback controller for inverted pendulum walking is derived so as to approximate the human perturbation responses. For this comparison, we use the averaged human pelvic marker state as a proxy for the CoM state. First, from the highly perturbed steps in the human data, we determine a step-to-step map; this step-to-step map describes the state transition from one highly perturbed mid-stance, say, step n, to the next mid-stance, namely the (n + 1)th step: Δ[Xpelvis(n+1)X˙pelvis(n+1)Y˙pelvis(n+1)]mid-stance=J1⋅Δ[Xpelvis(n)X˙pelvis(n)Y˙pelvis(n)]mid-stance,2.1 where Δ refers to changes in the states from the mean state across all mid-stances with the same stance foot (that is, all left mid-stances or all right mid-stances) and J1 is a 3 × 3 matrix. Two such matrices J1 are derived, both using ordinary least squares [21]: one for the transition from the left mid-stance to the right mid-stance and another for the transition from the right mid-stance to the left mid-stance. These step-to-step maps characterize the cumulative effect of the human walking controller over the mid-stance-to-mid-stance period.The position of one stance relative to the previous stance foot is characterized by two numbers: a step-length Yfoot and a step-width Xfoot, defined as the distance between the successive stance feet (in the belt frame) in the AP and the ML directions, respectively. We computed the mean step length and step width using unperturbed steps for each trial and deviations from these trial means were measured for all perturbed steps. Then, we computed the best linear mapping from mid-stance state deviations to step width and step length deviations corresponding to the next foot placement (as in [21]), given by: Δ[Xfoot(n+1)Yfoot(n+1)]=K⋅Δ[Xpelvis(n)X˙pelvis(n)Y˙pelvis(n)]mid-stance,2.2 where K is a 2 × 3 matrix quantifying the sensitivity of the foot placement to the mid-stance state deviations. Again, two such mapping were computed, one for left foot placements and another for right foot placements.The foot placement and push-off impulse controller for inverted pendulum walking are derived so as to best approximate the step-to-step map and foot placement dynamics directly derived from human data (J1 and K). A push-off impulse controller cannot be derived directly from human data since humans use forces spread out over the duration of a step rather than use a discrete impulse. In the simple inverted pendulum model of human walking, the complex modulation of ground reaction forces is captured simply by the control of the push-off impulse. The biped has three scalar control variables per step, namely, the fore-aft foot placement, the sideways foot placement, and the push-off impulse. We assume that these control variables are determined by a once-per-step linear controller of the following form: Δ [Xfoot(n+1)Yfoot(n+1)Ipush-off(n+1)]=J4⋅Δ [XCoM(n)X˙CoM(n)Y˙CoM(n)]mid-stance.2.3 That is, the deviations in the mid-stance state on the nth step results in the modulation of the immediately following foot placement and push-off impulse. The elements of this matrix J4 are the control gain unknowns we seek.First, we compute the linearized step-to-step map for the inverted pendulum biped model, written as the sum of two terms, one due to the passive inverted pendulum dynamics and another due to the effect of the controller: Δ [XCoM(n+1)X˙CoM(n+1)Y˙CoM(n+1)]mid-stance=J2⋅Δ [XCoM(n)X˙CoM(n)Y˙CoM(n)]mid-stance +J3⋅Δ [Xfoot(n+1)yfoot(n+1)Ipush-off(n+1)].2.4 Here, J2 and J3 are properties of the inverted pendulum model, and can be derived directly from the inverted pendulum model by linearizing its dynamics about the non-planar periodic motion. The sensitivity matrix or Jacobian J2 is derived by a finite difference approximation, by computing how a small perturbation in each state direction, namely [XCoM(n),X˙CoM(n),Y˙CoM(n)] at one mid-stance, grows over one step for inverted pendulum walking in the absence of any feedback control, that is, Δ[Xfoot(n+1),yfoot(n+1),Ipush-off(n+1)]=0. Thus, the J2 matrix captures the ‘passive dynamics’ of the inverted pendulum walking model. The matrix J3 is derived, again by a finite difference approximation, by computing the effect of small deviations in each of the control variables, namely [Xfoot(n+1),yfoot(n+1),Ipush-off(n+1)], in the absence of any initial state perturbations at the initial mid-stance, that is, Δ[XCoM(n),X˙CoM(n),Y˙CoM(n)]=0.Substituting our linear controller definition (equation (2.3)) into this linearized inverted pendulum model dynamics (equation (2.4)), we write the step-to-step state transition map for the model as: Δ [XCoM(n+1)X˙CoM(n+1)Y˙CoM(n+1)]mid-stance=(J2+J3⋅J4)⋅Δ [XCoM(n)X˙CoM(n)Y˙CoM(n)]mid-stance.2.5 We solve for the control gain matrix J4 by solving a nonlinear least-squares problem. Specifically, we minimize the weighted squared difference between the model and experimental versions of foot placement controller gains and the mid-stance-to-mid-stance state transition maps. That is, the matrix error terms we seek to minimize are: e1 = J4(1 : 2, : ) − K for the foot placement controller gains and e2 = J1 − (J2 + J3 . J4) for the mid-stance-to-mid-stance state transition maps. This gives us 15 scalar error terms in the objective function, six from the foot-placement gain error e1 and nine from the mid-stance-to-mid-stance map error e2. The squares of these 15 scalar error terms are scaled by the error variances of the corresponding elements of K and J1. The sum of the squares of these scaled error terms is minimized. A more formal representation of the optimization problem can be found in electronic supplementary material, S3.Most humans are approximately left-right mirror symmetric, but not exactly, at least as inferred from steady unperturbed walking dynamics [21,38]. That is, the regression coefficients in these linear models for left to right transitions are similar in magnitude to those for the right to left transitions, except for the signs of coefficients that couple sideways terms (i.e. step width and sideways CoM position or velocity) to forward terms (i.e. step length forward velocity and push-off impulse) or vice versa being reversed. The coefficients are mirrored with respect to the sagittal plane. To understand this mirroring, consider a rightward push when the right foot is in stance and a leftward push when the left foot is in stance. Both perturbations should both affect the magnitude of step width in the same way; however, a larger step width in left-stance means placing the right foot further right, and a larger step width in right stance means placing the left foot further left [21]. Here, for simplicity, we constrained the inferred controllers to be exactly mirror symmetric. To do this, we first produced a mean mirror symmetrized left-to-right version of K and J1 by averaging the left-to-right regression and the mirrored version of the right-to-left regression. This new mirror symmetrized K and J1 are then used in the optimization to derive the inverted pendulum controller. As an alternative to symmetrizing the gains by taking the mean, we could directly fit a single mirror-symmetric controller to all of the data, which gives approximately the same results. All variables and parameters listed in the results below are non-dimensional, normalized using body mass m, leg length ℓ and the acceleration due to gravity g. The ± and ∓ signs represent mirror symmetry in the control gain values, with the top sign representing the gains for left-to-right transitions and the bottom sign for the right-to-left transitions. The typical inverted pendulum walker has a zero step width planar motion [34,39]. Such planar inverted pendulum walkers cannot reasonably approximate the average human walking trajectory, which has non-zero step width and non-planar body motion [40]. Thus, here, to capture the 3D aspects of human walking, we use a non-planar 3D inverted pendulum walking with non-zero step width. Specifically, we obtained the nominal inverted pendulum walking motion of the simulated biped as having the same average step length (0.66 ± 0.03), step width (0.23 ± 0.04), and forward speed (0.39 ± 0.02) as the human subjects, in the absence of external perturbations. Figure 4 shows that this nominal motion closely resembles the mean unperturbed motion for our human subjects, even though we did not fit the detailed motion. Figure 4. Comparing human data to simulated biped. (a) Data from a human subject in the medio-lateral perturbation experiment are shown on the left-panel, compared with analogous motions of the inverted pendulum biped on the right panel. The nominal trajectory of the pelvis and the position of the foot are shown in the transverse plane (top down view). Two perturbed trajectories and their respective foot positions are also shown. For the simulated biped (right panel), we show top-view walking trajectories with and without a rightward perturbation applied to the initial mid-stance state. The trajectory and corresponding foot positions can be seen. The two trajectories chosen had both a rightwards position and velocity deviation at mid-stance, so as to be more distinct from the unperturbed trajectory. (b) Data from a human subject in the anterior–posterior perturbation experiment are shown in the left panel, compared with motions from the inverted pendulum model in the right panel. The nominal variation of the pelvis forward velocity as a function of gait phase can be seen. Two perturbation recovery trajectories are also shown. For the simulated biped (right panel), we show fore-aft velocity trajectories for gaits with and without a backward perturbation applied to the initial mid-stance state. The variation of the CoM forward velocity as a function of gait phase can be seen. More perturbed trajectories are shown in electronic supplementary material, figure S3. (Online version in colour.)
Given that we fit the model to the data, it may seem that this qualitatively similarity is just a consistency check. However, while we fit the biped model to the data, we do not fit full shapes of trajectories. We just fit the values of three variables (speed, step length and step width) for the nominal gait. Thus, that the shape of the trajectories for the body is similar for the data and the model both for the nominal trajectory and for perturbation responses is not inevitable, and relies on the 3D inverted pendulum walking model being a reasonable model of walking, capturing the essential CoM mechanics. For instance, the more common planar inverted pendulum model does not capture the top view kinematics of the CoM. Perturbations applied through pulls at the waist successfully moved subjects away from their nominal motion. When a rightward pull is applied to the subject, their subsequent pelvis trajectory and foot position are shifted in the direction of the pull by a margin much greater than 1 s.d. of normal variability during unperturbed walking, as can be seen in figure 4a. When a backward pull is applied to the subject, the pelvis forward velocity slows down by a margin much greater than 1 s.d., as can be seen in figure 4c. This is a consistency check: our goal with the perturbation experiment was to generate state deviations sufficiently far away from nominal human motion variability and we find that these large deviations were actually achieved, giving about 4900 total steps across all subjects and experiments that satisfied the criteria for ‘highly perturbed’ steps outlined earlier. Figure 4a,c shows only two perturbed trajectories out of thousands; see electronic supplementary material, S4 for more such trajectories. The foot placement control matrix K derived directly from regression on the human data is given by the linear equations: Δ Xfoot= 1.86⋅Δ Xpelvis+1.96⋅Δ X˙pelvis±0.07⋅Δ Y˙pelvis, with R2=0.85 andΔ Yfoot=∓0.47⋅Δ Xpelvis∓1.04⋅Δ X˙pelvis+0.81⋅Δ Y˙pelvis,with R2=0.3, corresponding to K=[1.86, 1.96,±0.07;∓0.47,∓1.04, 0.81]. See electronic supplementary material, S5 for the confidence intervals for these calculations.These linear descriptions have high R2 values (R2 = 0.85) for sideways foot placement, suggesting that a much more complex model with more state variables or nonlinearities may not explain the foot placement much better. There is no a priori necessity for linear models to capture such a large fraction of the variability. By contrast, the fore-aft foot placement is less well explained by the linear model (R2 = 0.3). This may be because in the fore-aft direction, humans use more within-foot modulation of centre of pressure to correct CoM state perturbations, so that the necessity of stepping-based foot placement control may be less [17]. Furthermore, we speculate that the lower explanatory power of the linear model may suggest the use of more complex models, having nonlinearities or more state variables, or more sensorimotor noise in that direction (given that motor noise may scale with forces, which are greater in the fore-aft than in the sideways direction). This foot placement controller suggests that humans step in the direction of the perturbation, as determined previously in natural variability and for smaller perturbations [7,21]. Specifically, a rightward push during left stance will result in the next right foot placement to be more rightward of its usual position; if the push is leftward, the right foot placement is left of its usual position. Similarly, when pushed backward, the human takes a shorter next step, and by extrapolation, takes a longer step when pushed forward. Furthermore, there is coupling between sideways foot placement and fore-aft motion, and vice versa. In each case, ‘how much’ the subject changes their foot-position in the next step depends on their pelvis-state deviation and the coefficients (gains) in the linear equations. While the remarks in this and the next paragraph refer to the foot placement control based on torso state directly derived from human data (equations (3.1)–(3.2)), we find that the same properties hold for the final foot placement controller for the inverted pendulum model, discussed later in equations (3.3)–(3.4). The foot placement controller is qualitatively similar to the foot-placement controller that was derived using the same methods but using natural variability in unperturbed walking (as in [21]). That is, the same terms had large coefficients and the signs of the coefficients were the same. Specifically, performing a linear regression to predict coefficients derived from unperturbed walking to those from perturbed walking [21], we obtain a 96.2% R2 with a slope of 0.93, suggesting considerable correlation between the coefficients across the two conditions; the largest difference in coefficients is in the coefficient relating Xfoot and X˙pelvis (see figure 7). The quantitative differences between the foot placement gains derived perhaps suggests a small nonlinearity or a more complex underlying controller for larger perturbations. The qualitative similarity supports the use of natural variability to infer details of the controller. One makes different underlying assumptions when using small natural variability versus large externally applied perturbations, and that they produce qualitatively similar controllers suggests that the results are robust to such different assumptions. Fitting the controlled inverted pendulum walker to the human perturbation recoveries, we obtain the following relations for the posited feedback controller: Δ Xfoot=2.08⋅Δ XCoM+2.24⋅Δ X˙CoM∓0.3⋅Δ Y˙CoM,3.3 Δ Yfoot=∓0.55⋅Δ XCoM∓1.07⋅Δ X˙CoM+0.99⋅Δ Y˙CoM3.4 and Δ Ipush-off=∓0.05⋅Δ XCoM∓0.60⋅Δ X˙CoM−0.13⋅Δ Y˙CoM,3.5 corresponding to a feedback gain matrix:J4=[2.082.24∓0.3∓0.55∓1.07+0.99∓0.05∓0.60−0.13]. The impulse controller has the intuitive property that when pushed forward, the push-off impulse decreases and when pushed backward, the push-off impulse increases. More significantly, the push-off impulse is coupled to the sideways perturbations. When on a left stance phase, a push to the right results in a smaller subsequent push off and a push to the left results in a larger push off. The foot-position controller has the same qualitative interpretation as described earlier, but is slightly different quantitatively from that derived directly from human data, a trade-off made by the optimization to better approximate the step-to-step map. Electronic supplementary material, S6 describes how well this controller approximates human behaviour.The derived controller stabilizes the simulated inverted pendulum biped walking with a human-like step length, step width and speed. That is, small perturbations decay exponentially and the biped approaches the nominal periodic motion. The physical mechanisms by which the foot placement and the impulse controller achieve stability are intuitive: by stepping in the direction of the perturbation, the controller changes the leg force direction, thereby increasing the horizontal component of the leg force in a direction opposing the perturbation. This serves as a restoring force to counter the perturbation. The impulse controller acts in a similar way, increasing the impulse when the forward velocity is too low, thereby helping to move back toward the desired forward velocity. More specifically, perturbations to this simulated inverted pendulum biped produce recovery trajectories that qualitatively resemble corresponding human recoveries in our experimental data (figure 4a,b). More quantitatively, pooled across all the highly perturbed steps, after one complete step, the linear step-to-step map J1 approximates the sideways position with an average R2 of 0.58, the sideways velocity with an average R2 of 0.52, and the fore-aft velocity with an average R2 of 0.27 (see electronic supplementary material, S5). Overall, these R2 values suggest that the linearized model approximates the sideways trajectories better than the fore-aft trajectories. Additionally, as a consistency check, for the range of perturbations applied in the experiment, we found that the linearized mid-stance-to-mid-stance inverted pendulum dynamics in equation (2.4) is a good approximation of the full nonlinear inverted pendulum model (electronic supplementary material, figure S5). Given that we fit the model to the data, it may seem that this qualitatively similarity is again just a consistency check. However, while we fit the biped model controller to the mid-stance to mid-stance map derived from human responses, we did not fit full shapes of the response trajectories. Thus, that the shape of the perturbation response trajectories for the body is similar for the data and the model suggests that the model, despite being simple and having only a small number of parameters to tune for the fits, has successfully captured the key elements of the response. In fact, even the linear stability of the simulation is not inevitable and relies on the ability of the controlled inverted pendulum model to approximate the step-to-step map of the human. Thus, a priori, all three of the following were possible for the controlled inverted pendulum model: (a) the simulation is unstable, (b) the simulation is stable but is not sufficiently similar to humans, and (c) the simulation is stable and is similar to humans in its nominal gait and recovery transients. We computed an approximate ‘basin of attraction’ for the controlled biped (figure 5) by determining the set of perturbations that does not result in the biped falling down during the next 20 steps and converging asymptotically to the nominal periodic motion. Consider uniaxial perturbations, applied only along one of the three directions, xCoM, x˙CoM or y˙CoM, while the other two are unperturbed at mid-stance. In each direction, figure 5 shows that the controlled biped can handle 20–30 times higher state perturbations than the standard deviation of the natural variability we measured in normal ‘unperturbed’ steps (see electronic supplementary material, table S1) and about 7–10 times the standard deviations of the highly perturbed steps (see electronic supplementary material, S5). Compared to these large basins of attraction for the controlled biped, the uncontrolled 3D inverted pendulum biped is unstable, and thus has a zero basin of attraction. Figure 5. Basin of attraction. The biped is given an initial mid-stance disturbance in two out of three of the states and then simulated for 20 steps. Disturbance combinations where the biped does not fall down are marked for the (a) Y˙CoM=0 plane, (b) X˙CoM=0 plane, (c) XCoM = 0 plane. All quantities plotted are non-dimensional. (Online version in colour.)
Varying parameters one at a time, we find the range for each of the nine gains for which the biped remains stable. The absolute magnitude of all the eigenvalues is used to determine if the biped is stable. As we vary each of the nine gains in the inverted pendulum controller one at a time, we find that small, and in some cases even large, deviations in these gains do not cause the biped to become unstable. The exact range of errors for each of the controller gains can be seen in figure 6a. Figure 6. Robustness to parametric uncertainty. The biped is given an initial mid-stance perturbation. (a) An error is introduced in the value of each of the nine gain parameters. The eigenvalues for the 3 × 3 state transition matrix are measured. The range of gain errors for which the maximum absolute eigenvalue is less than 1 are denoted by the blue bars. (b) An error is introduced in one of the body parameters. The biped is simulated for 20 steps, the range of body parameter errors for which the biped does not fall down, i.e. the CoM remains above the ground, are marked with the black bars. All quantities shown are non-dimensional, so the leg length, gravity and mass errors admissible (as shown in b) are deviations from their nominal non-dimensional unit values, namely unity. Similarly, the errors admissible in the feedback gain terms are deviations from their nominal non-dimensional values given in equations (3.3)–(3.5). (Online version in colour.)
Figure 7. Unperturbed walking vs perturbed walking. The foot placement gains inferred from the perturbation experiments herein (horizontal axis) are quantitatively similar to those derived from unperturbed walking (vertical axis) in a previous study [10]. The similarity two sets of gains is characterized by the best fit line y = 0.93x − 0.07, with an R2 = 0.963. It appears that the largest relative difference in the gains is in the response of sideways foot placement Xfoot to deviations in mid-stance X˙pelvis. Dropping this gain in the comparison gives an R2 = 0.996 and a regression equation y = 1.05x − 0.03. Note that the gains from [10] have been non-dimensionalized here for direct comparison.
The system remains stable despite changes in body mass from −50% to +100%, leg length from −10% to +120% and gravity from −50% to +50%. The range of these errors can be seen in figure 6b. Such robustness to parameter changes will allow the biped to walk stably despite uncertainty about such parameters; for instance, such robustness may allow humans to walk stably with the same controller while carrying an unknown additional mass. The foot placement controller described here uses the lateral state of the CoM to determine step width similar to previously reported models [5,21,23]. Kim & Collins [27] describe a controller that uses both foot placement and push-off impulse to stabilize a biped model with more degrees of freedom than the one described here. Their controller allows this more complicated biped to tolerate a sideways impulse disturbance of ±6.3 N · s for a biped of mass 74 kg which equals a non-dimensional sideways CoM velocity disturbance of ±0.027 compared to a range of −0.20 to +0.26 for our simpler controlled biped. The goal of this paper was to use human data to describe a method for deriving general controllers for models of walking and apply that specifically to the simplest biped model. To the extent that this simple biped model captures something about human biomechanics, the controller derived may shed light on the human control of walking. Indeed, our controller model responds qualitatively similarly to a human. By following the same procedure with more realistic biomechanical models, potentially constrained by available information about the structure of neural feedback, we can obtain controllers for such more complex models, perhaps approaching that of the true controller. Specifically, these methods can be applied equivalently to any other model of human walking and any valid controller formulation for the selected model. A natural next model would be one that still uses massless legs, but includes a 3D upper body (pelvis), so that the controller needs to manage the orientation dynamics as well. We could also consider models that can manipulate the ankle or hip joints, where each joint torque is dependent on the current or past state of the body. Fitting much higher degree-of-freedom models to the data may not be feasible, as such fitting may require a higher dimensionality of perturbations than we achieved. Approaches based on the neuromechanics of walking [41] might be more suitable in these cases, where we have a small set of neural feedback parameters that can be tuned to best fit the perturbation recovery data. However, this requires us to characterize, using perhaps neural recordings, the structure of the neural controller. We have made other similar simplifications in our assumptions here, for instance, related to linearity of the control mechanisms, which could be generalized to nonlinear terms, and focusing on mid-stance state, which can be generalized to state at any point during the stance phase. Prior work [7,15,17,21] suggests that the perturbation phase (given a perturbation magnitude) may affect the subsequent recovery response, as we will examine in future work. This difference between perturbation phases may be due, in part, to how a perturbation at one gait phase grows or decays even due to feedforward control as the gait proceeds forward to another gait phase. This is certainly true for the inverted pendulum walking gait, where say, a rightward perturbation during the early part of a left stance phase may grow over the stance phase due to inverted pendulum dynamics; so, in this case, a given perturbation earlier in stance is equivalent to a larger perturbation later in the stance phase. Perturbation phase may also affect future foot placement due to there being considerable sensorimotor delays, between sensing and muscle force development via feedback control. Such delays imply that perturbations that occur just before the foot placement may not leave enough time for neurally mediated feedback control to occur. Complete simulations of bipedal walking, such as those demonstrated here for the simple inverted pendulum biped, could allow us to devise measures of stability which extrapolate the information from our model-based controllers. For instance, we could repeat this procedure for subjects with movement disorders, amputees, or the elderly [5,42], potentially with much smaller perturbation sizes, to characterize how their walking controller differs from those of young healthy non-amputee individuals. We could then predict probability of falls due to specific perturbations or compare the basin of attraction for controllers derived from any individual with those derived from subjects with a low fall risk. First, however, these measures would need to be validated either by deriving similar controllers from individuals known to have a high fall risk or by conducting experiments with even larger perturbations to determine whether the basins of attraction accurately measure the limits of stability for healthy individuals. Even after validation, we would still need to compare the effectiveness of such measures relative to other stability measures suggested by others [43]. The notion of stability of an equilibrium or a periodic motion cannot be captured by one number, and different metrics of stability need not be correlated. For instance, a system with a large basin of attraction can have lower robustness or attraction to the stable periodic motion. Obtaining the foot placement and mid-stance-to-mid-stance linear models for individual subjects indicates qualitative similarity across subjects with some quantitative differences in the coefficients (electronic supplementary material, figures S4 and S5). To meaningfully compare the inferred feedback gains from different subjects or subject populations or experiments, we would first need to accurately characterize the trial to trial variability of such gains for individual subjects: repeating over different days and repeating with different experimenters with similar marker-placement instructions. One potential way to reduce variability due to experimental practice or marker placement may be to obtain CoM position and velocity estimates by combining integrated ground reaction force data and marker data with an appropriate Kalman-like filter to avoid drift [33,44]. We had the subjects walk with their arms crossed to allow for applying sideways perturbations without being interrupted by the swinging arm. This restriction of arm movement prevents the subjects from using arms for balancing, which may potentially change how they control walking with their legs. If we repeated these trials with arm movement, we hypothesize that the qualitative features of control inferred here will be preserved but potentially with some quantitative differences in stepping and push-off strategies [45,46]. This hypothesis will be tested in future work, and is plausible on account of another study [21] allowing some arm swing (without perturbations) did infer similar strategies. Another simplification in the experimental protocol was that the fore-aft perturbations we applied were only backward, but not forward. This will not have an effect on the inferred controllers if we are still in the linear regime, whereas a nonlinear controller may show different response magnitudes for forward and backward perturbations, and indeed perturbations in other directions in the horizontal plane spanned by fore-aft and sideways directions. Finally, we only considered discrete perturbations to the pelvis and did not consider other kinds of perturbations or perturbations to other parts of the body. If such alternative perturbations eventually affect the CoM state, our inverted pendulum biped controller can make predictions for the subsequent control action, which can be tested in future studies. All the human-derived results presented here are across pooled sideways and backward perturbation experiments. The presence of small nonlinearities may partly explain differences in R2 values when foot placement linear models (as in equations (3.1)–(3.2)) were derived using just the sideways perturbation trials or just the backward perturbation trials. Specifically, the R2 for fore-aft foot placement was higher (0.52) when derived from the backward perturbation experiments and lower (0.25) when derived from the sideways perturbation experiments, compared to 0.3 when derived from all trials pooled. Conversely, the R2 for the sideways foot placement was lower (0.75) when derived from just the backward perturbation experiments, compared when all the data were used (0.85). The controller we have derived is applicable directly to level ground constant speed walking. Of course, real human locomotion involves walking with turning, on stairs and slopes, with changing speeds, running, etc. While our controller does not apply directly to these situations, the methods used herein can be repeated in those circumstances to produce a more general or at least task-specific or context-specific controller, which we hope to explore in future work. Furthermore, we hypothesize that the controller, given its robustness, is able to walk stably at least on small slopes and slightly uneven terrain. We have used the word ‘control’ throughout this article to characterize how the human subject and the mathematical model respond to perturbations, for instance, through the modulation of foot placement and the push-off impulse. To be clear, for the human, this control need not entirely be due to neurally mediated feedback control, but instead due to the interaction between the feedforward dynamics of the human body (given some fixed motor outputs to muscles, say) and sensor-driven feedback control, which could be spinal or cortical. Identifying the relative contributions of feedforward and feedback control may require experiments that measure motor outputs after a perturbations to see if they change, and measure degradation in locomotor performance after degrading sensory feedback. For the mathematical model itself, both the foot placement and the impulse controllers are entirely feedback controlled, as these quantities can be directly controlled in the model as inputs to the inverted pendulum dynamics. Human-derived controllers, such as proposed here, can have other practical applications. For instance, they could be used to inform the development of better feedback controllers for robotic prostheses and exoskeletons [31,32]. Current robotic prostheses and exoskeleton controllers, when based on human walking, usually only use information regarding the average motion or torques during walking, but not how humans respond to perturbations. Implementing a human-derived controller into such devices may make them more ‘natural’ to the user, and perhaps reduce the duration needed to learn to use such devices effectively. More generally, such human walking controllers could inform the design of other biomechatronic assistive devices or even passive objects such as backpacks, so as to maximize stability while walking with such devices or objects. Analogously, one could implement controllers derived from human walking directly onto walking robots [29,30]. Such human-inspired robot controllers can make the robots more human-like, walking and respond to perturbations like humans do. Our methods could also be applied in the context of human learning, both children learning to walk or other humans learning to walk under a new circumstance (e.g. wearing a new device or on a new surface), which will allow us to track how the controller and stability properties change and (presumably) improve during learning [32,47]. To be clear, there is an infinity of possible controllers that would stabilize the inverted pendulum gait (e.g. [25–28,30]). For any given stable gait, generically, a nearby feedback-gain set will also be stable, unless the system is at the boundary of stability (which is not the case here). Indeed, figure 6 shows the set of feedback gains that stabilizes the biped. In addition to these feedback controllers, we could imagine controllers with other architectures, for instance, using feedback from many previous steps than just the previous or current one. One general goal in robotics is to develop controllers that might result in bipedal walking robots that are more stable and more efficient than current bipeds with ad hoc controllers. To achieve stable and efficient locomotion, bipedal legged robot controllers need not necessarily track the human walking controller. However, it may be desirable for psychological or social reasons to control a robot similarly to how a human walks, that is, build robots that not just look like humans, but also behave in a similar way, up to the way they recover from perturbations. While biomimetism in robots may not always be optimal, it sometimes can be despite different cost functions for humans and robots (e.g. [1] showed such coincidence in simple models). It is an open question as to whether biomimetic control is reasonably efficient and robust, and one can only settle this question by implementing such controllers on actual hardware. This study represents a step towards, and provides a general method for, the development of such biomimetic controllers. We note that foot placement control with qualitative similarities with that obtained here is often used in complex robots, although not directly biomimetic [30,36]. Experimental protocols were approved by the Ohio State University IRB and all subjects participated with informed consent. V.J. and M.S. conceived the study. V.J. performed all the human subject experiments. V.J. performed all data analyses and computational simulations, partly in discussions with M.S. V.J. wrote the first draft and V.J. and M.S. edited further. All authors approve the final draft. We declare no conflicts of interest. The data and code will be available without restrictions through the figshare database. Code is available at: https://doi.org/10.6084/m9.figshare.9337445.v1 (doi:10.6084/m9.figshare.9337445) and the human data is available at: https://doi.org/10.6084/m9.figshare.9273491.v1 (doi:10.6084/m9.figshare.9273491). This work was supported by the National Science FoundationCMMI grant nos. 1538342 and 1254842. Thanks to Movement Laboratory members for helping apply sideways perturbations for a few subjects. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4585757. References
Page 5The aortic (AV) and mitral (MV) valves are coupled via a common fibrous continuity known as the aortic–mitral curtain or intervalvular fibrosa [1]. As a result, the physiological function of either of these two valves involves the complementary function of the other [2]. A better understanding of the aortic–mitral structural coupling and the ability to accurately assess the biomechanical changes in different treatment scenarios is important, especially in the context of heart valve intervention planning and post-procedural assessment, where these changes will have an impact not only on valve dynamics but also on left ventricular (LV) function. A prime example of aortic–mitral functional complexity is during the treatment of severe aortic stenosis (AS) in the presence of secondary significant mitral regurgitation (MR). The prevalence of moderate-to-severe concomitant MR in patients undergoing transcatheter aortic valve replacement (TAVR) ranges from 20% to 33% [3], with significant impact on prognosis [4,5]. Albeit, MR is inherently left untreated in this setting, and its severity may decrease, remain unchanged or even increase following the procedure [6,7]. As existing clinical data on the direction and magnitude of MR severity following TAVR have demonstrated conflicting evidence and constitute a current clinical dilemma, a quantitative engineering analysis of the underlying synchronized reciprocal behaviour and biomechanical AV-LV-MV coupling during and after TAVR can potentially provide the basis for an individualized treatment approach and better procedural planning. Patient-specific cardiac computer modelling can provide unique insights into valve function and disease states, as well as allow for exploration within a range of treatment options that cannot be performed via in vitro, animal and human studies alone [8]. For instance, Gao et al. [9] simulated MV dynamics over half of the cardiac cycle using a computational LV-MV model developed from cardiac magnetic resonance imaging. More recently, Caballero et al. simulated AV-LV-MV dynamics over the entire cardiac cycle under physiological and degenerative MR states [10,11]. While these computational studies have addressed some key challenges in modelling the complex biomechanical coupling between the heart valves and the blood flow in the left heart (LH), to the best of our knowledge, no patient-specific fluid–structure interaction (FSI) computational models have been developed to analyse the effect of TAVR on aortic–mitral coupling and MR severity for patients with severe AS and concomitant MR. The aims of this study are therefore: (1) to develop a robust patient-specific LH model with severe AS in the presence of significant functional MR, and validate the pre-TAVR LH dynamics with clinical echo data, (2) model the TAVR procedure and analyse the influence of stent implantation height on aortic–mitral structural coupling, and (3) simulate and validate the post-TAVR LH dynamics with the goal to investigate the effect of TAVR on MR severity. We believe that this work offers a detailed engineering analysis that could shed some light on the structural and haemodynamic impact of TAVR on concomitant MR. Further development of such computer models could support better evaluation and prediction of interventional treatments, and ultimately improve patient outcomes. A 71-year-old male patient with a mean aortic annulus diameter of 23 mm was referred for TAVR at Hartford Hospital (Hartford, CT). Pre-TAVR echo examination revealed classical low-flow, low-gradient severe AS, a bicuspid AV (BAV) with eccentric coaptation between the non-coronary leaflet and fused left and right coronary leaflets, with no raphe between them (Sievers type 0 L/R) [12]. Mild aortic insufficiency was detected. Moderate-to-severe functional MR was also found, with restricted posterior mitral leaflet (PML) motion and reduced leaflet coaptation, resulting in a posteriorly directed regurgitant jet. The LV wall thickness was normal, but the chamber was mildly dilated with severe global hypokinesis with regional variation. The left atrium (LA) was dilated despite a normal antero-posterior diameter. A 26 mm first-generation Edwards SAPIEN valve was successfully deployed via transapical approach. Moderate-to-severe MR was still present after TAVR. The patient-specific LH model was created from the pre-TAVR cardiac multislice computed tomography (MSCT) images after obtaining approval from the Institutional Review Board. The MSCT images, acquired with a GE LightSpeed 64-channel volume CT scanner, had an in-plane resolution of 0.82 × 0.82 mm and a slice thickness of 0.625 mm. Ten phases of the cardiac cycle were collected using an ECG-gated sequence. DICOM images were imported into Amira-Avizo (Thermo Fisher Scientific, MA) and 3D Slicer (www.slicer.org) software to segment the cardiac structures, which included the ascending aorta, aortic root, BAV, calcification, MV, chordae, LV and LA endocardial walls. HyperMesh software (Altair Engineering, Inc., MI) was then used to create a high-quality finite-element (FE) mesh. Figure 1a shows the patient-specific LH model. Two calcific deposits were found in the fused leaflet, as well as one in the coronary ostia close to the right coronary artery. Image segmentation and mesh generation techniques have been previously described in detail elsewhere [10,13]. Figure 1. (a) Patient-specific LH model reconstructed from the MSCT images, (b) balloon-expandable TAV model, (c) aortic and LA pressure waveforms and (d) LV volume waveform. (Online version in colour.)
An anisotropic hyperelastic material model, based on Holzapfel et al. [14,15], was adopted to characterize the mechanical response of most cardiac tissues, while the isotropic hyperelastic Ogden material model [16] was used to characterize the mechanical properties of mitral chordae and aortic–mitral curtain. Material parameters were determined by fitting in-house multiprotocol biaxial and uniaxial testing data of human cardiac tissues. Details on the implementation of the constitutive models as well as material parameter fitting can be found in the electronic supplementary material. A generic balloon-expandable TAV model, based on a 26 mm first-generation Edwards SAPIEN device was used in this study [17]. As seen in figure 1b, the TAV consists of three components: stainless steel stent, skirt and bovine pericardium leaflets. The generic TAV leaflet geometry and material parameters used in this study were obtained from previous studies [18–20]. In this model, the attachment line of the leaflets to the stent is curved from the commissures downwards to the leaflet cusps, having a scallop leaflet shape similar to that of the Edwards SAPIEN valve design. More details on the balloon-expandable TAV model can be found in the electronic supplementary material. The TAVR procedure was simulated in the patient-specific LH model through four major simulation steps [17,21]. To accurately evaluate the impact of TAVR on MR severity, the patient-specific myocardium surrounding the aortic root, aortic–mitral curtain, MV and proximal LV/LA endocardial walls was also reconstructed from the MSCT images and integrated in the LH model, as seen in figure 2b. Figure 2. (a) Crimped stent geometry at the three deployment heights, (b) LH models with myocardium after TAVR deployment and stent recoil. (Online version in colour.)
Initially, the nominal stent was positioned coaxially within the aortic root and centred into the aortic annulus. The crimped stent and balloon geometries were obtained by applying a radial displacement on two cylindrical-surface sheaths. The stent was crimped to an exterior diameter of 8 mm (24 Fr catheter), as seen in figure 2a, and positioned in a way that the TAV commissures had the same orientation as the native aortic sinuses. The BAV leaflets were pre-opened with a cone-shaped catheter to simulate the effect of the insertion of the delivery system across the AV. To quantify the impact of TAV deployment height on LH dynamics and MR severity, the axial positioning of the stent with respect to the aortic annulus plane was parametrized to replicate three clinical deployment configurations: (1) midway, denoted as post-TAVR 50%. The conventional manufacture's recommendation for the Edwards SAPIEN system is to position the mid-point (i.e. 50 : 50) of the stent at the aortic annulus plane [22]. (2) Post-TAVR 30%, defined as a slightly higher device implantation, with 30% of the stent located below the aortic annulus. (3) Post-TAVR 10%, defined as a high device implantation, with only 10% of the stent located below the aortic annulus plane. The aortic annulus plane was defined as the virtual plane formed by joining the points of basal attachment of the BAV leaflets [22]. The mechanical response of the fluid-filled balloon expansion was modelled by the surface-based fluid cavity method, capable of simulating the coupling between the deformation of the balloon and the fluid inside [17]. The balloon was pressurized for 0.25 s until it was fully inflated. This method was initially calibrated to ensure a correct nominal stent size after balloon inflation. The internal pressure of the balloon was decreased over a time period of 0.25 s. Therefore, the stent recoiled and stabilized, and the native BAV leaflets adopted a more natural open position. The TAV leaflets and skirt were not included during the stent deployment procedure, but were added after stent recoil. Their effect on the biomechanical interaction between the stent and the native aortic tissue has been shown to be negligible [23] The TAV leaflets were mounted in the deformed stent using a virtual assembly method [24]. Briefly, the nodes lying on the leaflet attachment and along the commissures were mapped onto the stent frame through a set of non-uniform imposed displacements, ensuring an appropriate final leaflet position. During TAVR modelling, the ascending aorta and myocardium were constrained at their distal ends allowing only rotational degrees of freedom. The balloon was constrained at its distal and proximal ends in order to mimic the bond to the catheter and avoid rigid body motion. The kinetic energy was monitored to ensure that the ratio of kinetic energy to internal energy remained under 10%. The resulting deformed LH models after stent recoil were used to assess post-TAVR LH dynamics using FSI modelling. A fully coupled FSI numerical approach that combines smoothed particle hydrodynamics (SPH) for the blood flow and nonlinear FE analysis for the heart valve structures was implemented in this study. FSI simulation set-up was performed as described in previous works [10,11]. Briefly, time-dependent pressure boundary conditions were applied at the two atrial inlets (pulmonary veins) and at the aortic outlet of the pre- and post-TAVR LH models. As seen in figure 1c, a pathological atrial pressure waveform was prescribed at the inlets, with an elevated V-wave due to the regurgitant volume in the atrial cavity during systole [25]. To match the patient's clinical data, the mean diastolic atrial pressure value was adjusted to be 12 mmHg, while the peak systolic and diastolic aortic pressure values were set to 100 mmHg and 63 mmHg, respectively. Endocardial wall motion was imposed as a time-dependent nodal displacement boundary condition based on the 10 cardiac phases from the MSCT images. A detailed description of the cardiac wall motion modelling procedure can be found in previous publications [10,13]. Figure 1d shows the time-varying LV volume change waveform obtained from the FSI model. Pressure boundary conditions and cardiac wall motion were kept the same for the pre- and post-TAVR LH models, simulating immediate post-operative LH dynamics, without considering the cardiac remodelling mechanisms that occur over time after TAVR. The patient's heart rate was approximately 60 bpm, corresponding to a cardiac cycle of 1 s. Note that FSI simulations begin at early systole, resembling the isovolumetric contraction phase. Two cardiac cycles were conducted and the results from the second cycle were analysed herein. SPH particle sensitivity [13,26] and FE mesh sensitivity [27] studies were previously performed. All FE and SPH-FE simulations were performed using Abaqus/Explicit. The regurgitant volume in the MV (RVMV) and the AV (RVAV) were obtained from the pre- and post-TAVR FSI models by integrating the negative MV systolic flow rate curve and negative AV diastolic flow rate curve over time, respectively. The RV was defined as the sum of the valve closing and the leakage volumes. Similarly, the stroke volume in the MV (SVMV) and the AV (SVAV) were obtained by integrating the positive MV diastolic flow rate curve and positive AV systolic flow rate curve over time, respectively. MR severity was graded using the regurgitant fraction criterion [28], RFMV = RVMV/LVSV, where LVSV is the total SV of the LV (SVAV + RVMV). AV effective orifice area was calculated as EOAAV=MSF/51.6ΔP, where MSF is the root mean square systolic flow rate, and ΔP is the mean systolic pressure gradient [29]. The three stent deployment configurations were analysed and compared during the TAVR procedure in terms of peak (SIMAX) and average (SIAVRG) maximum principal stress values in the BAV leaflets, sinus, calcification and MA, contact radial force between the stent and aortic root, and stent deformation. To facilitate comparison between different models and avoid the bias caused by local high stress concentration, only the 99-percentile values of the peak stress were evaluated [30]. Moreover, leaflet annular regions were not included in the average stress calculation in order to avoid boundary effects. The contact radial force between the recoiled stent and surrounding cardiac tissues was calculated as RF=∑i=1nsrfi,post, where ns is the total number of nodes in the stent, and rfi,post is the radial contact force at each stent node after recoil. Stent deformation was evaluated by quantifying stent eccentricity and recoil. Stent eccentricity, which assesses the conformity of the stent deformation after recoil, was calculated as the ratio of the maximum stent diameter to minimum stent diameter. Stent recoil was determined as (Dmin expanded − Dmin recoiled)/Dmin expanded, where Dmin is the minimum stent diameter during fully balloon expansion and stent recoil. These two metrics were calculated at three different cross-sections of the stent (i.e. bottom, middle and top) with the mean value reported. Additionally, post-TAVR tissue mechanics were evaluated by the average maximum principal stress values calculated in the BAV/TAV leaflets and MV leaflets during peak diastolic and systolic pressure, respectively. Chordae forces (Fchordae) at peak systole were also reported. The force experienced by a particular chordae group was calculated as the sum of vectors representing the tension in each individual chorda attached to that chordae group. Aortic–mitral structural coupling during TAVR was evaluated in terms of the geometrical parameters shown figure 4a. The following measurements were obtained during systole: (a) aortic annulus and MA areas, (b) antero-posterior (AP) distance, defined as the distance between mid-anterior and mid-posterior MA points, (c) anterolateral–posteromedial (AL-PM) distance, (d) inter-commissural (CC) distance, (e) MA height, defined as the maximum vertical distance between the highest and lowest points of the saddle-shaped MV, (f) aortic–mitral angle, defined as the angle between the planes of the MA and the aortic annulus, (g) aortic–mitral distance, defined as the centroid distance between the MA and the aortic annulus, (h) aortic annulus motion, defined as the longitudinal excursion of the aortic annulus during the TAVR procedure, and (i) anterior and posterior MA motion, defined as the posterior displacement of the MA during the TAVR procedure. Table 1 displays the computed haemodynamic parameters of the patient-specific LH model throughout the cardiac cycle before TAVR. Numerical measurements are compared to the patient's available pre-TAVR echo data. Generally, there is good quantitative agreement between the simulation results and the clinical data, which demonstrates that the SPH-FE FSI modelling framework can accurately simulate the patient-specific LH pathological dynamics. FSI simulation of BAV function revealed the classical low-flow low-gradient AS found clinically, with an EOAAV less than 1 cm2, mean systolic pressure gradient less than 40 mmHg, and decreased LV ejection fraction (LVEF) less than 50%. Similarly, FSI simulation of MV function demonstrated moderate-to-severe MR, which was consistent with the patient's clinical echo examination.
Figure 2a shows the crimped stent geometry at the three deployment heights, while figure 2b shows the final deformed geometries after balloon deflation and stent recoil. No severe leaflet overhanging or potential risk for coronary obstruction was observed. The shortest distance between the coronaries and the native leaflets was found to be 7.5 mm for the post-TAVR 10% case, which was between the left coronary ostia and the fused leaflet free edge. No evident gaps between the recoiled stent and the native annular tissue were observed in the post-TAVR LH models, which was later confirmed in the FSI blood flow simulations by the absence of paravalvular leak (PVL). The stress distribution on the native BAV leaflets after stent recoil is presented in figure 3. Similar stress distribution patterns were observed for the three implantation configurations, where peak values were located in regions in contact with the higher portion of the stent for the post-TAVR 30% and 10% models, and in the leaflet-root attachment region close to the commissures for the post-TAVR 50% case, as shown by the red circles in figure 3. Another leaflet region that experienced high stress values was the leaflet belly region in contact with the large calcification deposit due to the stiffer material properties. For the aortic root, high stress values were found at the leaflet-root attachment lines, especially in the fused leaflet commissures. Figure 3. Stress distribution (MPa) in the BAV leaflets after stent recoil. Red circles denote regions of peak stress values. A maximum stress value threshold of 4.5 MPa was applied such that higher stress values were displayed in grey, facilitating comparison between models. (Online version in colour.)
As presented in tables 2 and 3, the highest deployment model (post-TAVR 10%) resulted in higher peak stress values in the leaflets, sinus and calcification regions. The highest peak stress in the anterior MA, however, was found for the midway implantation model (post-TAVR 50%), a direct result of a larger aortic–mitral curtain tissue in contact with the stent. The contact radial force between the recoiled stent and surrounding cardiac tissues is listed in table 2. The post-TAVR 10% model gave the highest contact force, with a value of 108 N, while the post-TAVR 50% case gave the lowest contact force, with a value of 98 N. Stent eccentricity and recoil values are also presented in table 2. All three stents showed a similar eccentricity, maintaining a nearly uniform circular pattern. Stent recoil, however, was lower for the midway implantation model (3.3%) than for the highest implantation model (6.2%).
We found an important reduction in the MA height from pre- to the post-TAVR 50% model (15%); however, as the stent implantation height increased, the MA height increased instead; up to 8% for the post-TAVR 10% case. Additionally, the aortic–mitral angle decreased between 5% and 10% following TAVR, with a smaller systolic angle in the highest implantation model than in the midway implantation model. Although the MA area (approx. 3.2%), AP distance (approx. 4.3%) and aortic–mitral distance (approx. 4.5%) decreased in all post-TAVR models, there were no evident changes when comparing these before and after TAVR. Similarly, both AL-PM distance (approx. 3%) and aortic annulus area (approx. 2%) had a slight tendency to increase after TAVR. In regards to the aortic annulus motion, it was found that this anatomical structure was displaced approximately 3.3 mm towards the LVOT during TAVR, and this motion seemed to decrease as the stent implantation height increased. Moreover, the anterior MA was also displaced approximately 2.5 mm posteriorly during TAVR, again, this motion decreased as the implantation height increased. Finally, figure 4b shows the aortic–mitral complex before and after TAVR during peak systolic flow. Although it is evident that the regurgitant orifice area decreased in all post-TAVR models, MR was still present. Moreover, the post-TAVR 50% model exhibited the smallest regurgitant orifice area, which was later confirmed by the quantification of the RVMV. Figure 4. (a) Aortic–mitral geometrical parameters, (b) aortic–mitral coupling at peak systole. (Online version in colour.)
Figure 5 shows the flow rate waveforms across the AV an MV throughout the cardiac cycle for the pre- and post-TAVR LH models. The negative aortic flow at end-systole (figure 5a) indicates the retrograde blood flow into the LV during AV closure, while the negative systolic mitral flow (figure 5b) indicates the backflow of blood into the LA due to MV leaflet closing and MR. Diastolic inflow rates (positive) across the MV were approximately the same for all models, since LV size and motion remained unchanged immediately post-TAVR. As expected, the pre-TAVR LH model had the lowest peak aortic flow and forward SV (SVAV), as seen in figure 5a. Following TAVR, correction of the AV obstruction led to an immediate reduction in the LV systolic pressure, which decreased the pressure gradient across the AV and MV, and therefore led to a reduction in the RVMV and an increase in the SVAV. Figure 5. Flow rate (ml s−1) across the (a) AV and (b) MV throughout the cardiac cycle. (Online version in colour.)
As shown in table 4, improved LV function and haemodynamic success of the procedure was found in all post-TAVR LH models, as confirmed by the reduction of MR and AV/MR peak velocities, and by the increase in the EOAAV. The greater degree of MR improvement (10%) was for the post-TAVR 50% model; which, based on the RFMV criterion, can now be classified as moderate MR. When compared with the pre-TAVR model, the degree of MR severity remained unchanged for the post-TAVR 30% and 10% models.
When comparing the pre- and post-TAVR aortic flow rate waveforms shown in figure 5a, a notable change in the onset time of BAV/TAV closure was found. The TAV device had a distinct slower closure and higher closing volume compared to the native BAV leaflets, with the highest RVAV obtained for the post-TAVR 10% model. Pre-procedural mild aortic regurgitation, found clinically in the patient's echo examination, was also detected in the pre-TAVR computational model, as shown by the negative diastolic aortic flow (figure 5a). Aortic regurgitation was resolved following TAVR, both in the real clinical case and in the FSI simulations. Moreover, the absence of PVL in all post-TAVR LH models matched the post-operative echo findings. Additionally, figure 6 shows the intraventricular velocity streamlines coloured by velocity magnitude during peak systole. Due to the restricted PML motion, the pre-TAVR model displayed a posteriorly directed regurgitant jet, which qualitatively matched the Doppler colour echo image shown in figure 6. The regurgitant jet structures were similar between pre- and post-TAVR states, with an eccentric ‘wall-hugging' jet that impinged the postero-lateral atrial wall. The strength and velocity of the jet, however, decreased after the procedure, with a similar MR peak velocity for all post-TAVR models (table 4). Figure 6. Pre-TAVR Doppler colour echo image, and pre- and post-TAVR FSI velocity streamlines showing regurgitant jet structures at peak systole. Note the different velocity scales between pre- and post-TAVR models. (Online version in colour.)
Overall, when compared to the patient-specific post-TAVR echo data, numerical results corresponded well to the echo measurements (table 4), which demonstrate the accurate predictive capabilities of our patient-specific modelling methodology, that is, the simulated TAVR procedure using the FE analysis, followed by the simulated post-TAVR LH dynamics using the FSI analysis. Nevertheless, some differences between the echo and FSI data were found, especially regarding the AV pressure gradients. These differences, which will be later discussed, are in line with the inherent uncertainties in the echo flow recordings and the image-based computer simulations. Table 5 presents the average stress values calculated for the BAV/TAV leaflets and MV leaflets during peak diastolic and systolic pressure, respectively. It was found that the average stress in the AML was at least 20% lower following TAVR, potentially due to the morphological changes caused by the posterior displacement of the anterior MA during the procedure. Due to the restricted motion of the PML observed clinically, the average stress in the PML did not show significant changes between pre- and post-TAVR states. Although the average stress in the AML was generally higher than in the PML before and after TAVR, it was found that the peak mitral leaflet stresses were actually located at the tethered PML chordae insertion regions, near the leaflet free-edge and basal chordae locations.
Since this patient had a postero-lateral regurgitant gap with PML tethering near the anterior papillary muscle (PM), the LH models experienced high PML chordae forces when compared to AML chordae tensions. There was an imbalanced force distribution between the two PM, with anterior PM force being higher than posterior PM force. Due to the anatomical changes in the MV and the mitral apparatus force redistribution following TAVR, AML strut tension increased at least 50%, while AML marginal and basal chordae tensions decreased between 17% and 36%. Similarly, PML basal tension decreased at least 15%, while PML marginal chordae showed no evident changes in their tension. It is important to note, however, that the total AML and PML chordae tensions were largely unchanged during pre- and post-TAVR states, and between the different three deployment models, which suggest that mitral leaflet tethering was not improved immediately after TAVR, but that there was a force redistribution between the different components of the mitral apparatus, in particular concerning changes in tension between marginal, basal and strut chordae. The main contribution of the present study was a comprehensive analysis of TAVR impact on aortic–mitral coupling, MR severity and LH dynamics in a rigorously developed and validated patient-specific cardiac computational model. Specifically, this work presented an engineering mechanics study that
Moreover, the new knowledge learned from a clinical perspective includes the following:
To the best of our knowledge, this is the first validated patient-specific computational model that simulated the LH dynamics before and after TAVR during the entire cardiac cycle. Comparison between simulation results and available pre- and post-procedural echo data demonstrated good quantitative agreement and predictive capabilities, with some computed variables agreeing better with the echo measurements than others. First, it is generally noted that differences in terms of the SVAV, RVMV, LVEF, E wave, A wave, E/A ratio and MR peak gradient and velocity were mostly below 10% (tables 1 and 4). It is also noted that the errors in the estimation of the AV peak and mean gradients for the pre-TAVR model, and of the AV peak velocity and EOAAV for the pre- and post-TAVR models were below 20%. As with any image-based computational model, there exist sources of geometric and numerical uncertainties. In this study, for example, the generic TAV model used does not completely match the design of the first-generation Edwards SAPIEN valve clinically implanted in the patient. Moreover, it is important to be aware of the inherent limitations and variability in the measurement of haemodynamic variables using echo [31]. Echo examination is highly dependent on the experience of the operator, the technical quality of the study and on the patient's acoustic window [32]. Patient-specific modelling of TAVR in a BAV patient under various procedural scenarios provided quantifiable information about device anchoring and the interaction between the stent, the aortic–mitral curtain and the native valve. The highest deployment model (post-TAVR 10%) led to higher peak stresses in the BAV leaflets, sinus and calcification, as well as higher contact radial force, stent recoil and average leaflet stress. These results are in agreement with an FE study by Bianchi et al. [33] that found that after stent recoil the high deployment model resulted in higher leaflet and sinus stresses, as well as higher contact forces than the midway deployment model. In our study, the peak stresses were generally found in the leaflet region in contact with the upper stent struts, in the leaflet-root attachment lines, and in the transition region between calcification and leaflet tissue. Altogether, these results could indicate a higher risk for tissue damage and potential for aortic root injury or rupture for the highest stent positioning, as compared to a slightly lower or midway implantation configuration. This is especially important in the BAV space, as rates of annular rupture have been reported in some series to be as high as 5.3% using the balloon-expandable SAPIEN XT valve [34]. Moreover, under BAV disease, the TAV is usually implanted higher (+4 mm above the annulus) and anchored at the tightest part of the BAV commissures, with an often higher final implantation depth due to the anchoring effect of the calcification [35]. Future parametric studies that investigate the effect of additional implantation strategies and parameters, as well as the influence of newer-generation TAV design in the BAV population would be useful to draw more definite conclusions. Device apposition and, therefore, a correspondent measure of stent anchoring can be approximated by measuring the contact radial force, pressure or area between the stent and its surrounding tissue [21,36]. Previous in vitro and computational studies have aimed to improve the understanding of the relationship between radial force and TAVR performance [37,38]. Up to now, however, the value of this critical force for TAVR-in-BAV was unknown. Our simulation results showed that the radial force for the three implantation models ranged between 205 and 230 N, and 98 and 108 N during fully stent expansion and after stent recoil, respectively. These forces, which increased as the implantation height increased, are markedly higher than the radial forces for a tricuspid AV reported by Egron et al. [38] and Wang et al. [21] between 100 and 150 N during fully stent expansion. This difference in contact radial force between BAV and tricuspid AV anatomies may be important clinically, since TAV oversizing is common in TAVR-in-BAV in order to prevent significant PVL, especially with the older-generation TAV models. While radial dilatation is usually desired for proper device anchoring and to avoid dislocation of the implant, an excess in radial force may also lead to impairment of the conduction system, leading to the left bundle branch or even complete atrioventricular conduction block. Indeed, PVL and high pacemaker implantation rates have been one of the main limitations of TAVR-in-BAV [39], with reported pacing rates of up to 25.5% for the balloon-expandable TAV [40]. It has been postulated that the higher incidence of cardiac conduction disorders and permanent pacemaker implantation rates in TAVR-in-BAV are related with difficulty in valve positioning and asymmetric expansion due to the irregular leaflet shape, heavy calcification and the inability to achieve a coaxial position during valve deployment [40]. For the patient-specific case studied herein, a symmetrical expansion of the TAV stent was obtained, as supported by a stent eccentricity value close to one for the three deployment models (table 2). This positive outcome can be explained by the less complex BAV type present in this patient, which was Sievers type 0 L/R without aortic root dilatation. The absence of a raphe and excessive bulky calcification, especially in the LVOT, allowed an apparently safe stent positioning with a circular uniform expansion, with no post-operative PVL found in this patient, agreeing with the clinical findings. Following TAVR, several clinical studies have found the aetiology and severity of MR to have varying effects on short- and long-term mortality [7,41,42]. Less is known, however, on the changes in MR severity post-TAVR, with single-centre and multicentre registry studies often giving conflicting results [6,7,41,43]. Given the paucity of high-quality data on this topic, the American Heart Association's guidelines and the European Society of Cardiology have opted not to make recommendations on concomitant MR treatment in patients undergoing TAVR. Nevertheless, with the previous knowledge of physiological aortic–mitral reciprocal behaviour [44], some echo studies have aimed to identify the structural changes in the aortic–mitral complex that may influence MR degree following TAVR [43,45,46]. Although these imaging studies have greatly enhanced our understating of the aortic–mitral coupling in this setting, the use of 2D/3D echo only allows a semi-quantitative analysis of the AV-MV anatomy and coupled function. Taking into account the fast-paced developments in transcatheter valve therapies, there is a need for more robust quantitative methods for objective and accurate assessment of AV-MV coupled dynamics. The patient-specific computational model developed in this study provided a quantitative engineering analysis of the relationship between MR severity and the 3D structural changes of the aortic–mitral continuity throughout the cardiac cycle following TAVR in a virtual human beating heart. In agreement with previous echo studies [45,46], our analysis showed that during the TAVR procedure the MA height decreased in the post-TAVR 50% model compared with the pre-TAVR model. Interestingly, with a higher stent implantation configuration, we found that the MA height increased instead; up to 8% for the post-TAVR 10% model. The aortic–mitral angle also decreased, with a narrower angle in the highest implantation model than in the midway implantation model. Narrower distances between the centre of the aortic and mitral annuli matched changes in the aortic–mitral angle. The reports by Shibayama et al. [43] and Vergnat et al. [46] are also consistent with the present study, showing that the MA area, AP distance and AL-PM distance did not significantly change after TAVR. A critical finding of this study was that during TAVR, the aortic annulus was displaced longitudinally approximately 3.3 mm towards the LVOT, while the anterior MA was displaced approximately 2.5 mm posteriorly, with this motion decreasing as the implantation height increased. Particularly, the posterior displacement of the anterior MA suggests an actual mechanical effect of the TAV stent on the aortic–mitral curtain. Although the acute effects of afterload reduction cannot be entirely excluded, mechanical compression is likely a contributor to the small MR reduction seen after TAVR using a balloon-expandable valve. It remains to be seen whether the use of a self-expanding stent frame, which has a different radial force and a more prominent protrusion into the LVOT results in more or less structural perturbations to the aortic–mitral continuity. This is the subject of a study we are currently undertaking. Immediate LV pressure reduction secondary to a reduction in afterload was an additional mechanism for the early improvement in MR. In this study, however, while the RVMV decreased in all post-TAVR models, no significant differences were reported, in part due to the lack of PML tethering improvement (table 5). Overall, resolution of BAV flow obstruction led to a sharp drop in the LV systolic pressure, and subsequently, a lower transmitral pressure gradient (table 4); which in turn reduced the driving force of the MR. Both the peak and mean pressure gradients across the AV were similarly reduced among all post-TAVR models, with a trend toward slightly better TAV performance haemodynamic metrics and reduction of the RVMV for the midway implantation model (post-TAVR 50%). Although not studied herein, reduction in MR severity in the late post-procedural period may be secondary to a regression of myocardial hypertrophy and positive remodelling of LV shape, especially in functional MR [47]. Several limitations should be taken into consideration when interpreting our findings. First, in this study only one patient-specific anatomy was examined, and as such the specific results regarding the TAVR biomechanics and its impact on MR severity cannot be assumed to represent the entire population. However, the validity of the results stands due to their comparative nature. In the future, we plan to extend this modelling framework to a larger cohort of patients and newer-generation commercially TAV types. Second, while, on one side, cardiac tissues were realistically described as hyperelastic and anisotropic, on the other side, patient-specific cardiac tissue material properties were not available. Thus, the mechanical response of the tissues was defined from an extensive human tissue database obtained from in-house multiprotocol biaxial and uniaxial tests. Third, since this study did not consider the zero stress–free configuration of the aortic wall, a rigid body constraint was assigned to the aortic root/ascending aorta wall elements during the pre- and post-TAVR FSI simulations; however, deformable tissue properties were attributed during the TAVR FE simulations. For the same reason, aortic wall pre-stress state was neglected during the TAVR FE analysis. This assumption, which will be tacked in a future study, could lead to uncertainties on the computed stresses as well as in the tissue compliance [48–50]. The pre-stress assumption of the native leaflets might have a small impact on the solution since the transvalvular pressure is very small during TAV deployment. Fourth, post-operative MSCT images were not available, thus it was not possible to compare the deformed TAV stent geometries from the simulations with post-TAVR clinical images. However, the current study still enabled us to compare different implantation configurations and evaluate the impact of stent depth on several clinically relevant parameters. Finally, the current SPH-FE FSI modelling framework involves a high computational cost. The FE TAVR simulations required approximately 3 days to run while the pre- and post-TAVR FSI simulations required approximately 5 days to run one cardiac cycle in an Intel Xeon E5-2670 cluster. As a result, the present modelling methodology cannot be used in a clinical setting and currently, is only suitable for a research environment. Nevertheless, among other advantages, SPH is easy to parallelize. The ability to run both FE and SPH codes on GPUs will significantly reduce the running time in the near future and avoid the need for a computer cluster [51,52]. In this work, we performed a comprehensive computational engineering analysis to investigate the impact of TAVR on MR severity, aortic–mitral coupling, TAVR-in-BAV device performance, and pre- and post-TAVR LH dynamics in a retrospective real clinical case. Our results demonstrated that the biomechanical mechanism of MR evolution after TAVR is clearly multifactorial. The structural coupling of the AV-MV is a relationship that undergoes measurable changes following TAVR as the balloon-expandable stent compresses the aortic–mitral continuity. Moreover, we showed that correcting the AS abruptly reduces LV systolic pressure, which results in a lower transmitral systolic pressure gradient, which in turn reduces the pathological retrograde flow through the MV. In conclusion, albeit a single real clinical case, this study offered a detailed engineering analysis that could shed light on the underlying mechanisms of TAVR impact on MR. Further development and validation of the computational modelling techniques in a large cohort of patients could eventually enable the use of such techniques for an individualized treatment approach and ultimately support improved clinical outcomes. All relevant data are within the paper and as electronic supplementary material. A.C. contributed to conceptualization, formal analysis, investigation, methodology, project administration, validation, visualization, writing—original draft, writing—review and editing. W.M. contributed to investigation, writing—review and editing. R.M. contributed to resources, writing—review and editing. W.S. contributed to conceptualization, funding acquisition, investigation, project administration, resources, supervision, writing—review and editing. All authors gave final approval for publication and agree to be held accountable for the work performed therein. W.S. is a co-founder and serves as the Chief Scientific Advisor of Dura Biotech. He has received compensation and owns equity in the company. The remaining authors have nothing to disclose. This work was supported in part by the NIH HL127570 grant. A.C. was, in part, supported by a Fulbright-Colciencias Fellowship. FootnotesElectronic supplementary material is available online at http://dx.doi.org/10.6084/m9.figshare.c.4591631. References
Page 6Cardiovascular diseases (CVDs) are the leading cause of mortality in the world. In Europe, 3.9 million deaths a year are attributed to CVDs, with coronary heart disease (CHD) and stroke the primary culprits [1]. CHD is characterized by the development of atherosclerotic plaque, consisting of deposits of cholesterol and other lipids, calcium and macrophages, within the arterial wall. This causes a progressive reduction in the lumen available for the blood flow (stenosis), hardening and loss of elasticity of the arterial tissue. The use of endovascular devices such as stents and balloons to treat advanced atherosclerotic lesions is now commonplace in the clinic. However, one of the major limitations of these interventions is the development of restenosis [2–7]. Restenosis is understood to comprise three main mechanisms: elastic recoil (in the short term), vessel remodelling and neointimal hyperplasia (in the longer term). The first and second mechanisms are typical in the case of balloon angioplasty, whereas stent deployment usually promotes the formation of neointimal hyperplasia, associated with smooth muscle cell (SMC) migration and proliferation and extracellular matrix (ECM) deposition [7]. Coronary revascularization now generally involves the use of a stent in more than 70% of cases, leading to reduced restenosis rates in comparison to balloon angioplasty alone. With the advent of drug-eluting stents (DES), the incidence of restenosis has been dramatically reduced to approximately less than 10–12% of all angioplasties [8]. However, DES do not completely remove this problem. Since the precise mechanisms behind restenosis after stenting, so-called in-stent restenosis (ISR), are still not fully understood, it therefore remains a significant clinical challenge to predict which patients will develop ISR. Experimental studies have established a strong correlation between the level of arterial injury caused by the device and the following neointimal thickness and lumen diameter reduction at the stented area [9,10]. Novel stent designs and stent-deployment protocols that minimize induced vascular injury are therefore needed. However, the optimum stent design and the ideal drug release strategy still remain in question despite technological advances [11]. Complementary to the wide variety of experimental studies, computational analysis has emerged as a useful method for designing new medical devices in order to minimize ISR. In the last two decades, many computational models of stent deployment have been developed to study the stress–strain level that the device induces within the arterial wall [12–15]. Several mathematical and numerical models have also been developed to try to understand drug release from stents and subsequent redistribution in the arterial tissue [16–20]. Less attention, however, has been directed to modelling the biological response to treatment. In recent years, mechanobiological models have emerged to relate mechanics to the complex biological response. These have primarily made use of discrete agent-based models (ABMs) or cellular automata (CA) methods, in some cases combining with finite-element analysis. For example, Zahedmanesh et al. developed an ABM to investigate the dynamics of SMCs in vascular tissue engineering scaffolds [21] and an ABM of ISR which allows a quantitative evaluation of the ECM turnover after stent-induced vascular injury [22]; Boyle et al. demonstrated that nonlinear growth could be simulated with a cell-centred model [23] and simulated ISR by modelling a combination of injury and inflammation with SMCs represented as discrete agents [24,25]; and Keshavarzian et al. [26] developed a mechanobiological model of arterial growth and remodelling. Evans et al. [27] highlighted the importance of using a multiscale approach to describe computationally the main physical and biological processes implicated in ISR, introducing the concept of the complex autonoma (CxA) models, based on a hierarchical aggregation of coupled CA and ABMs. Tahir et al. [28] focused on the initial phase after stenting and developed a cellular Potts model from which they hypothesized that deeper stent deployment allows easier migration of SMCs into the lumen. Most recently, Zun et al. [29] presented a three-dimensional multiscale model of ISR with blood flow simulations coupled to an agent-based SMC proliferation model and demonstrated qualitative agreement with in vivo porcine data. In the aforementioned examples, discrete models attempted to represent individual cells in the form of a lattice governed by a set of rules, in contrast to the continuum approach where the evolution of populations of cells and other species such as growth factors (GFs) are described through partial differential equations (PDEs). In terms of continuum models, Rachev et al. [30] proposed a theoretical continuum model to describe the main mechanisms of the coupled deformation and stress-induced arterial tissue thickening observed at the regions close to an implanted stent, comparing the results obtained with experimental data documented in the available literature [31]. However, this phenomenological model does not take into account, for example, the mechanisms implicated in the SMC proliferation or the ECM synthesis. Other continuum mathematical biological models typically comprise a series of coupled diffusion–reaction equations for describing the biological interaction between several species. Such models have been developed to describe phenomena such as neointimal hyperplasia formation [32], atherosclerotic plaque formation [33–36], fibrotic tissue formation surrounding medical implants [37,38] and, more recently, ISR [39] and arterial physiopathology [40]. In contrast to discrete models where cell behaviour is usually described by a set of rules, continuum models offer a mechanistic description, featuring physical parameters which may in principle be measured. In addition, continuum models often have a lower computational cost than ABMs, and naturally allow for modelling diffusion of species and coupling with the mechanical aspects of the problem. In this paper, we develop a model that allows us to simulate the restenosis process following the insertion of a stent into a coronary artery. The key novelty of our model is that it adopts a continuum approach to describe the sequence of events following damage to the arterial wall, and is therefore formulated in terms of densities/concentrations of a number of important species. This is advantageous because it allows us to assess how the evolution of the various species affects the overall healing process, and more importantly, how variations in the associated parameters and initial conditions influence the process. Diffusion–reaction equations are used for modelling the mass balance between biological species in the arterial wall. The main species considered to play a key role in the process are SMCs, endothelial cells (ECs), matrix-degrading metalloproteinases (MMPs), GFs and ECM. The parameters used in the model to define the biological interaction between the different species have been adapted from experimental data available in the literature. Unlike any of the existing continuum models, we are able to simulate the time-course response of six different biological species involved in ISR after the initial mechanical damage, while at the same time simulating tissue growth. Our primary aim is to gain insight into the physical mechanisms of tissue remodelling post-stenting. Simulating patient-specific cases is beyond the scope of this work: historically, mathematical and computational models in the stents domain have been formulated in idealized scenarios to gain insight before moving onto more realistic patient-specific geometries. With this in mind, we employ a simplified geometry. Notwithstanding, we do compare our results to clinical data and use our model to assess the impact of geometric variations on the final outcome through consideration of a series of commercial stents as well as different inter-strut distances and levels of strut embedment, thereby enabling us to relate our findings to stent design. ISR is an immeasurably complex multiscale system involving a large number of species and an intricate cascade of biological processes (figure 1). Indeed, much of the biology is still unknown. Due to its complexity, in this model we include only what we believe are the predominant species and processes. Specifically, we consider three types of cells (contractile and synthetic SMCs and ECs) and three extracellular components (GFs, MMPs and ECM). We consider the following behaviour: cell types can proliferate, migrate, differentiate and die (apoptosis) while the extracellular components can be produced or degraded. Since the processes considered take place predominantly in the intima and media layers, in the following model the arterial wall refers to these layers, with the adventitia considered as the outer boundary. Furthermore, blood flow in the lumen and plasma filtration in the tissue have not been included. We refer the reader to §5 for a full discussion of the limitations of this work. Figure 1. Schematic of ISR. In summary, prior to stent deployment, SMCs exist in a quiescent and contractile phenotype predominantly in the media layer of the tissue and ECs populate the uninjured arterial wall (a). After stent deployment (b), endothelial denudation, atherosclerotic plaque compression (often with dissection into the media and occasionally adventitia), and unphysiological stretch of the entire artery occur [7]. Platelets are then deposited at the injured area and GFs are released. Local increases in stationary mechanical strain, which occurs following stenting, lead to up-regulation of MMP by SMCs and degradation of collagen in the ECM. Endothelial damage and denudation, among multiple factors, lead to the phenotype switch of medial SMCs from a contractile to a synthetic state [3]. Synthetic SMCs proliferate in response to GFs [24] first in the media and then these begin to migrate towards the injured area (c,d). Simultaneously, ECs migrate from the lateral edge of the damaged blood vessel surface [2] (e). Moreover, synthetic SMCs secrete ECM components, such as collagen and proteoglycans, which constitute the neointimal lesion [24]. Between two and three weeks after the stenting procedure, synthetic SMCs begin to revert to the contractile phenotype [2]. Neointima typically increases up to three months after the procedure, with little change to six months and a gradual reduction between six months to three years [6] (e,f). (Online version in colour.)
An isotropic hyperelastic constitutive model based on a Yeoh strain energy function (SEF) [41] was considered to describe the stress–strain response of the arterial wall. We assume the same mechanical response for the intima and the media layer. Assuming incompressibility of the tissue, the Yeoh SEF can be written as Ψ=c10(I1−3)+c20(I1−3)2+c30(I1−3)2,2.1 where c10 = 17.01 kPa, c20 = −73.42 kPa and c30 = 414.95 kPa are the hyperelastic material constants and I1 is the first strain invariant of the Cauchy–Green deformation tensor. The coefficients of the hyperelastic model were identified from fitting the experimental results obtained from Holzapfel et al. [42] in specimens of human coronary arteries for the media layer in the circumferential direction using the software for calibration of hyperelastic material models (HyperFit, www.hyperfit.wz.cz).The stent is modelled as an elasto-plastic material with a Young’s modulus of E = 200 GPa, Poisson’s ratio of ν = 0.28 (representative of biomedical grade stainless steel alloy 316L) and plasticity described by isotropic hardening J2 flow theory with the tensile stress–strain curves taken from the literature, including a yield strength of 264 MPa and an ultimate tensile strength of 584 MPa at an engineering plastic strain of 0.247 [15]. Although not completely elucidated, the stimulus triggering the cascade of inflammatory events leading to neointimal formation appears to come from the endothelial damage caused immediately after balloon dilatation and stent placement [43]. We do not explicitly account for all of the mechanical factors that initiate the process which leads to vessel injury and endothelial dysfunction. Following Zahedmanesh et al. [22], we assign a level of injury to the arterial wall, d, in a continuous range from 0 to 1. Due to the absence of reliable mechanical data in the literature on the stress–strain levels known to cause arterial injury, the experimental data of in vitro tensile tests up to failure of 13 human left anterior descending coronary arteries [42] were used to obtain a qualitative estimation of the damage in the tissue. In particular, we define a piecewise linear function to prescribe the initial arterial damage as a function of von Mises stress, σvm, calculated by a finite-element (FE) simulation of the stent expansion: d0={0if σvm≤σvm,infσvm−σvm,infσvm,sup−σvm,infif σvm,inf<σvm<σvm,sup1if σvm≥σvm,sup,2.2 where d0 is the initial damage in the tissue due to the stent placement. This function assigns a value of zero for the damage in areas of the arterial wall in which von Mises stress is lower than σvm,inf and a value of 1 for the damage in those regions in which von Mises stress exceeds σvm,sup (figure 2).Figure 2. Initial damage as a function of von Mises stress in the arterial wall. In this work, the value selected for the inferior limit of the von Mises stress is 165 kPa, corresponding to three times the value of the physiological circumferential tension calculated by Laplace’s Law, for a given blood pressure of 100 mmHg (13.3 kPa). For the superior limit of the von Mises stress, a value of 446 kPa was chosen, based on the mean value of the ultimate tensile stresses for the media layer in the circumferential direction reported by Holzapfel et al. [42]. (Online version in colour.)
Moreover, damage, d, is assumed to decrease continuously with time at a rate directly proportional to the MMP concentration, cmmp [24]. The damage evolution equation is therefore ∂d∂t=−kdeg,d dcmmp⏟Degradation,2.3 where d is a continuous function of space and time, d(r, z, t), to describe the damage level, kdeg,d is the degradation rate of damage and cmmp(r, z, t) is the concentration of MMP. Equation (2.3) may be considered as a simple description of healing.After mechanical arterial injury, platelets, leucocytes and SMCs, among others, release several types of GFs: platelet-derived GF (PDGF), epidermal GF (EGF), insulin-like GF (IGF), transforming GF (TGF) and fibroblast GF (FGF) [2,3,5,7]. However, since their specific roles in the inflammatory phase are not completely understood [2], the term GF includes the combined action of all of them. Denoting the concentration of GFs in the arterial wall by cgf(r, z, t), and assuming an initial value of cgf,0 [44], their behaviour is modelled as follows: ∂cgf∂t+∇⋅(−Dgf∇cgf)⏟Random motion=kprod,gf d(1−cgfcgf,th)⏟Production−kdeg,gf(cgf−cgf,0)⏟Degradation.2.4 We are assuming that the GFs experience random motion (i.e. diffusion). Although the diffusion coefficient, Dgf, may in general depend on position, we take it here to be constant [32]. The right-hand side of equation (2.4) considers a dynamic balance between degradation and production. GF production is assumed to follow logistic growth: the concentration increases proportionally to the production rate, kprod,gf, and level of damage, d, until some threshold value, cgf,th, is reached. To simplify the mathematical model, we consider that vascular damage is the only trigger of GF production. Degradation, on the other hand, is assumed to occur at a constant rate, kdeg,gf, and in proportion to the ‘distance’ from the initial value. Human MMPs are a family of 26 members of zinc-dependent proteolytic enzymes which are considered to be the normal and physiologically relevant mediators of ECM degradation [45,46]. Given that MMP-2 cleaves a wider range of ECM constituents [22], the term MMP specifically refers to MMP-2 in this model. ECs, SMCs, fibroblasts and infiltrating inflammatory cells are able to produce MMPs. In this work, we assume that mechanical damage resulting from stenting upregulates MMP production only by SMCs [46], while at the same time MMP is reduced at a constant rate [24]. The MMP evolution equation is defined as follows: ∂cmmp∂t+∇⋅(−Dmmp∇cmmp)⏟Random motion=(kprod1,mmpccsmc+kprod2,mmpcssmc)d(1−cmmpcmmp,th)⏟Production−kdeg,mmp(cmmp−cmmp,0)⏟Degradation,2.5 where cmmp is the concentration of MMPs, Dmmp is the diffusion coefficient to simulate the random movement of MMP molecules, kprod1,mmp and kprod2,mmp are the production rates of MMPs by contractile and synthetic SMCs, respectively, and kdeg,mmp is the constant degradation rate of MMPs. The parameter cmmp,0 represents the initial concentration of MMP in the tissue [47], while cmmp,th is the threshold MMP concentration.The ECM is the non-cellular component of the arterial wall that provides physical scaffolding for the cellular constituents and is responsible for cell–matrix interactions [48], playing a crucial role in the development of the pathogenesis of restenosis. In particular, ECM components are involved in the regulation of SMC phenotype: degradation of ECM after stenting promotes the transition of SMCs from a quiescent/contractile to an active/synthetic phenotype, whereas its synthesis leads to the opposite [49]. In this work, the behaviour of ECM has been directly related to the behaviour of collagen which constitutes the major component of mature restenotic tissue [7]. For this reason, collagen is assumed to be the only component of the ECM in our model. Collagen is secreted by synthetic SMCs [2] and degraded at a rate proportional to the amount of MMPs [7,24], so its behaviour can be defined as follows: ∂cecm∂t=kprod,ecmcssmc(1−cecmcecm,th1)⏟Production−kdeg,ecm(cmmp−cmmp,0)(1−cecm,th2cecm)⏟Degradation,2.6 where cecm(r, z, t) is the concentration of collagen, which is produced at a rate kprod,ecm and degraded at a degradation rate kdeg,ecm. The initial concentration of collagen is cecm,0 [47]. We assume that the natural synthesis of collagen is equal to the age-related degradation rate, therefore these terms have not been taken into account in this equation. Moreover, we assume that ECM kinetics are dominated by production and degradation and that random motion of ECM is negligible.SMCs are the most prominent cell type found in intimal hyperplasia. In our model, SMCs can exist in one of two phenotypes: contractile or synthetic. Before stent implantation, SMCs are in a quiescent/contractile phenotype (do not proliferate or synthesize matrix) within the uninjured tissue. We can define the behaviour of the contractile SMCs as ∂ccsmc∂t=−kdiff,csmcccsmcKssmc,ecm⏟Differentiation from cSMC to sSMC+kdiff,ssmccssmcKcsmc,ecm⏟Differentiation from sSMC to cSMC,2.7 where ccsmc(r, z, t) is the contractile SMC concentration, kdiff,csmc is the differentiation rate from contractile SMCs to synthetic SMCs, kdiff,ssmc is the differentiation rate from synthetic SMCs to contractile SMCs and Kcsmc,ecm and Kssmc,ecm (figure 3) are functions defined to modulate the SMC differentiation as a function of the ECM concentration:Kssmc,ecm=−(e−((cecm,th/cecm)−1)−|(cecm,th/cecm)−1|−1)2.8 andKcsmc,ecm=−(e−((cecm/cecm,th)−1)−|(cecm/cecm,th)−1|−1).2.9 Figure 3. Behaviour of the switch functions defined by equations (2.8) and (2.9). (Online version in colour.)
In this model, contractile SMCs may differentiate into synthetic SMCs and vice-versa. The function Kssmc,ecm contributes to the emergence of the synthetic phenotype, so when cecm < cecm,th the function Kssmc,ecm takes positive values, SMCs are activated and switch to a synthetic phenotype starting to migrate, proliferate and synthesize ECM, while if cecm ⩾ cecm,th then Kssmc,ecm is equal to zero (no differentiation). On the other hand, Kcsmc,ecm contributes to the emergence of the contractile phenotype, so when cecm ⩽ cecm,th then Kcsmc,ecm is zero and if cecm > cecm,th the function Kcsmc,ecm takes a positive value and synthetic SMCs switch back to the contractile phenotype. At the beginning of the simulation cecm = cecm,th so Kssmc,ecm and Kcsmc,ecm are both equal to zero. We assume that contractile SMCs are initially present only in the media, and at a density of ccsmc,0 [50]. Random motion is not taken into account for this cell type since contractile SMCs are considered quiescent and do not migrate. Moreover, contractile SMCs are considered unresponsive to GFs [2]. The synthetic SMC evolution equation is as follows: ∂cssmc∂t+∇⋅(−Dssmc∇cssmc)⏟Random motion=kdiff,csmcccsmcKssmc,ecm⏟Differentiation from cSMC to sSMC−kdiff,ssmccssmcKcsmc,ecm⏟Differentiation from sSMC to cSMC+kprolif,ssmc(cgf−cgf,0)⏟Proliferation−kapop,ssmccssmc⏟Apoptosis,2.10 where cssmc(r, z, t) is the synthetic SMC concentration, Dssmc is the diffusion coefficient of synthetic SMCs in order to simulate the migration process, kprolif,ssmc is the proliferation rate of synthetic SMCs in response to GFs and kapop,ssmc is the apoptosis rate at which synthetic SMCs die. The initial density of synthetic SMCs in the arterial wall, cssmc,0, is assumed to be equal to zero [38].Along with SMCs, ECs constitute the main cell type within the vasculature. ECs perform a wide variety of significant functions, e.g. cell migration and proliferation, remodelling, apoptosis and the production of different biochemical substances [51], as well as the control of vascular function [52]. Moreover, most of the mechanical responses to flow in the arterial wall, such as shear stress and stretch, directly affect ECs indicating that these cells have specific mechanotransducers capable of transforming mechanical forces into biological responses. A thin single layer of ECs forms the endothelium, which in normal conditions, besides being a permeability barrier between the blood flow and the arterial wall, promotes vasodilatation and suppresses intimal hyperplasia by inhibiting inflammation, thrombus formation and SMC proliferation and migration [53]. However, at sites of injury caused by the stent, the endothelium is denuded [7]. This de-endothelization is considered to be one of the most important mechanisms contributing to restenosis [54]. In this work, it is assumed that the endothelium is denuded between stent struts, considering that only a small amount of cells survive in this region. The behaviour of the ECs may be modelled as ∂cec∂t+∇⋅(−Dec∇cec)⏟Random motion=kprolif,eccec(1−ceccec,0)⏟Proliferation,2.11 where cec(r, z, t) is the density of the endothelial cells, Dec is the diffusion coefficient of the ECs to simulate their migration from the lateral edge of the damage blood vessel surface [2] and kprolif,ec is the EC proliferation rate. ECs can exist only in the intima or subendothelial spaces (SES). The initial concentration of ECs is assumed to be near to zero at sites where the SES is injured and cec,0 [55] at the intact SES. In order to ensure a tractable mathematical model, the equation which governs the behaviour of the ECs is assumed to be independent of the rest of the presented coupled PDEs.We follow the continuum framework for growth of biological tissue developed by Garikipati et al. [56] to describe the tissue growth that leads to restenosis. This formulation considers mass transport and mechanics coupled due to the kinematics of volumetric growth. Accordingly, the balance of mass in the system must satisfy ∂ρoi∂t=Πi−∇⋅Mi,2.12 where the index i is used to indicate an arbitrary species; ρoi are the concentrations of the species as mass per unit volume in the reference configuration; Πi are the sources/sinks related to migration, proliferation, differentiation and apoptosis of the cells and synthesis and degradation of the substances; and Mi are the mass fluxes of the i arbitrary species. The operators ∇(∙) and ∇⋅(∙) denote the gradient operator and the divergence of a vector in the reference configuration, respectively. The total material density of the tissue (ρo) is the sum of all the individual species concentrations (ρoi), i.e. ρo=∑iρoi. These densities, ρoi, evolve if local volumetric changes take place as a result of mass transport and inter-conversion of species. That implies that the total density in the reference configuration, ρo, also changes with time, i.e. as species concentration increases, the material of a species swells, and conversely, shrinks as concentration decreases [56]. Assuming that these volumetric changes are locally isotropic, we can define the following growth deformation gradient tensor: Fgi=(ρoi)/(ρorigi)I, where ρorigi represents the original density of a species in the reference configuration and I is the isotropic tensor of second order. Taking this into consideration, under the small strain hypothesis we can write∇⋅vi=ρoiρorigi,2.13 where v is the velocity of the material points [36]. Since the primary components of restenotic tissue are ECs, SMCs and collagen (here represented by ECM), we neglect volume contributions from the other species. Therefore, the isotropic growth that leads restenosis can be finally determined as∇⋅v=∂Δcec∂tVec+∂Δcsmc∂tVsmc+∂Δcecm∂t1ρecm,2.14 where Δcec, Δcsmc and Δcecm are the variations of concentrations of ECs, both contractile and synthetic SMCs and ECM, respectively, with respect to the initial concentration of these species before the restenosis process initiation. The parameters Vec and Vsmc are the volume of an EC and an SMC, respectively, and ρecm is the collagen density (table 3). To calculate the volume of the individual cells, the shape of the ECs is assumed to be spherical [22] and the SMC shape is assumed to be ellipsoidal or spindle-shaped [57]. Therefore, the volume of each cell type may be estimated as follows:Vec=43πrec32.15 andVsmc=43πrsmc2lsmc,2.16 where rec is the typical radius of an EC, and rsmc and lsmc are the typical radius and the length of an SMC, respectively as shown in table 3. This growth process has been defined following the method described for the development of atherosclerotic plaque in [36] and the fibrosis process after the implantation of an inferior vena cava filter in [38], respectively.A two-dimensional axisymmetric geometry corresponding to an idealized representation of a straight stented coronary artery segment is considered in all simulations (figure 4a). The baseline computational geometry (table 1) is similar to that introduced by Mongrain et al. [16], also employed by Vairo et al. [18] and modified by Bozsak et al. [19] to study the stent drug release and redistribution in the arterial wall. In order to be able to relate our findings to stent design, we also assess the impact of geometric variations on the final outcome. Specifically, we consider geometrical parameters for three commercial stents, Resolute (Medtronic), Xience (Abbott Vascular) and Biomatrix (Biosensors), whose strut dimensions were obtained from Byrne et al. [59], as well as varying the numbers of struts (ns), inter-strut spacing (ISS), expansion diameter and level of strut embedment. In total, 9 different geometrical configurations are simulated, as summarized in table 2. Figure 4. (a) Baseline model geometry. The arterial wall is modelled as a multilayer structure distinguishing two different domains: intima and media layer. The adventitia is not modelled as a distinct layer but rather as a boundary condition at the outer surface of the media. The initial luminal radius, rl, and the thickness of each wall layer, δj, in the unloaded geometry of the vessel are listed in table 1, based on typical physiological values found in the literature. The stent implanted in the arterial wall is represented by 10 circular struts each of 0.125 mm radius, rs, half-embedded in the tissue and located 0.7 mm centre-to-centre distance, ws, simulating a small lesion of 7 mm. We note that the problem is actually symmetrical about the r-axis, i.e. half way between the 5th and 6th struts. (b) Details of the FE mesh of the computational model. (Online version in colour.)
The following boundary conditions were applied to the arterial wall mechanical model: (a) at the lumen–arterial wall interface, Γet,i and Γet,d, a constant pressure of 100 mmHg is used to simulate in vivo physiological conditions and (b) a displacement of 0.2 mm of the stent struts domain, Ωs, against the wall is prescribed in order to achieve an extra dilatation of the vessel that is usual in stenting techniques. For the biological species evolution, flux and concentration continuity at the SES–media interface, Γiel, is prescribed. A zero flux boundary condition, −n⋅(−D j∇c j)=0, where n is the unit normal vector to the corresponding exterior boundary and j denotes each layer of the arterial tissue, was applied at the following boundaries: lumen–wall interface, struts–wall interface, Γs, arterial wall inlet, Γj,inlet, and outlet, Γj,outlet, and outer surface of the media, Γeel, in case of GFs, MMPs, ECM and SMCs or SES–media interface in case of EC. All computational boundaries and domains are shown in figure 4 for the baseline model. The commercial software package COMSOL Multiphysics 5.3 (COMSOL AB, Burlington, MA, USA) was used to create the computational geometry and to solve numerically, by means of the finite element method (FEM), the mechanobiological model detailed in §2, which is composed of three coupled systems: (1) a steady system which simulates the mechanical expansion of the stent used for the estimation of the level of damage within the tissue, (2) a transient PDE system which simulates the temporal evolution of the biological species in the arterial wall and, finally, (3) a stationary mechanical analysis to simulate the tissue growth that leads to restenosis. The computational domains (stent struts, intima and media) were meshed using quadratic Lagrange triangular elements, resulting in an overall fine mesh with approximately 150 000 elements (figure 4b). A mesh sensitivity analysis was carried out in order to investigate the model mesh independence, testing a series of meshes with different mesh densities. Mesh independence was obtained when there was less than 2% change in the mean concentration of the biological species within the arterial wall for successive mesh refinements. The time-advancing scheme used in the transient problem was a backward differentiation formula with variable order of accuracy varying from one to five and variable time stepping. Both stationary and transient problems were solved using a direct linear solver (MUMPS) with relative and absolute tolerance assigned at 10−4 and 10−3, respectively. Reference values of the parameters included in the governing equations are summarized in table 3. They correspond to the rates of production, degradation, proliferation and differentiation, the diffusion coefficients of the biological species, the initial concentrations and the threshold values taken into account. Wherever possible, the model input parameters were derived directly from experimental data available in the literature, but in some cases estimation was necessary to ensure that the evolution of the species was broadly consistent with the time-course of restenosis described in some experimental studies [2,3,5,7].
A sensitivity analysis of the 28 parameters indicated in table 1 was performed in order to evaluate the effect of varying each input parameter involved on the evolution of the restenosis process and to test the robustness of the results of the computational model. This is of particular importance because of the absence of a complete set of experimental data and the variability seen in many of the parameters. Computations were carried out for four different values for each parameter apart from the reference value, RV, which is shown in table 3. The first two values were considered ± half the reference value of the selected parameter and the other two were given by increasing and decreasing by one order of magnitude, as can be seen in table 4.
The cascade of events occurring within the arterial wall after the stenting procedure and the consequent response of all the biological species are detailed in this section. The level of damage, local concentrations of GFs, MMPs and ECM and local densities of the SMCs and ECs have been evaluated over time at different points of the computational domain, shown in figure 5a, in order to show the restenosis evolution and the stability of the model. Points A and C are located close to a central stent strut in the media and in the denuded SES between struts, respectively. Points B and D are situated far away from the stented area in the media and SES, respectively. Moreover, in order to study the evolution of the solute dynamics in the stented domain, where the behaviour of the system will be more affected by the level of damage in the tissue, the distribution of all the species at five different times of the simulation is shown. Figure 5. (a) Deformed computational geometry and (b) von Mises stress distribution and corresponding quantification of the initial damage in the arterial wall. Points A and C are located close to a central stent strut in the media and in the denuded SES between struts, respectively. Points B and D are situated far away from the stented area in the media and SES, respectively. (Online version in colour.)
The deformed baseline computational geometry is shown in figure 5a. The final radii after deployment in the stented and unstented domain were 1.7 mm and 1.5 mm, respectively. Moreover, the total wall thickness was reduced to approximately 0.4 mm in the stented area. Figure 5b displays the von Mises stress distribution in the arterial wall. The scaling of the stress and the corresponding quantification of the initial damage has been performed is detailed in §3.2. The local evolution of damage at points A and B is shown in figure 6a. At point B, the level of damage is equal to zero at every time, demonstrating that the model does not evolve over time in regions distant from the device. For this reason, local concentrations of substances and local densities of cells at points located far away from the damage caused by the stent remain approximately constant with time. At point A, the level of damage in the tissue decays exponentially with time until the healing process is complete. Figure 6b shows the distribution of the damage in the stented domain over time. Initially, the highest level of damage is found close to the stent struts and to the regions between the struts, where the endothelium has been denuded. As time proceeds, damage decreases continuously to zero in the radial and longitudinal directions towards the adventitial boundary and the unstented domain, respectively. The literature suggests that wound healing is variable in duration. Comparing with our results it can be observed that, from day 90, damage is essentially negligible across the entire baseline computational domain, indicating that the healing process is largely complete in agreement with the classical healing response documented in Forrester et al. [2]. This healing process is mainly governed by the evolution of the concentration of MMP as observed in equation (2.3) coupled with the remaining diffusion–reaction equations. Figure 6. Local evolution of the damage over time at two different points within the media (a) and distribution of damage in the stented area of the arterial wall at five different times after stent implantation (b). Point A and point B are located close to a central stent strut and far away from the effect of the stent in the media, respectively. (Online version in colour.)
The results for the GF evolution are shown in figures 7a and 8a. At damaged vascular sites (point A), the local concentration of GFs initially increases abruptly peaking approximately between two and three weeks after stent implantation. Following this peak, the concentration decays exponentially over time returning to the physiological baseline value, contributing to the stabilization of the synthetic SMCs, in agreement with the temporal sequence of GF expression documented in Forrester et al. [2]. As the GF production depends directly on damage (equation (2.4)), at points far from the stented area (point B), the concentration of GFs does not vary over time. Figure 7. Time-varying local concentration/density profiles of (a) GFs, (b) MMPs, (c) ECM, (d) contractile SMCs, (e) synthetic SMCs and (f) ECs at two different points within the arterial wall. Points A and C are located close to a central stent strut in the media and in the denuded SES between struts, respectively. Points B and D are situated far away from the stented area in the media and SES, respectively. (Online version in colour.)
Figure 8. (a–e) Evolution of the distribution of all biological species in the media layer of the arterial wall at different times of the simulation. The substance concentration and cell density are shown in mol m−3 and cell m−3, respectively. (Online version in colour.)
Figures 7b and 8b show the evolution over time for the MMPs. At the beginning of the simulation the initial concentration of MMP is set to a homeostatic value of 3.83 × 10−7 mol m−3 [47]. The local concentration of MMP at point A starts increasing, mainly due to its production by the synthetic SMCs, until reaching its maximum value at day 9, which is consistent with the time course of expression of MMP-2 observed by Bendenck et al. [60]. As MMP synthesis depends directly on the level of damage in the tissue, which continues decreasing over time, the effect of the degradation term of equation (2.7) is greater than the effect of the production term and, therefore, the concentration of MMPs starts decreasing until reverting to its normal physiological levels [61]. The collagen variation over time is shown in figures 7c and 8c. At early times post-deployment, the local concentration of ECM at point A is degraded as a consequence of the increase of MMPs until reaching a minimum value of 4.02 mol m−3 after 2 days. From this point, its concentration starts increasing over time, mainly because of the differentiation of the contractile SMCs and the proliferation of the synthetic SMCs, producing neointimal thickening in the weeks after injury [7] until an equilibrium value of approximately 7.8 mol m−3 is reached at approximately day 50. Figures 7d, 8d and 7e, 8e illustrate the evolution of the contractile and synthetic SMCs, respectively. At the beginning of the process, the local density of the contractile SMCs at point A decreases due to the differentiation into a synthetic phenotype until reaching a minimum value of 2.85×1013 cell m−3 one to two weeks after the stenting procedure. Meanwhile, the synthetic SMCs are immediately activated and start proliferating in the media [3]. The local density of synthetic SMCs increases to a value of 3.92×1012 cell m−3 between two and three weeks after injury [3], coinciding in time with the peak in the concentration of GFs and demonstrating that locally produced GFs are a major stimulus for SMC migration and proliferation, in agreement with Forrester et al. [2]. After this time, synthetic SMCs begin to revert back to the contractile phenotype. This process continues over several months and is paralleled by a change in the ECM [2]. It can be noted that the evolution in the concentration of the ECM components, mainly collagen, plays a key role in SMC differentiation, as discussed in §2.3.3, since when the ECM concentration is sufficiently low, synthetic SMCs appear, and when it is sufficiently high, contractile SMCs appear. ECs only exist within the intima and their behaviour is governed mainly by proliferation and migration. The time-varying local density profiles of this cell type are shown in figure 7f. It can be seen that at points where the endothelium has been denuded by the stent (i.e. point C), ECs start proliferating at the beginning of the restenotic process. Between days 100 and 150, they cease proliferation, reaching, at approximately day 180, an equilibrium density of 5×1011 cell m−3 [55] along the length of the SES, corresponding to homeostatic conditions and in agreement with the temporal response described by Forrester et al. [2]. It can be considered then that the endothelium has been completely restored. However, after this time restenotic events continue since both SMC proliferation/migration and ECM deposition do not necessarily cease at this time. The results for the sensitivity analysis performed involving those parameters that have the greatest influence on the behaviour of the system are shown in figures 9 and 10. The discussion is included in the electronic supplementary material. Figure 9. Influence of varying damage degradation rate on the evolution of the damage. Refer to the electronic supplementary material for a discussion. (Online version in colour.)
Figure 10. Key results from the sensitivity analysis. The plots show the effect of varying several parameters on the concentration of GFs, MMPs and ECM as well as the density of contractile and synthetic SMCs for eight different cases. Computations were carried out for four different values for each parameter apart from the reference value, RV, which is shown in table 3. The values of 2RV and RV/2 were considered ± half the RV of the selected parameter; 10RV and RV/10 were given by increasing and decreasing by one order of magnitude the RV (table 4). The results shown for point B correspond exclusively to the RV (baseline model). Refer to the electronic supplementary material for a discussion. (Online version in colour.)
The volumetric growth of new tissue into the lumen of the vessel in response to the mechanical injury caused by the medical device after 300 days is shown in figure 11, for the baseline model. It should be noticed that the restenotic tissue grows considerably between struts. The degree of occlusion can be measured in terms of the diameter or area of restenosis [62]. In this work, the change in the cross-sectional area of the model is used as a measure of measuring arterial restenosis. The percentage of stenosis is calculated as (1 − Ar/Aref) · 100, where Ar and Aref are the area of the lesion and of the reference site, respectively. Therefore, the resulting degree of restenosis after 300 days is approximately 25% for the baseline set of parameters simulated in broad agreement with the clinical data presented in Nobuyoshi et al. [63]. Figure 11. Evolution of the volumetric growth at different times of the simulation for the baseline model. (Online version in colour.)
In figure 12, we consider the impact of different geometric configurations on the % restenosis for the various cases considered in table 2. The model produces different levels of restenosis depending on the geometric parameters, and our results are in qualitative agreement with clinical observations. Specifically, the results of our simulations show that % restenosis:
Figure 12. The impact of different geometric configurations (table 2) on von Mises stress distribution and volumetric growth after 300 days. (Online version in colour.)
We would like to emphasize that there are a number of limitations in this work, as we now discuss. Whilst the continuum approach that we have adopted is advantageous in that it allows us to assess how the evolution of the various species affects the overall healing process, the disadvantage is that it is extremely difficult to measure spatiotemporal cellular densities and GF concentrations experimentally. Certainly, we are not aware of how such validation may be obtained in vivo. In short, we do not believe that the data currently exist to validate the predictions of the model at the level of individual concentrations/densities in space and in time. Therein lies the major advantage of the continuum model proposed here, i.e. the ability to simulate the evolution of species which we cannot get insight into from in vivo experiments. Notwithstanding, we have compared our results to restenosis data (at the tissue level) and have shown that our model is in broad agreement with what is seen in the clinic. We acknowledge that there is a known link between the level of shear stress and restenosis. Indeed, most haemodynamics models that assess different stent designs and patient-specific geometries set out to predict quantities such as wall shear stress (WSS), oscillatory shear index and time-averaged WSS, because these can be related to clinical outcome. Keller et al. [64], for example, investigated the correlation between mechanical and fluid stresses and the magnitude of restenosis, and found that while a linear correlation is not obtained when these stimuli are considered separately, there is a closer correlation when the combined action of these stimuli is considered. While such correlations are useful for providing insight into the influence of force on restenosis, what is less clear is precisely how the transmission of fluid forces affects the behaviour of the cells and other species involved in the healing response. The most sophisticated model of restenosis to date [29] does include the effects of flow, but in in an indirect way. Specifically, Zun et al. assume that neointimal growth is dictated by nitric oxide (NO) production, which is governed by average WSS. Therefore, Zun et al. calculate WSS from a steady laminar flow simulation and pass this to their biological solver, where the level of NO dictates the ensuing biological response through a set of rules. In the absence of data resolving the underlying physics of how the transmission of force generated by flow affects the healing process in vivo, the influence of flow has been neglected in the present analysis. Certainly, the force generated by flow would influence the parameters of our model: in this sense, the sensitivity analysis could be seen to indirectly incorporate the effects of varying flow. Concerning the geometry of the FE model, a two-dimensional axisymmetric geometry corresponding to an idealized representation of a straight segment of a healthy coronary artery has been considered in this study. This could be improved using more realistic geometries of the artery, such us coronary arteries with curvature, bifurcations, the presence of atherosclerotic plaque or derived directly from three-dimensional patient-specific geometries. However, such models would considerably increase the cost of the numerical simulations and are left for future work. In this work, following the approach presented by Zahedmanesh et al. [22], the modelling of the initial damage in the tissue after stenting is considered in a very simple way, which is based on the ultimate tensile stress–stretch response for each layer of human coronary arteries [42]. However, to our knowledge, there is no experimental data published in the literature which directly relate the levels of stress–strain due to stent deployment to the arterial wall injury and the sequence of events associated such as GF and MMP production, synthesis, number of proliferating SMCs, etc. The biological parameters involved in this model were obtained from a wide range of in vivo and in vitro experiments from the published literature on different blood vessels from human and animal models. However, due to the important structural and functional differences between arteries [48], these parameters could vary from artery-to-artery, species-to-species and patient-to-patient, or even from one lesion to another in the same patient. Additionally, some of these parameters have had to be estimated to be consistent with the time course of the restenosis process found in different studies [2–5,7]. Moreover, all the parameters have been taken as constant and uniform through the whole artery, but in fact they are very likely to change during the process. Finally, only the most important biological species have been taken into account. Other species involved in the coronary restenosis process such as platelets, monocytes, different classes of GFs (PDGF, EGF, IGF, TGF, FGF), mesenchymal cells, fibroblasts, collagen subtypes, proteoglycans, fibronectin, etc., have been grouped or omitted. Moreover, only the main biological processes have been considered in this model. Other processes such as activation and migration of the mesenchymal stem cells, release of GFs by fibroblasts or ECM synthesis by ECs were not included. A mathematical and computational model which successfully captures the main characteristics of the restenosis process after stent implantation in a healthy coronary artery has been presented in this work. A continuum approach has been taken into account for modelling the behaviour of the different biological species involved in ISR, resulting in a PDE system of several coupled diffusion–reaction equations, solved numerically by means of the FEM. Mechanical damage, which is quantitatively estimated as a function of the von Mises stress levels obtained in the arterial wall after a FE simulation of the stent expansion, is considered as the stimulus needed to start the process. Our results confirm that ISR depends on multiple factors, with the ECM dynamics and SMC proliferation the primary contributors to its pathogenesis. In addition to this, the sensitivity analysis carried out for the different model parameters, as well as providing information on the stability of the model, provides us with an understanding of how changes in one parameter will affect the behaviour of the whole system. The value of kdeg,d has a significant impact on the healing rate. Moreover, it was shown that the rates of production, degradation, differentiation and proliferation taken into account highly affect the local levels of concentration/density of the species involved in the process and the temporal response of the system. The apoptosis rate of synthetic SMCs, diffusion coefficients and initial conditions also influence the evolution of the model, although to a lesser extent. In conclusion, in spite of the simplifications and limitations we have discussed, the model developed is able to capture some of the underlying mechanisms and patterns of ISR. Moreover, the results obtained are in good agreement with clinical hypotheses relating to ISR occurrence. Therefore, this model can be considered as a step forward to a better understanding of this phenomenon. The data associated with this paper consist of the mathematical models and their numerical solution as detailed in the text. Conceived and designed the study: M.A.M., S.M. and E.P. Development of mathematical model: J.E., M.A.M., S.M. and E.P. Computational implementation of model: J.E. Writing, review and editing: J.E., M.A.M., S.M. and E.P. We declare we have no competing interests. The authors gratefully acknowledge funding provided by the Spanish Ministry of Economy, Industry and Competitiveness under research project number DPI2016-76630-C2-1-R and grant no. BES-2014-069737. FootnotesElectronic supplementary material is available online at http://dx.doi.org/10.6084/m9.figshare.c.4586102. References
Page 7Infrastructure failures are often inevitable following either natural or man-made disasters including hurricanes [1], earthquakes [2] and ensuing tsunamis [3], ice storms [4], and terrorism incidents [5]. The economic prosperity, security and public health of our society are extremely vulnerable to these accidental, weather-related and human-instigated events [6,7]. Events such as 11 September 2001, hurricane Katrina in 2005, the Haiti earthquake in 2010, etc., showed the cataclysmic aftermath of such hazards. Notably, US Pacific Northwest is highly prone to an M9.0 Cascadia subduction zone earthquake [8,9]. The last mega-earthquake occurred in 1700, and there is a 40% chance of recurrence within the next 50 years [10]. This fact calls for our understanding of network robustness behaviour and action on critical infrastructure protection. Roadway networks play important roles in transporting people and goods efficiently and safely, evacuating people from the site of a disaster and importing critical resources to affected sites. Patuelli et al. [11] suggested that physical constraints are likely to restrict the topology of road networks and make them nearly planar, which also makes them extremely sensitive to failures. Therefore, systematic understanding and accurately measuring network robustness under disruptions is of great significance in achieving a resilient critical infrastructure system. Various studies [12–16] defined their own metric in analysing network robustness. From a network science perspective, the robustness of a network is often characterized by the value of the critical threshold analysed using percolation theory and defined as the largest connected cluster size during the entire attack process [17]. Essentially, network robustness describes the ability of a system to maintain its performance after a disruption, and accessibility to the critical facilities (i.e. shelters, hospitals and police/fire stations), all of which are an essential part of post-disaster roadway performance. Therefore, in this paper, transportation network robustness is measured by the integrated size of the clusters that are connected to critical facilities, which is formulated as a robust component. With this definition, we assume that if a node is contained in a robust component, the cars can reach the critical facilities. Empirically, it represents the ability of the transportation system to withstand hazard-induced infrastructure failure without reducing access to critical facilities. Assessing robustness generally consists of determining the system behaviour that results from each possible network disruption state. Engineering models use reliability [18–20] or travel cost [21,22] to measure network robustness under disturbance. However, these models use daily operational travel demand in the analysis, which provides limited insights on post-disaster transportation network robustness assessment. Besides, this paper focuses more on the existence of post-disaster network access to critical facilities rather than normal travel performance. In addition, link disruption in the model is arbitrary, which is far from the reality of infrastructures exposed to different levels of hazard risk. Hazard vulnerability should be included in the model in order to provide an accurate measurement of post-disaster network robustness. Furthermore, access to critical facilities is essential in a network’s functionality and therefore, should be included in the network robustness assessment framework. Existing research [23] uses centrality measures to quantify link importance in accessing specified emergency services. However, such a model omits the impact of collective network disruption on access to critical facilities. Also, the centrality measure focuses on the shortest path to the destination, while in reality, all paths should be considered in evaluating network access to critical facilities. In post-disaster network robustness assessment, disruptive events are described by removing one or more network components from the system, which can be modelled with a percolation approach [24,25]. Extensive research has been primarily focusing on the percolation modelling of network robustness [17,26–31]. However, there are several limitations in conventional percolation modelling of infrastructure network robustness. First, existing research mainly uses generating function methods to derive network robustness. However, due to the degree correlation in spatial networks, generating function methods fail to measure the robustness of the infrastructure network accurately Dong et al. [32]. Second, network disruption is commonly described by random failures, localized failure or targeted attack. However, the infrastructures’ vulnerability to natural hazards varies across the network. For example, low-lying roads in a valley and roads on steep hills are more vulnerable to landslides than roads on flat ground. Thus, a probabilistic link-removal scheme should be considered to provide a realistic post-disaster road network robustness assessment. Third, the traditional giant component assumes the largest connected component of the network functions after the disruption. However, nodes in infrastructure networks such as water, gas and electricity, are not autonomous but rather rely on resources feeding them, without which network performance will deteriorate. In a disastrous event such as an earthquake or a deliberate attack, the network may be broken into different components and having access to critical facilities in their local neighbourhood is essential for individuals in each component [20]. For example, if a community’s access to a hospital is cut off by an earthquake, injured people cannot receive timely medical treatment and their health and safety will be at risk. Therefore, investigating network robustness and considering the network’s access to critical facilities is essential to avoid functional isolation in disasters, and accessibility to critical facilities needs to be reflected in the robustness measurement. Figure 1a shows that when a network is fully connected, the giant component has access to critical facilities. When the network is disrupted, although the left giant component is more connected and contains the most nodes (as in figure 1b), it does not connect to the critical facilities, and therefore may likely fail without access to critical resources. When critical facilities are entirely isolated from all other components (as in figure 1c), the network will paralyse due to lack of resources regardless of the size of the cluster. Figure 1. Network access to critical facilities. Blue nodes represent the ordinary nodes in the network, while the red node represents the critical facility. Critical facilities can be defined as the facilities that contain essential resources which support the normal functionality or restoration of the system, e.g. a lifeline warehouse, hospital, fire station and restoration centre. In (a), the network is fully connected to the critical facility. In this case, the largest connected component (or giant component) contains the critical facility. In (b), the network is broken into different clusters. The largest connected component contains most of the nodes; however, it does not include the critical facility, only the second largest component connects to the critical facility. In (c) the critical facility is isolated; without access to critical resources, the network would fail. (Online version in colour.)
This paper is largely motivated by the fact that network robustness assessment based on the conventional percolation approach alone is not sufficient to represent the performance of transportation networks in the post-disaster setting and that hazard-induced link failure probability is lacking in the engineering studies. To bridge the gap between engineering simulation methods and theoretical analysis, this paper contributes to the state-of-art design by proposing a new robustness metric: robust component—the component that has access to critical facilities—in incorporating post-disaster access to critical facilities in network robustness measurement. Further, in order to overcome the limitation of applying generating function in infrastructure network robustness characterization due to degree correlation, and also integrate hazard vulnerability into the post-disaster transportation network robustness assessment, we proposed a probabilistic network disruption percolation modelling and simulation framework to accurately measure network-level robustness behaviour in the face of disaster-induced massive disruption. This paper enables a comprehensive assessment of network robustness, one that considers network access to critical facilities in the face of disastrous events. The result allows decision-makers to devise mitigation strategies to different hazards by focusing on the critical area to counter the disastrous effects, especially in areas such as emergency response planning, evaluation of the location of additional critical facilities and infrastructure prioritization. The remainder of the paper is organized as follows: §2 presents a literature review of the related research. In §3, the methods adopted in this paper are discussed in detail, and a comparison study is conducted on a simulated ER network and road network. Section 4 presents network robustness in considering access to hospitals by using earthquake-induced probabilistic failure scenarios. Following that, §5 provides the major findings of this paper. Finally, the paper concludes with a discussion in §6. Investigating the robustness of disrupted networks is often associated with the measurement of vulnerability, reliability and accessibility [33]. Vulnerability refers to the degree of inability of a system to function due to disruption [33]. Essentially, the concept can be considered as the reciprocal of robustness [34]. Poorzahedy & Bushehri [19] define the network reliability for the (k, s) origin–destination (OD) pair user group as the probability that the network may hold a suitable condition for those users after an incident. Murray & Grubesic [35] define reliability as the probability that a given element in a critical infrastructure system is functional at any given time. Mattsson & Jenelius [36] refer to reliability as the probability that a system can maintain its satisfactory operation over the long run. Murray and Grubesic [35] suggest that reliability analysis mainly focuses on the possibility of maintaining the performance of critical infrastructure elements. Reliability analysis is often classified into three categories: connectivity reliability (the probability that a node remains connected), travel time reliability (the probability that a trip between nodes is made within a specified time interval) and capacity reliability (the probability that a network can successfully accommodate a given level of travel demand) [20,35]. Poorzahedy and Bushehri [19] proposed a measure of link importance based on consumer surplus for solving the problem of network performance in case of incidents. This study incorporated the link survival probability after catastrophic events and provided a heuristic solution to solve large-scale network problems. Chen et al. [18] introduced capacity reliability to measure the performance of a transportation network under traffic disturbance. The proposed performance measure took network capacity and traffic demand into consideration. Chen et al. [21] further extended the research by providing a network accessibility measure that considered the consequences of link failure expressed in terms of travel time and calculated them based on the travel cost increase. However, the methodology was tested on a rather small hypothetical network (five nodes). To assess a transportation network of tens of thousands of nodes, the performance of the method is not guaranteed. Overall, these reliability analyses require the travel demand from the OD pair. However, such real post-disaster demand data are limited and therefore are impractical to implement in a real-life study. Network robustness measures a system’s capability to withstand an unexpected internal or external event or change without degradation in performance [37,38]. Chopra et al. [39] presented a resilience analysis on the London metro system that considered network topology, spatial organization and passenger flow. The results identified the particular sources of structural and functional vulnerabilities that needed to be mitigated for improving the resilience of the London metro network. Nagurney & Qiang [22] proposed two relative total cost indices to assess road network robustness when the links are disrupted or travel behaviour is altered. Measures such as link travel cost [22], traffic delay [40] and the ratio of pre- and post-disaster condition [41] are also commonly used to investigate the transportation network robustness behaviour. It is worth noting that, depending on the performance that we focus on, e.g, network connectivity, efficiency, travel time, etc., the metric that is used to characterize the network robustness would change. For example, Albert et al. [42] introduced connectivity loss in a study of the structural vulnerability of the North American power grid. Similarly, Dueñas-Osorio & Vemuru [43] adopted connectivity loss in investigating cascading failure in power systems. Hines et al. [44] used connectivity loss to measure vulnerability in an electric power blackout risk analysis. In addition, Crucitti et al. [13] used the network efficiency measure to analyse the structural vulnerability of the Italian GRTN power grid. Kinney et al. [14] also adopted the efficiency concept to measure the impact of cascading failure on the North American power grid. Although network connectivity and people’s commute behaviour are studied extensively [45–48], disrupted network connectivity, especially network access to critical facilities, is rarely studied. However, post-disaster access to critical facilities including hospitals, police stations, fire stations and rescue is essential to the health, safety, post-disaster response and recovery of our society. As the roadway infrastructure provides essential access to these facilities, it should be included in the measurement of network performance. Investigating access to critical facilities in a disrupted network is often associated with accessibility analysis. Accessibility is generally defined as ‘the relative ease of reaching various services, destinations and/or activities from a particular origin [49,50]. Redondi et al. [51] studied the accessibility of the airport by using the shortest path length, the minimum number of non-stop flights in their case, to measure airport network connectivity. Bigotte et al. [52] formulated a mixed-integer model for integrated urban hierarchy and transportation network planning. The model enables identification of the links that should be improved or new links that should be built to increase network accessibility. Grubesic & Murray [53] explored network interconnection by removing the vital nodes, and Church & Scaparra [54] studied operation efficiency loss as the identified facilities are destroyed, which are all similar to a targeted attack scenario. Network disruption in natural hazards, however, is determined by the network’s geographical exposure instead of its geographical or functional importance. Novak & Sullivan [23] introduced the critical closeness accessibility measure for evaluating the accessibility of emergency services on a road network. It can also identify the critical links that are important in terms of facilitating system-wide access to emergency services. It measures accessibility on a link-by-link basis, which essentially assigns an accessibility value to the individual links in the network that accounts for the empirical information such as the spatial distribution of critical link/node, road network topology and road characteristics such as road type, capacity, volume and travel speed. Nevertheless, it fails to capture network-level accessibility behaviour, which is the key factor of state, regional and municipal transportation agencies’ funding and policy decision-making. Dong [55] used a total accessibility matrix to measure the topological structural robustness of the supply chain network. However, the calculation of the accessibility matrix does not include the impact of the collective network disruptions such as earthquake-induced failure. Although it provides an index for each node, it does not show where this node can access, which is of great importance in emergency rescue. Therefore, these methods provide limited insight into the post-disaster network access to critical facilities. On the other hand, there is rich literature on transportation network robustness assessment. Sullivan et al. [56] introduced a scalable system-wide performance measure called network trip robustness to compare networks of different sizes, topology and connectivity levels. Erath et al. [57] presented a framework that investigates the robustness of the transportation system to natural hazards. This approach can accommodate transportation-related failure consequences, including congestion effects. In addition, Chen et al. [21] used network-based measures to assess disrupted transportation networks, which consider the consequences of one or more link failures in terms of network travel time or generalized travel cost increases as well as the behavioural responses of users due to the failure in the network. However, existing research on transportation network robustness analysis faces several challenges. First, the travel demand under normal operation is used in the model to characterize the degraded network performance. As the travel behaviour in a post-disaster setting is expected, the existing analysis offers limited insights into post-disaster network accessibility to critical facilities. Second, the road disruptions considered in these studies are selected arbitrarily. However, in real disaster scenarios, the geographical exposure of infrastructures to hazards are different and the condition of infrastructures due to ageing varies across the network. In a post-disaster network access analysis, these ought to be incorporated into the analysis. Third, the centrality measures in nature, such as closeness and betweenness, measures the shortest distance between an OD pair [58]. However, in a post-disaster scenario, we focus on investigating whether or not an individual site has access to any of the critical facilities. In order to accurately measure the performance of a transportation network post-earthquake considering the network’s access to critical facilities, we propose to use a network topology-based metric robustness component to assess post-disaster network robustness. We define network robustness in this paper as a transportation system’s capability to withstand hazard-induced infrastructure failure without degradation in providing access to critical facilities. The robustness investigated in this paper attempts to tackle the question of how earthquake-induced failure will impact a network’s access to critical facilities such as hospitals. Despite missing traffic information, the comprehensive topological and geometrical data can provide engineers with a comprehensive first-step assessment of network robustness in the face of a catastrophic event such as an earthquake, hurricane, ice storm, flooding or other natural disasters. In addition, we integrated infrastructure probabilistic geographical exposure to network failure in order to generate a realistic post-disaster network robustness assessment. More importantly, instead of only considering a predefined OD pair, we examine the individual site’s access to all of the designated critical facilities on the network. This enables a comprehensive evaluation of network-level robustness behaviour. From a network science perspective, we are investigating whether there is a path between a site to any selected critical nodes. Intuitively speaking, the robust component can be interpreted as ‘given that ϕ proportion of the network is disrupted, the RCS part of the network still maintains its performance in terms of having access to critical facilities’. Existing research in assessing the robustness of real-world networks shares some common features [59]: (1) simulating or obtaining empirical data for a network (e.g., generating a network from random graph, mapping the real network to obtain the data); (2) measuring the investigated network’s structural features; (3) conducting random failure or a targeted attack on the network and (4) assessing the aftermath performance (static, dynamic) of the network. In particular, road network robustness analysis normally models road infrastructure as a network with links (roads) and nodes (intersections) in order to investigate the network disruption and its impacts on society [60]. A wide variety of network robustness measures have emerged from recent research. Figure 2 presented the major types. Figure 2. Selected performance measure types for modelling network robustness [61–69]. (Online version in colour.)
The largest connected component (i.e. the giant component) in percolation theory is commonly used in physics and computer science [70]. Cohen et al. [71] studied the robustness of scale-free networks with power-law degree distribution, i.e. the Internet, by measuring the giant component (spanning cluster) after network breakdown. Solé et al. [16] assessed network robustness by measuring the giant component on the European power grid under targeted attacks and determined the transition threshold. Motter & Lai [15] used the largest connected component method to measure network robustness after cascading failure occurs. Li et al. [72] measured the largest connected component in spatially constrained Erdös–Rényi networks to determine the impact of spatial constraints on network robustness. The above studies assume the largest component will be functioning after the disruption. However, we argue that accessibility to critical facilities is essential for post-disaster survival. For example, a giant component without access to medical services cannot be considered as functioning from an emergency response perspective. The proposed robust component measure overcomes this challenge by including important nodes into the critical component measuring. Inspired by the similarities between human and insect infrastructures, Middleton & Latty [73] reviewed the literature on resilience in three key social insect infrastructure systems: transportation networks, supply chains and communication networks, and then described how systems invest in three pathways to resilience: resistance, redirection or reconstruction. This finding demonstrates that we can learn from social insect research and then develop analytical and simulation tools to study human infrastructure resilience based on their findings. Percolation theory is a powerful tool that allows the analysis of network robustness. There is a rich body of literature focusing on percolation modelling of network robustness through generating function methods. Most of the studies focus on theoretical networks such as the ER network and scale-free network [29,74–77]. However, due to the degree correlation in the infrastructure network, the generating function method is incapable of capturing network robustness behaviour [32]. Since this degree correlation is inevitable, because our infrastructure network is spatially embedded, a simulation-based method is more desirable in characterizing the percolation process in infrastructure network robustness analysis. Furthermore, most theoretical methods assume an infinite size of the investigated network, which is not the case in infrastructure network robustness research [74]. Although Radicchi [29] studied percolation in the real interdependent network, his model still focuses on the theoretical networks. This paper proposes a percolation modelling and simulation framework that captures the spatial complexity of the road network in order to generate an accurate assessment of network robustness behaviour. Regarding another modelling spectrum, extensive research has focussed on the random failures [32,78], localized attacks [79–81] and targeted attacks [16,82,83]. However, in reality, the probability of multiple road failures is largely dependent on the built environment it is exposed to. For example, landslides are widespread in regions that have steep slopes, weak soil and significant precipitation or storm events. Probabilistic failure based on the link’s exposure to hazards is desired for a case study on the real-life disaster. Therefore, this paper integrates infrastructure failure probability into the percolation modelling framework and presents a study on network robustness under the influence of earthquake-induced probabilistic failure. The percolation process is parametrized by the probability, p, that a node or an edge is present or functioning in the network. The functional nodes/vertices are considered occupied and p is called the occupation probability. When p is large, the network tends to be more connected. As p decreases, there comes a point where the giant component breaks apart. This point is called the percolation threshold. The formation or dissolution of a giant component is called a percolation transition [28]. Here, we use the notation ϕ = 1 − p to represent the proportion of links that are removed from the network. Giant component size (GCS) is commonly used as a measure of network robustness because percolation assumes that the largest connected cluster will maintain its functionality after network disruption [28]. However, this is not a valid assumption since the largest connected component is not guaranteed to be functioning if there is no access to the necessary resources. For example, if an injured person has no access to a hospital, a cluster containing said hospital cannot be considered functional from a healthcare point of view, even if it maintains the largest possible size. In terms of hazards, robustness represents the degree to which a system is able to withstand an unexpected internal or external event or change without degradation in performance [37,38]. Therefore, we propose a new network robustness measurement: robust component. Robust Component. In a graph G, two vertices u and v are considered connected if there is a path from u to v, which is denoted as ρ(u, v) = 1. Given a network of size N, containing K critical facilities, the connected component of k can be represented as Ck={vi∣ρ(k,vi)=1, ∀i=0,1,…,N}. Following this, the robust component of a network with occupation probability ϕ can be defined as ℜϕ=⋃k=0,1,…,KCk3.1 Intuitively, the robust component is the union of the nodes that connect to at least one critical facility. In other words, since the original network is fully connected, every node has connections to all of the critical facilities. After the imposed link failure on the network, a node is considered failed when it loses connection to all of the critical facilities. A robust component will significantly help to maintain network performance in an unexpected disruptive event. A percolation illustration of the robust component is presented in figure 3. Algorithm 1 shows the designed algorithm for calculating the robust component size (RCS). Figure 3. Percolation simulation of the robust component under varying attack sizes. Green dots represent the selected critical facilities (randomly generated for demonstration), and red links represent the links that are connected to the critical facilities. In (a), as 10% of the edges are destroyed (ϕ = 0.1), 85% of the network still has access to the critical facilities. In (b), an increase of 5% more edge failure (ϕ = 0.15) would lead to 80% of the network being within reach of critical facilities. In (c,d), as nodes continue to be removed from the network, the destruction effect escalates. As 40% of the edges are removed, the majority of the network loses access to the critical facilities. (a) ϕ = 0.1, (b) ϕ = 0.15, (c) ϕ = 0.25, (d) ϕ = 0.4. (Online version in colour.)
To demonstrate the performance of the robust component in measuring the network robustness considering the network access to critical facilities, two types of network are used to experiment: The ER network and the Portland road network as shown in figure 4. The ER network is generated through Networkx module, and the Portland road network is provided by Portland Metro, the road network GIS shapefile can also be obtained from Metro [84]. Two networks are of the same size (5147 nodes) and same mean degree (2.97). There are two ways to present a graph: the prime approach (vertices are intersection/joints, edges are links/interactions) and the dual approach (vertices represent links/interactions, edges represents intersections/joints). Both approaches are investigated through the ER and Portland road networks, and their degree distributions are presented in figure 4. As we can observe, the road network shows a strong spatially embedded feature. A peak shows at the degree of 3 and 4. The road network here only contains major arterial roads and minor roads are not included in the Portland Metro’s traffic analysis, which results in some four-way intersections turning into a T-intersection. Also, the massive amount of ramps that connect arterial roads to highways contribute to the high frequency of degree-3 intersections. On the other hand, the random network shows a smooth degree distribution which fits into a Poisson distribution with a mean degree of 2.97. Figure 4. Network configuration and degree distribution of random network and spatially embedded road network. (a) ER network, (b) Portland Metro road network. (Online version in colour.)
According to the two aforementioned methods of constructing the network, the percolation process can be classified into two categories: node percolation, which removes the node, and edge percolation, which conducts edge removal. In a road network failure percolation case, roads are the objects that tend to be destroyed. Therefore, in this paper, network percolation is conducted through derived link removal based on the failure strategies. In particular, a node is considered failed only when it loses all alternative connections to all the critical facilities, in other words, all the possible routes from the node to critical facilities are impassible. Since the size of the robust component is dependent on the network’s connection to designated critical facilities, the number of critical facilities will impact the robust component’s performance during link percolation. To investigate network robustness, we increased the number of critical facilities (k) and recorded the size of the robust component in each case. At each k, 100 simulations are conducted and each simulation represents a random failure scenario. The average RCS is calculated and presented in figure 5. To compare with the conventional robustness measures, the giant component, the size of the largest connected cluster, is recorded as well. Figure 5. Degree distribution of the (a) ER network and (b) Portland Metro road network in prime and dual approach. (Online version in colour.)
Comparing network robustness under both GCS and RCS in figure 5, we can see that in the case when access to critical facilities is of significance to disaster recovery, GCS gives an incorrect assessment of network robustness, which can lead to further false disaster mitigation strategy planning. The value of the critical percolation transition threshold, where the RCS diminishes to zero, is normally used to define the robustness of the network [24,72,77,85]. As shown in figure 5, the critical threshold ϕc varies between the ER network and the Portland Metro road network. Low ϕc means that very few link removals are required to destroy a network’s functionality. In other words, the network is more vulnerable to failure. Looking at k = 1, we can conclude that under the random removal scenario, the Portland road network is more vulnerable to the loss of critical facilities than the ER network. As k increases, the value of ϕc becomes nearly identical for the ER and Portland road networks. However, there still exists a difference in the percolation process. Purely using the critical percolation transition threshold ϕc neglects the case when the network is severely damaged but not completely destroyed [77]. To complement this, Schneider et al. [70] proposed a systematic measure, called the robustness measure R, to estimate the robustness of the network. R=1N∑Q=1Ns(Q),3.2 where N is the total number of the nodes in the network, and s(Q) is the fraction of nodes in the largest connected component after removing Q = Nq nodes. The 1/N normalizes the result so that the robustness of networks of different sizes can be compared [24,77]. From a geometry perspective, R describes the area under the percolation curve.Figure 6 shows the comparison between the ER network and the Portland road network. The simulation results show that when the number of critical facilities is very low, the ER network exhibits more robust behaviour than Portland road network, despite the fact that they have the same mean node degree. The difference in robustness behaviour is largely due to the spatially embedded feature of the road network as the random network possesses a better-balanced degree node/link distribution and the nodes have more redundancy in coping with the link removal. The comparison shows that at k = 1000, the robustness behaviour of ER network and Portland Metro road network are very similar. This comparison shows the extent to which the infrastructure network is different from the theoretical network and suggests that when the network evolves to a certain level, the spatial network can also obtain similar properties to the theoretical networks. Figure 6. Robustness comparison between the ER network and Portland Metro road network with varying numbers of critical facilities. (Online version in colour.)
A Cascadia subduction zone earthquake posts a great threat to the Pacific Northwest region as an estimated M9.0 earthquake will severely damage the infrastructure and affect community’ access to the necessary resources for post-disaster recovery. In this paper, we use Portland, OR, as the study site to investigate the impact of earthquake-induced infrastructure failure on the network’s access to hospitals. It is worth mentioning that the proposed methodology can be applied to road networks in other cities in different disaster scenarios. We have previously investigated the network robustness in a random critical facility and random failure scenario. However, in real life, the location of critical facilities is normally decided based on the geographical features or the needs of the surrounding areas. Emergency medical services (EMS) personnel and hospitals are the community-based resources that are responsible for injuries during the initial disaster response. Robustness towards disasters varies from community to community and is dependent on the availability of EMS and hospital resources. Therefore, we evaluate network robustness considering the access of communities to hospitals post-disaster. Figure 7 shows the Portland Metro network and area with hospitals. There are 5147 nodes and 7646 links included in the network. Initially, there were 22 hospitals in our scope of study [84]. To reduce overlapping, 20 hospitals were used for the simulation. It is worth mentioning that in reality, hospitals are not built at the intersection of the roads, but close to a major intersection. Therefore, the location of the closest intersection to the hospital is used to represent the hospital’s location. Figure 7. Portland Metro area transportation network. (Online version in colour.)
Because road networks are spatially embedded, their vulnerability relies heavily on their structural nature and surrounding geographical features. For example, bridges are more vulnerable during an earthquake and roads on top of faults are prone to landslides. Figure 8 shows the geographical exposure of the Portland Metro network to natural hazards, i.e. earthquake-induced landslides and liquefaction. This information can be obtained from O-HELP [86]. Since the probability that the links that are most prone to failure varies across the network, uniform link removal in a traditional percolation process is inadequate in characterizing the stochastic nature of network disruption. In this paper, we propose a probabilistic approach in removing links in order to approximate the destructive effect of natural hazards on the network. Figure 8. Geographic exposure of Portland Metro transportation routes to natural hazards. (a) Landslide probability. (b) Liquefaction probability. (c) Bridge location. (d) Link disruption probability integration illustration. (Online version in colour.)
Link failure probability (Pf) in this model is comprised of three elements: probability of failure by landslide Pl (figure 8a), probability of failure by liquefaction Pq (figure 8b), and the probability of failure if the link is a bridge Pb (figures 8c and 9). It is worth mentioning that all of the earthquake-induced hazards considered in this paper can severely damage the road. When roads are disrupted, they are likely to be impassible. Restoration time is variable and depends on many different factors. In this paper, we specifically focus on the immediate impact of network failure. First, we determine the probability that a bridge will be damaged. As each bridge has four possible damage states with different probabilities, we consider a bridge as failed only when it experiences moderate or complete damage. Then, the overall link failure probability Pf=Pl+Pq+Pb−Pl ∗ Pq−Pq ∗ Pb−Pl ∗ Pb+Pl ∗ Pq ∗ Pb. Once we obtained the link failure probability, we used that probability to conduct a weighted selection to determine the probabilistic failure sequence at each iteration. First, we constructed a cumulative probability range based on the probability we obtained. Then we randomly generated a number in the range of [0, ∑(Pf)], and it will fall into one of the intervals in the constructed cumulative probability range. The corresponding link of the range will be considered as failed. In this case, the larger the Pf, the wider the range, and the more likely it will be selected. Once all the links are iterated through, we will have the probabilistic failure sequence. Based on ϕ, we can select the links to be removed at each step and simulate the percolation process on the Portland Metro network. Using the proposed robust componentequation (3.1) and algorithm 1, the size of the robust component is recorded throughout the simulation and presented in figure 10. Figure 9. Bridge fragility curve illustration. (Online version in colour.)
Figure 10. Network robustness of Portland Metro network under earthquake-induced probabilistic failure. Black dots represent the 100 simulation conducted, and red squares represent the average performance. (Online version in colour.)
Since link removal is probabilistic, a Monte–Carlo simulation was conducted to capture the overall performance. The Monte–Carlo simulation allows the generation of different link failure scenarios and produces more comprehensive and accurate results. Here, ϕ is increased in intervals of 1%, and at each ϕ, 100 simulations were conducted. Figure 10 shows a two-phase transition in the RCS. First, we need to note that robustness variation amplifies in the range of [0, 0.49], which results from the uncertainty in links’ exposure to natural hazards. This is because 48% of the nodes on the network are exposed to earthquake-induced hazards, i.e. landslide, liquefaction, and bridge failure. Based on the generating scheme of probabilistic failure, the links exposed to these hazards are more likely to be removed from the network. Therefore, the simulation first converges at ϕ = 0.49. The high variance at the range of ϕ ∈ [0.18, 0.38] suggests that the impact of earthquakes at this scale is hard to predict. The corresponding hazard mitigation plan and post-disaster recovery effort should prepare for the worst scenario so that post-earthquake access to hospitals can be maintained at its highest level. Similarly, when all the hazard-prone links are removed, the percolation at a range of ϕ ∈ [0.49, 1.0] is analogous to the random failure on the road network as the rest of the links have equal probability of being removed. The network achieved an overall robustness R = 0.392. Although network failure is less likely to reach 80%, monitoring of RCS percolation helps to evaluate network access to hospitals in the face of an M9.0 earthquake and in devising mitigation strategies accordingly. The robustness component characterizes the contribution of the ability of critical facilities to provide resources and maintain the functionality of the network. Removing one critical facility will lead to the degradation of robustness performance. This decrease in robustness, in another way, highlights the importance of maintaining network access to the critical facilities and enhancing network robustness, and it can be used to identify a component’s criticality. The lower the robustness (the higher the vulnerability) when the identified critical facility is missing, the more critical it is to the network's robustness. Using this logic, we iterated through the list of hospitals in the Portland Metro region and identified the most critical hospital in terms of ensuring network access to EMS after an earthquake. Figure 11 shows the hospital’s criticality when the selected hospital is removed. The nodes represent the crossing of the roads, and the colour indicates the criticality of the node to the overall network robustness. We can observe that the hospital located in northern downtown (Legacy Good Samaritan) is the most critical. This makes sense because figure 8a shows that Legacy Good Samaritan is located at a region (Portland hill fault) that is highly prone to landslides, and that its failure will lead to a severe shortage of medical care in the surrounding neighbourhoods. In designing a critical infrastructure protection plan, the route to Legacy Good Samaritan should be retrofitted to make sure that part of the community can have access to the hospital. Figure 11. Hospital criticality under earthquake-induced failure. (Online version in colour.)
Network robustness if a new node is included (i.e. a new hospital is built) can be used to determine the significance of the node. Iterating through the network, figure 12 presents the identified optimal placement of a future hospital. The nodes are the intersections of roads, and colour identifies the hot spots for the future site of the hospital. As we can observe, southwest of Portland is one potential location for a future hospital. This is because figure 7 shows that no hospital is built in this region. Despite the fact that the southwest is an earthquake-prone area, the construction of a new hospital that can withstand high magnitude earthquakes will enhance network robustness. Figure 12. Identified optimal location for the future hospital. (Online version in colour.)
Post-disaster network access to a critical facility can significantly impact the robustness of a network. In this paper, we proposed a robust component measurement using a percolation simulation framework to assess network robustness by encapsulating the impact of network access to critical facilities. Two types of network are used: the random network and the Portland Metro road network. The results demonstrate that without considering network access to critical facilities, the conventional giant component will falsely estimate network robustness and lead to inefficient hazard mitigate strategies. In addition, increasing the number of critical facilities will enhance network robustness, but the marginal benefit decreases after a certain threshold is reached. Also, when the number of critical facilities reaches 1000, Portland Metro network shows a very similar robustness performance to a random network of the same size and mean degree. Rather than random failures, network disruptions in real life are often influenced by the geometric features of the network. Therefore, we conducted a probabilistic earthquake-induced failure of the Portland Metro road network and used hospitals as a case study. The percolation of RCS shows a two-phase transition decided by the proportion of the links exposed to earthquake-induced hazards. The robustness shows great variation between ϕ ∈ [0.18, 0.38] and ϕ ∈ [0.5, 0.8]. The depicted transition in robustness can help us to create effective mitigation plans and informed policies to minimize the loss of network access to hospitals, in different scenarios under an M9.0 earthquake. To transform the research findings to network design and inform the stakeholders about critical infrastructure protection, we used the current framework in devising a strategy to protect existing infrastructures and to allocate resources on newly built infrastructure. To maximize network robustness, we iterated through the hospitals and derived the robustness that reveals the criticality of each hospital. We found that Legacy Good Samaritan is the most critical hospital for maintaining network robustness and should be particularly protected. Furthermore, we also identified the optimal placement for a future hospital. We identified the region that will enhance network robustness by providing community access to a hospital. Through the simulation, we identified that southwest of Portland city centre is the potential location for future hospital facilities. Limited access to critical facilities will make a poorly connected network vulnerable to network disruption. This suggests that the number and location of critical facilities can significantly influence network robustness. Therefore, network robustness should be characterized based on interdependencies with critical facilities and not as an inherent property of the transportation network alone. In this paper, we incorporated post-disaster access to critical facilities into network robustness to accurately assess the network condition, and to provide future critical infrastructure protection strategies and new development schemes. It is worth mentioning that the simultaneous link-removal on the network is to approximate network disruption resulting from catastrophes such as an M9.0 earthquake or flooding like Hurricane Harvey. The network will suffer from extensive loss of connectivity due to the link failure. To counter such damage on the network, different strategies can be applied, for example, emergency response planning, evaluation of the location of additional critical facilities, and infrastructure prioritization. The proposed simulation framework can not only assess network robustness in consideration of post-disaster access to critical facilities but also can identify critical infrastructure and future infrastructure sites. The results show us the infrastructure criticality hot spot on a network. With this information, we can emphasize hazard mitigation planning in the critical areas, prioritize protection of critical facilities, set up a temporary emergency response centre for effective post-disaster recovery and build a new facility or relocate existing facilities to mitigate the disastrous effect. Network robustness can be analysed via a number of different approaches. However, comprehensive empirical data are hard to obtain, which limits the implementation of a majority of the methods. This paper enables an assessment of network robustness in considering post-disaster network access to critical facilities through the use of topological and geometrical data. The parameter in calculating robustness involves network structure, critical facilities, and natural hazards mapping, which can all be obtained from agencies and state departments of transportation. For example, spatial networks from GIS files can be extracted into node/link graphs. Hazards such as the hurricanes, ice-storms and flooding can also be explored by calculating the link exposure to disruption through the use of the proposed simulation framework. Despite the use of hospitals as the representative critical facilities, other types of critical infrastructure can also be investigated in the future, such as material warehouses, resource repositories and equipment centres. Therefore, the proposed robust component can be generalized in a wide range of scenarios and help cities to evaluate the robustness of their road networks. Dong et al. [32] discovered that most cities share very similar road structures using a giant component as the robustness metric. Therefore, although the exact percolation transition threshold would vary, the robustness patterns are very similar. Beyond the road network, the proposed assessment framework can be also applied to other critical infrastructure networks such as electricity networks or water distribution networks. For example, power lines have to connect to the generator and distributor in order to maintain functionality. Failure of the water pumps would lead to water shortage in the communities. These phenomena can all be investigated through the proposed robust component-based percolation modelling approach. Transportation planning can not only identify the existing or future critical facilities and inform resource allocation but can also be beneficial by applying a robustness component lens to assess post-disaster accessibility to resources and services. For example, in the event of urban flooding, flooded neighbourhoods will certainly lose transportation to essential services such as food and pharmacy. On the other hand, neighbourhoods that survived flooding may also be isolated from the critical services as roads become inundated. The proposed metric and framework allow us not only to measure the direct impact of network disruption but also to identify the network components that are indirectly affected by disasters. We can thus identify the critical roads for future infrastructure development and hazards mitigation planning to alleviate the societal impact of disastrous network disruption. In addition, different subpopulations of a community use, access, and rely on the infrastructure and respond to disaster impacts in different ways. The proposed infrastructure network robustness assessment framework can also be combined with social vulnerability to identify the focal area for urban planning and critical infrastructure protection. The network data used in this paper can be obtained from OpenStreetMap, and the natural hazards data can be obtained from OHELP (https://ohelp.oregonstate.edu/). S.D. and H.W. conceived the study; S.D collected and analysed the data, conducted the experiment, and drafted the manuscript; H.W. and J.G. provided constructive comments on the methodology and paper revision; H.W., J.G. and A.M. advised and edited the study. All authors participated in the design of the study, helped draft the manuscript and gave final approval for publication. This research is supported by the National Science Foundation through grant CMMI Award no. 1563618 and no. 1826407. J.G. was partially supported by the Knowledge and Innovation Program no. 1415291092 at Rensselaer Polytechnic Institute. Any opinions, findings, and conclusion or recommendations expressed in this research are those of the authors and do not necessarily reflect the view of the funding agencies. The authors are grateful for the insightful comments of the anonymous reviewers. FootnotesReferences
Page 8Biomineralized organisms show an incredible diversity of complex microstructural forms and structure–property relationships [1–6]. A more complete realization of these naturally occurring structures provides not only a better understanding of an animal's ecology [7–10] but also supports bioinspired development of human-made materials [11–16]. Acorn barnacles (order Sessilia) are sessile marine arthropods that often inhabit the high-energy intertidal zone and have adapted structurally, compositionally and architecturally to challenging abiotic conditions, as well as the threat of diverse predators [17]. The calcareous exoskeleton (shell) of barnacles is well studied structurally, for example, the specific calcite crystal orientations in the operculum of Balanus amphritrite (=Amphibalanus amphitrite [18,19]); the high mechanical strength and adhesive properties of the baseplate in A. amphitrite, A. reticulatus and Balanus tintinnabulum ([9,20–23]); the involvement of extracellular matrix molecules in exoskeleton biomineralization in the giant barnacle Austromegabalanus psittacus [24]; and the structurally sound nanomechanical properties of the exoskeleton of A. reticulatus [25]. However, little is understood about how macro–micro–nanoscale structures, particularly in the shell, are linked. Correlative imaging provides an opportunity to discover the multiscale interactions and mechanisms involved in the structure of complex systems at varying length scales [26–29] and, specifically for barnacles, provides an opportunity to correlate optical, analytical, structural and mechanical information [30] for the first time. Here, we have coupled numerous systems at various length scales: X-ray microscopy (XRM), scanning electron microscopy (SEM), light microscopy (LM) and focussed ion beam microscopy (FIB-SEM) to ascertain the macro-to-nanoscale structure, crystallographic orientation and mechanical properties of wall-plate joints in the parietal exoskeleton of the barnacle Semibalanus balanoides. Semibalanus balanoides is the commonest intertidal barnacle on British coastlines [31] commonly outcompeting other barnacle genera [7], although the structural properties of its shell are relatively poorly understood compared with other species (e.g. B. amphritrite), as are the morphological properties of the wall-plate joints, with just two previous studies outlining basic details [7,17]. The shell of S. balanoides comprises six interlocking joints, where the shell originates from within an existing organic cuticle. These joints are located in a particularly active and dynamic region of the barnacle shell and provide a waterproof seal and structural integrity in the face of extreme conditions of the physically harsh intertidal zone [8]. This work identifies the specific macro–micro–nano features of the wall-plate joints in both two and three dimensions through connected correlative imaging and establishes how these features are linked at varying length scales. A greater understanding of how these complex structures function provides valuable biomechanical information for biologists as well as the broader bioinspiration topic. Acorn barnacles are sessile organisms that attach to hard substrates via either a calcified base plate or an organic membrane [20], and biomineralization of the calcareous shell is mediated by the mantle epithelium via secretion of a calcium matrix [32]. The conical-shaped exoskeleton is composed of four, six or eight wall plates depending on the species [25], which overlap at sutures or joints (figure 1a); parts of the plate overlapping internally are called alae (wings), and parts that overlap externally are called radii (‘rims' [33]; figure 1b). The wall plates grow both upwards towards the apex and outwards as the internal soft-bodied organism grows inside [20]. As with other crustaceans, barnacles moult the chitinous exoskeleton surrounding their main body periodically to grow, but the calcareous shell is not shed during this process [33]. Figure 1. Morphological structure of the barnacle Semibalanus balanoides. (a) Transverse view of the six wall plates that make up the barnacle conical structure. Alae between adjacent wall plates are highlighted by red arrows, radius on neighbouring plates by black. (b) Longitudinal internal view through adjacent wall plates (adapted from Murdock and Currey [17]). Insets illustrate morphological differences of the ala at different points of the interlock. (Online version in colour.)
Barnacle specimens were collected from the intertidal region at Bracelet Bay, Swansea, UK (51.5660° N, 3.9801° W). Samples were subsequently vacuum impregnated into a 32-mm-wide resin block and ground and polished to reveal a transverse section. The sample surface was coated with a 10-nm layer of carbon to ensure sample conductivity in the SEM and FIB-SEM. Individual plates were also detached from the exoskeleton and attached with adhesive to wooden pins for imaging using XRM. All preparation, subsequent analysis and imaging occurred within the Advanced Imaging of Materials (AIM) Facility, College of Engineering, Swansea University (UK). LM and SEM were used to obtain general two-dimensional (2D) information on barnacle morphology. LM images were obtained using a Zeiss SmartZoom and a Zeiss Observer Z1M inverted metallographic microscope. SEM images were collected on a Carl Zeiss EVO LS25 with a backscatter detector at 15 kV, 750 pA and a working distance of 10 mm. As well as the carbon coating, copper tape and silver paint were added to the sample surface to aid charge dissipation. A JEOL 7800F FEGSEM and a NordlysNano EBSD detector controlled via Aztec (Oxford Instruments) software were used to obtain crystallographic information. The phase selected for EBSD indexing was calcite [34], and patterns were collected at 15 kV with a step size of 0.2 µm. A relatively high number of frames (five frames per pattern) were collected using 4 × 4 binning to give a camera pixel resolution of 336 × 256 pixels and a speed of 8 Hz. A Zeiss Xradia Versa 520 (Carl Zeiss XRM, Pleasanton, CA, USA) was used to carry out high-resolution XRM non-destructive imaging; this was achieved using a CCD (charge coupled device) detector system with scintillator-coupled visible light optics and a tungsten transmission target. Initial scans of the barnacle region block were undertaken with an X-ray tube voltage of 130 kV, a tube current of 89 µA, an exposure of 4000 ms. A total of 1601 projections were collected. A filter (LE4) was used to filter out lower energy X-rays, and an objective lens giving an optical magnification of 4 was selected with binning set to 2, producing an isotropic voxel (three-dimensional (3D) pixel) size of 3.45 µm. The tomograms were reconstructed from 2D projections using a Zeiss Microscopy commercial software package (XMReconstructor) and an automatically generated cone-beam reconstruction algorithm based on the filtered back-projection. Individual plates were also scanned (not in the resin block); these were collected using the 4X objective lens at 60 kV and 84 µA, with an exposure time of 12 000 ms and a resulting (isotropic voxel size) of 0.5 µm. A filter (LE1) was used to filter out low-energy X-rays, and 1601 projections were collected. The scout and zoom methodology was used to create high-resolution regions of interest (ROIs) within the sutures. Targeted navigation to ROIs was achieved using Zeiss Microscopy correlative Atlas 5 (3D) software package on the Zeiss Crossbeam 540 FIB-SEM. This method enables a live 2D SEM view to be combined with other data and information from previous sessions or relevant characterization techniques on the same area or volume of interest; this is achieved by importing and aligning other 2D datasets (e.g. LM images, EBSD maps) and 3D data (XRM stacks) to accurately correlate and locate ROIs for further nanoscale imaging and characterization (figure 2). Initial overlay is achieved by manually aligning the live SEM image with the imported data and ‘locking in' the imported data to the current SEM coordinate system. This correlative microscopy approach is especially useful when ROIs may be internally located within a subsurface area of the specimen and allows samples to be accurately studied at varying length scales by combining information from multimodal sources. Figure 2. Schematic of the multimodal and multiscale correlative workflow using LM, XRM, SEM and FIB-SEM. These techniques can be correlated using Zeiss Microscopy Atlas 5 (3D) software. (Online version in colour.)
Specific ROIs in the barnacle shell were studied using a Zeiss Crossbeam 540 FIB scanning electron microscope (FIB-SEM, gallium source; Carl Zeiss Microscopy, Oberkocken, Germany). The sample stage was tilted to 54° to allow the sample to be perpendicular to the FIB column; the ion beam energy was 30 kV in all cases. The FIB and SEM beams are then aligned at 5 mm working distance at the co-incidence point. Within the Atlas 5 (3D) correlative workspace, it is possible to identify ROIs for further study, and then with the same interface, 3D nanotomographic volumes are prepared and collected (figure 3). A template is set up over the ROI that outlines the numerous steps in the milling process (figure 3a). A 10 × 10 µm platinum layer was deposited using a gas injection system and the 700 pA FIB probe; this is to protect the ROI sample surface from damage during the milling process. 3D tracking marks (which enable automatic alignment and drift correction during an automated run) are milled onto the first platinum pad using the 50 pA FIB probe, and then a second platinum pad is deposited on the top (again at 700 pA) creating a ‘sandwich’ of protection and alignment layers (figure 3b). A trench is then milled using the 7 nA probe to create a cross-sectional surface through the ROI to a depth of approximately 15 µm (figure 3b,c). The cross-sectional surface of the trench is polished using the 700 pA probe. Once the sample preparation is complete, automated tomographic milling and slice generation can take place (figure 3c). The run is set, so the length of the protected platinum pad is milled to create a 3D volume. Each slice (10 nm thickness) is milled by the 1.5 nA probe using the FIB and simultaneously imaged by the SEM; parameters for image acquisition with the SESI detector include 1.8 kV, 300 pA, 10 µs dwell time and a 12 nm pixel size. Once the run has completed overnight (approx. 8 h), the slice images are reconstructed to create a 3D volume (figure 3d) and segmented and visualized via other specialized tomographic software (e.g. FEI Avizo, Hillsboro, USA; ORS Dragonfly, Montreal, Canada). Quantified data can be found in electronic supplementary material, S1. Figure 3. Stages of the FIB-SEM automated milling process using Zeiss Microscopy Atlas 5 (3D). (a) An overlay is created for each part of the milling preparation. (b) Once a platinum pad has been deposited over an initial platinum pad and the milled reference marks creating a ‘sandwich’, a trench is milled to reveal a cross-sectional face (c). (d) Acquired images can then be stacked together to create a tomographic volume. (Online version in colour.)
SEM reveals the microscale 2D morphology of the barnacle (and specifically the alae; figure 4). Alae generally have rounded tips and slot into a groove in the neighbouring plate with organic material separating the two plates (figure 4a–d). The microstructure in the 40–70 µm closest to the tip of the ala appears to have a different morphological texture and more voids than other alae regions and the opposing plate (figure 4c,d). The voids are of two types: transverse banding, which is parallel to the ala tip orientation, and elongated grooves/channels, which are perpendicular to this (figure 4c,d). The elongated grooves/channels and transverse banding appear to be of varying size, shape, elongation and thickness (figure 4c,d) and may represent pore networks. By contrast, the neighbouring plate and the area behind the ala tip appear smooth and featureless (figure 4c,d). Figure 4. SEM imaging and EBSD crystallography of the barnacle and ala. (a) SEM image of an individual barnacle in transverse section. (b) View of three interlocking plates and ala (red arrows). (c,d) Close up of two alae (inset boxes in b), revealing microstructure transverse banding and perpendicular elongated grooves/channels at the tip. (e,f) EBSD maps of ala in (c,d), illustrating elongated grain orientations at the tip of the ala, and granular grains behind the tip and on the adjacent plate. Blue arrow illustrates inside edge coarse grains. Elongated grains appear to correlate with the porous area of the ala. Scales in (e) and (f) correspond to (c) and (d), respectively. (Online version in colour.)
EBSD inverse pole figure maps of the ala tips show a microstructure with a highly segregated bimodal grain size (figure 4e,f). The grains at the tip of the ala closest to the plate joint are elongated and radiate 50–70 µm downward into the ala structure perpendicular to the curve of the join (figure 4e,f). The individual 3D hexagonal crystal diagrams for each elongated grain show that in each case, the c-axis [0001] is parallel to the long axis of the grain and perpendicular to the line of the join of the plates (figure 4e,f). The grains in the adjacent area below the elongated grains, and in the adjoining upper plate, are around 10–20 times smaller at 3–5 µm and have a more equi-axed structure with no obvious texture. In both EBSD images, there are also regions of coarser grains within the equi-axed areas behind the ala tip on the inside-facing edge of the ala (blue arrow; figure 4e); however, these are not elongate or organized like those in the tip (figure 4e,f). We have reconstructed the entire barnacle in 3D (figure 5a; electronic supplementary material, S2) as well as individual plates (figure 5b) illustrating morphological variations in the ala through the length of the exoskeleton. We observe the protruding ‘tab' of the ala towards the apex where it slots into the neighbouring plate (figure 5a(ii)); in 2D image slices (figure 5a(iii)(iv)), the ala appears as a finger-like protrusion with a rounded tip. Towards the base and the ala sutural edge, the ala recesses and creates a flat junction with the neighbouring plate (figure 5a(ii)); in 2D, the ala appears more angular and has an almost square tip (figure 5a(ii)(iv)). In addition to the 3D morphological change in the ala through its length, we also observe networks of elongate channels, grooves and bands that are also visible in the 2D surface imaging (figure 4c,d). We propose that these are related to the pore networks observed in figure 5. Pores appear black in 2D stack images because they exhibit a lower X-ray attenuation compared with the surrounding calcium carbonate exoskeleton (figure 5a(iii)(iv)). The pores appear to ‘fan' perpendicularly to the ala edge, the same as the textures in 2D (figure 4c,d). The micropores are only found at the tip and are not present in other areas of the ala. Pores also change shape, orientation and location through the length of the ala; towards the apex, they fan around the entire ala tip (figure 5a(iii)); however, towards the ala sutural edge, they are on one, inner side only (figure 5a(iv)). Figure 5. XRM of the barnacle. (a) (i) 3D XRM image of the barnacle. (ii) XRM reveals changes in the morphology of the ala through its length. Also indicated are the directions upon which the ala meets the neighbouring plate (yellow arrows). Also identified are pore networks and how these change through the ala length (iii,iv). (b) Segmented XRM ala pores (i–iii). From local thickness analysis in Fiji, pores appear to be thickest on the inside edge of the plate, nearest the soft body of the organism. Purple = thin, yellow = thick. (iv) Simplified illustration showing the change in pore geometry through the length of the ala; the pores (blue lines) are parallel to the direction of the ala, which continues down the ala length. Once at the ala sutural edge, the pores change direction, instead running from top to bottom (direction illustrated by red arrows). Thick blue lines indicate areas of thickening. Image reconstructions occurred in Drishti (a) and ORS Dragonfly (b). Segmentation of pores occurred in Zeiss Microscopy Intellesis software. Pore thickness map produced by Local Thickness plugin in Fiji/ImageJ software. (Online version in colour.)
Segmentation of the pores using Intellesis machine learning segmentation software (Zeiss Microscopy, Oberkocken, Germany) reveals the morphology of pores in 3D through the length of the ala. Nearest the apex, the pores form fan-like networks, which continues down into the ala length (figure 5b(i)(ii)). However, towards the ala sutural edge and base, the morphology and orientation of the pores change and are instead parallel to the ala surface running from top to bottom; a simplified diagram of this is shown in figure 5b(iv). In addition, there is a widening of pores on the inside edge of the plate nearest the soft-bodied organism (figure 5b(iii)). Despite the identification of pores, no grain boundaries, crystal structure or segregated greyscale variations were observed via XRM imaging, therefore requiring further characterization via other techniques (e.g. FIB-SEM, EBSD, SEM). From targeted FIB-SEM nanotomographic milling through Atlas 5/3D (§2.4.1), it is possible to study the ala pore networks at a higher resolution to establish nanoscale features and relationships. We have compared ala pore networks with those on the opposing plate (figure 6) to establish exoskeleton variations in pore structure. 10 × 10 µm FIB-SEM nanotomographic volumes reveal variations in pore morphology and alignment between those on the ala and those on the neighbouring plate (figure 6). Pores on the ala (figure 6d,e) are numerous (962 in this volume), have pore diameters up to 1.56 µm and are composed of mostly shorter and singular pores. This is compared with those on the opposing plate (figure 6b,c), which are less numerous (594 in this volume) and are dominated by thicker and longer connected pores. Ala pore directionality (figure 6d,f) follows EBSD crystallographic orientations (figure 4e,f); however, dominant orientations on the opposing plate (figure 6b,c) do not appear to be related to the crystallographic structure (figure 4e,f). Further analysis to the porosities via Avizo software indicates similar trends in elongation and pore diameter between the opposing plate and the ala (figures 6f(i)), with a larger spread for values of pore width (figure 6h) and more spherical pores (figure 6g) in the ala. This illustrates that individual pores and pore networks vary in structure (and possibly function) across the barnacle shell. No crystal structure, crystal boundaries or phase variations were observed from FIB-SEM images. Figure 6. (a) Locations of milled volumes on the ala and opposing plate. (b,c) Reconstructed and segmented pore volumes on the opposing plate, illustrating a 17/197° orientation. (d,e) Reconstructed and segmented pore volumes on the ala, illustrating a 105/285° orientation. (f–i) Histograms highlighting statistical variations in the pores between the ala and opposing plate. (Online version in colour.)
This work represents the first correlative multimodal and multiscale study of barnacle morphological and mechanical structure across multiple dimensions. Correlative microscopy overcomes the multiscale ‘needle in a haystack' challenge of working in complex 3D volumes and has proved successful for accurately locating specific regions of study in human-made materials; examples include lithium ion batteries [35] and corrosion in magnesium alloys [36]. Additionally, the technique is well established across a range of applications in the life sciences [37–40]. The advantage of using a multimodal and correlative approach is that each specialized technique can provide information relating to a specific feature or structure and that correlation across dimensions can thus inform how features and structures are linked, particularly in hierarchical materials. Increasing the resolution is important for identifying and improving the accuracy of measurement of features at the micro- to nanoscale (e.g. the voids in figures 5b and 6) and in three dimensions reveals characteristics that might not be identifiable in one or two dimensions alone (e.g. pore orientations in figure 5). The correlative workflow improves our understanding of barnacle shell structure (figures 6 and 7) where specific regions are accurately located to the nanoscale. The workflow outlined here can be used in other bioinspiration studies (e.g. mollusc shells) to correlate macro- to nanoscale shell structures, which ultimately improves the understanding of form and function as well as the application for human-made materials. Figure 7. Correlation of 2D and 3D over macro–micro–nano scales and multimodes to inform about barnacle exoskeleton morphology. (Online version in colour.)
This work reveals that the ala tips of S. balanoides exhibit a distinct crystallographically graded biological material (figure 4). We propose the elongated crystal growth at the tip of the ala compared with the more equi-axed grains behind the tip and on the opposing plate (figures 4e,f) represent a growing front in a region of active biomineralization. During biomineralization, calcium carbonate generally forms prismatic, sheet nacreous, lenticular nacreous, foliated, cross-lamellar and homogeneous crystal morphologies [41]. Only prismatic and homogeneous crystals were identified here in barnacles (figure 4e,f). Elongated crystals in Semibalanus alae have previously been identified from a single study of B. balanus and S. balanoides [17] that was limited to two-dimensional study (LM and SEM) and left their origins unexplained, and elongate crystals have been identified in B. amphitrite [19,21]. Different crystallographic orientations, in particular elongate, prismatic columns associated with organic materials are common in a variety of biomineralized molluscs [6,42,43] but remain largely unidentified in barnacles. The ordering of calcite in the scutum (one of the two plates that guard the apical opercular opening) is significantly disordered compared with the calcite in the wall plates in A. amphitrite [19], and the calcitic microcrystals in the wall plates of this species show almost no orientation [19,21], while those in the base plate of A. amphitrite shows some preferred alignment [21]. This suggests that there may be some variations between barnacle genera other than Semibalanus balanoides. Elongated, prismatic calcite columns growing perpendicularly to the shell surface are present in shells of various molluscs (including oysters and scallops [42,44]) and other arthropods (specifically the Mantis shrimp [45]), which are surrounded by up to 3 µm-thick organic membranes and vaterite columns in the shell of the bivalve Corbicula fluminea [6]. This indicates that different crystallographic orientations, in particular elongate, prismatic columns associated with organic materials are common in a variety of biomineralized molluscs, but however remain largely unexplained and undescribed in barnacles and may form an important part of the shell structure. Several hypotheses, either independently or in union, may explain the crystallographic elongation at the tip of the ala in S. balanoides:
We have identified and examined numerous porous channels in the barnacle alae. We considered that ala pores may represent crystallographic boundaries (figure 4e,f) as they have the same orientation (figure 6d,e); however, further study via FIB-SEM showed that the pores sit within the grains (some crystals are up to 10 µm wide, the same as the entire FIB-SEM volumes; figures 4e,f, 6d). This could be a factor of the EBSD resolution; however, it is likely that the pores exist independently of crystallographic structure while maintaining the same orientations. This hypothesis is supported by elongate pore networks in the more equi-axed crystals of the opposing plate and behind the ala tip (figure 6b) and the lack of observed crystallography in the cross-sectional face during FIB-SEM milling (figure 5c), suggesting that the locations for FIB-SEM nanotomographic milling were small enough to be considered intragrain. Exoskeleton/shell pores are common in many groups of biomineralized marine organisms including gastropods [56], bivalves [6,57] and within the exoskeleton of some barnacles (particularly in base plates [25,50]). It has been suggested that the most important adaptive breakthrough in balanoid barnacles, and their competitive success over Chthamalus barnacles, is a tubiferous wall structure that enables the fast exoskeletal growth and colonization of free space [7]. We propose that the ala pores in the exoskeleton of B. balanoides are organic-rich areas, possibly involved in biomineralization. The involvement of organic material in the biomineralization of specific crystal structures and orientations could have a bearing on the function of the ala pores and may represent channels/canals, which hold or deliver biomineralization products to specific areas of the exoskeleton. Organic membranes are known to influence the pattern of columnar prismatic layers in numerous mollusc shells [43], so it is possible that organic channels (or, ala pores) running through the barnacle structure contribute towards the delivery of and biomineralization of calcium carbonate. The organic layer separating the ala tip and neighbouring plate may play a part in this. A layer of organic cuticle exists between the ala and the neighbouring plate in Balanus balanus [17], which may explain the concentration of pores and elongated crystals near the ala tip. The pores, however, are not all elongate channels, and some pores, particularly in the opposing plate, being more spherical in shape (figure 6g). Pores in different parts of the exoskeleton may therefore have different functions, possibly acting as channels in the ala to deliver organic material for biomineralization and to hold pockets of organic material in the opposing plate. Barnacle wall-plate mineralization occurs through cell-mediated Ca2+ uptake, storage and mobilization to the mineralization front [32] and pore canals assist in transporting components necessary for calcification (including Ca2+ and organic molecules). Voids/pores in the aragonitic platelets of mollusc nacre contain increased amounts of carbon [58] and tube-like shell pores containing organic material are also present in limpets [57], while the organic intertile layer in abalone is anchored by the growth of minerals through pores [59]. The pores forming ‘canal' networks in the wall plates of large sessile barnacles (Austromegabalanus psittacus [24]) has yet to be ascribed a function. Longitudinal canals in the wall plates and radial canals in the base plate of B. amphitrite are lined with mantle epithelium, and biomineralization is facilitated by salt-rich secretions from the junction between the wall and the base plates [32]. Some barnacles also possess microducts/pores in their base plates to facilitate secretion of adhesive [25]. Exoskeleton pores seem to be used for the transport of organic material and biomineralization, although the role of proteins and other macromolecules in the biomineralization process is still poorly understood [19], and future studies should aim to quantify this. Despite the presence of probable organic pore networks and specific crystallographic orientations in many genera and species of barnacle [17,24,25,29], it is possible that the features discovered in this study are unique to S. balanoides. The alae and wall plates of other barnacles, for example B. balanus, are considerably different to those of S. balanoides [17], consequently their crystallographic structures may also differ. Shell morphology is also highly phenotypically plastic within a barnacle species and can change according to wave exposure [60], predation [61] and, especially, crowding [62]. Hummocks of tall, columnar barnacles are common under high population densities, while squat, conical growth forms with much thicker (2–5 times in S. balanoides) walls dominate in solitary individuals or low population densities [62]. Whether these growth forms differ in microstructural and mechanical properties may warrant investigation, although the substantial difference in shell strength between B. balanus and S. balanoides has been attributed to distinct variation in shell architecture rather than mechanical properties of the wall plates [17]. The range of crystal sizes and shapes, as well as reinforcement by organic-rich channels, will all contribute to the mechanical properties of the barnacle shell. For example, the probable organic pores and specifically oriented crystallographic structure of the ala tip may also act as a strengthening mechanism in a region of active growth [20] and high stress [60,61]. The presence of organic material within the biomineralized structure also has important implications for strength and toughness. Crossed-lamellar structures, composed of aragonite and a small amount of organic material, are the most common microstructures in mollusc shells and possess a fracture toughness and elasticity much higher than pure carbonate (calcite) mineral [58,63–66]. Indeed, removal of this organic material from abalone shell greatly contributes towards its mechanical decline [67]. The ala in S. balanoides is non-geometric through its length (figures 1 and 5a,b; [17]), suggesting that it is potentially not the strongest design for an interlocking joint. Further tests are required to establish the hardness and strength of different regions of the barnacle, in particular, the alae, and the effect the elongated crystal structure and organic-rich pores of specific orientations have on mechanical strength. Understanding the morphology and structure of biomaterials can contribute towards the design and manufacture of human-made materials [2,68,69]. Similar discrete bimodal grain sizes are observed in manufactured materials for aerospace, such as the ‘dual microstructure' of some nickel superalloy-based gas turbine disks [70]. To improve the material properties and in-service behaviour, the material is specifically designed to have distinct microstructures in different regions of the disk. A fine-grained microstructure is produced in the bore, providing a higher proof strength and fatigue life, whereas a coarse-grained microstructure in the rim results in greater creep life [71]. Location-specific microstructures in different regions of the part are tuned to the environmental conditions in which they are exposed for optimized design and life. But this level of modification to different parts of the microstructure requires multiple complex steps, including heat treatments at temperatures in excess of 650°C [72]. It is possible the barnacle alae dual microstructure with specific crystallographic orientation of the elongated grains perpendicular to the loading/contact surface is a functional characteristic, with highly adapted microstructural features driven by the evolutionary processes. In comparison to the processes required to produce the nickel superalloy, the barnacle achieves a highly complex microstructure in ambient conditions, dynamic tidal conditions and with the chemistry and temperatures imposed on it by the environment. Additionally, the interlocking nature of the barnacle joints described here, combined with the variation in crystallographic organization and pore structure, could contribute towards the development of materials that require movement and expansion while remaining strong, such as expandable pressurized containers or submersible structures. Another potential could be the utilization of barnacle-like designs in additive manufacturing. In recent years, functionally graded additive manufacturing has developed its capabilities of fabricating materials layer by layer and by controlling morphological features (such as porosity) to create structurally and mechanically distinctive materials [73]. A correlative approach to understand the morphological, chemical, and structural characteristics of natural biomaterials outlined in this study could therefore contribute greatly to the development of future bioinspired materials. Here, we show the advantages of using multimodal, multidimensional and multiscale correlative microscopy techniques to identify the morphological, microstructural and crystallographic properties of the shell of the barnacle S. balanoides. The barnacle shell is composed of six interlocking calcium carbonate wall plates with alae (electronic supplementary material, S2), finger-like protrusions acting as a contact point of potential high stress for the joining of adjacent plates. 2D imaging via LM and SEM indicate that the tip of the ala contains a series of pores. EBSD results indicate a crystallographically-graded texture of the biomineralized calcium carbonate, where elongate grains on the ala's outer edge are oriented perpendicularly to the surface of the joint and the c-axis rotates with the radius of the ala (the same orientation as the pores). 3D imaging via XRM enables the segmentation of the pores and the realization that pores are only visible within the very tip of the ala, their orientations change through its length and there is pore thickening on the inside (soft bodied) edge of the plate. Further analysis of the nanoscale structure of the pores through FIB-SEM illustrate that the pores are probably organic channels and pockets, which are involved with the biomineralization process. These properties indicate the macro–micro–nano scale features of the barnacle exoskeleton, particularly at the ala, could be useful for bioinspiration for human-made materials. Furthermore, correlative imaging allows the targeting of specific ROIs across different imaging techniques and length scales and greatly increases the amount of information that can be acquired from imaging in purely two dimensions, bridging the materials science of structure–property relationships with the biological form and function. Additional files supporting this article have been uploaded as part of the electronic supplementary material, S1–S4. XRM scans (tiff stacks) of whole barnacles mounted in resin and individual plates are available from the Dryad Digital Repository at: https://doi.org/10.5061/dryad.jc40j5v [74]. We declare we have no competing interests. Authors acknowledge AIM Facility funding in part from EPSRC (EP/M028267/1), the European Regional Development Fund through the Welsh Government (80708), the Ser Solar project via Welsh Government and from Carl Zeiss Microscopy. We thank two anonymous reviewers for insightful reviews and comments on this manuscript. Additional thanks go to James Russell and Imogen Woodhead for aid during SEM imaging and preliminary pore analysis, respectively, both from Swansea University, and Tobias Volkenandt and Stefanie Freitag from Carl Zeiss Microscopy (Germany). FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4571471. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 9Gastric mixing and emptying are modulated by gastric motor functions, including the peristaltic and tonic contractions of the stomach and opening and closure of the pylorus. A number of studies have been conducted to investigate the relationship between gastroduodenal motility and the emptying of gastric contents [1–6]. Closure of the pylorus is coordinated with antral contractions [7,8]. When a peristaltic contraction reaches the terminal segment of the antrum, the terminal antrum contracts near simultaneously, called terminal antral contraction, and the pylorus closes near the onset of the terminal antral contraction. Gastric contents in the terminal antrum are then forced back into the proximal antrum. The antrum then relaxes, and the pylorus begins to open. Flow from the antrum to the duodenum is driven by a pressure gradient across the pylorus, and flow through the pylorus has been observed even without contraction of the distal antrum [9–12]. A question in gastrointestinal physiology is to what extent gastric emptying is produced by peristaltic contractions of the antrum, tonic contractions of the fundus or the hydrostatic pressure difference between the antrum and the duodenum. These factors may appear simultaneously in vivo, and it is experimentally difficult to isolate and define the relative contribution of each factor to gastric emptying. Pyloric opening and closure are not always coordinated with antral contractions. Impaired coordination may alter flow patterns, potentially promoting or impeding gastric emptying, or contributing to retrograde flow from the duodenum to the antrum and bile reflux. Surgical procedures such as pyloromyotomy and pyloroplasty also impair pyloric function. Hence, it is important to understand what happens when pyloric function is diminished or compromised. Advanced medical imaging techniques [13,14] have enabled us to visualize the distribution of gastric contents in the stomach [15,16] and the shape and velocity of peristaltic contractions of the antrum [17–19]. However, information on gastric flow available from medical imaging is limited; hence, the relationship between gastric motility and gastric mixing and emptying has not been fully clarified. Computational modelling and simulations of gastric flow have been performed, enabling integrative and quantitative in silico studies that cannot be easily performed experimentally [19–23]. We previously developed a computational fluid dynamics model of gastric mixing, using an anatomically realistic geometry of the stomach [24]. We have also applied this model to successfully quantify gastric mixing produced by antral contractions [25,26]. Here, we extend this model to quantify the relationship between gastric motility and gastric mixing and emptying, particularly focusing on impaired coordination of the pyloric closure with antral contractions. An anatomically realistic gastroduodenal model based on data from the Visible Human Project [27,28] was used. The total volume of the stomach was approximately 650 ml, and the average diameter of the antrum was approximately D = 50 mm. The diameter of the pylorus was 9 mm when open. Here, we concentrated on gastric mixing and emptying produced by antral contractions, and we omitted tonic contractions of the stomach. Each antral contraction consisted of three phases: peristaltic contraction, terminal antral contraction and antral relaxation. A peristaltic contraction initiated at the mid-corpus, and travelled towards the terminal antrum. Parameters of the contraction wave were identified based on published magnetic resonance imaging (MRI) studies [19]. For healthy subjects, the propagation velocity of the contraction wave was V = 2.5 mm s−1, and the cycle of the contraction was T = 20 s. The contraction width was 18 mm, and the contraction ratio (degree of occlusion) increased as the wave progressed towards the distal antrum. The terminal antral contraction refers to a segmental and near-simultaneous contraction of the terminal region of the antrum [7]. When peristaltic contractions reach the terminal antrum, terminal antral contraction occurs, followed by antral relaxation. Because the propagation velocity increases at the terminal antrum [25,29,30], the terminal antral contraction was modelled as an acceleration of the propagation velocity and an increase in the contraction width at the terminal antrum. When peristaltic contractions reached 30 mm from the pylorus, the contraction width linearly increased from 18 to 54 mm. The propagation velocity, the contraction ratio and the contraction width are shown in figure 1. Figure 1. Parameters of contraction waves. (a) Propagation velocity, (b) contraction ratio and (c) contraction width.
Carlson et al. [7] investigated the coordination between pyloric closure and the terminal antral contraction. The pyloric canal was open during 28% of the total period of observation, and pyloric closure occurred at nearly the same time as the onset of the terminal antral contraction. To model impaired coordination of pyloric closure with antral contraction, we defined two parameters: the duration of pyloric closure, TC, and the delay of pyloric closure from the onset of terminal antral contraction, TD. For example, the duration TC/T = 0 corresponded to an impaired function of the pylorus, where the pylorus was unable to close or remained permanently open as a result of surgical procedures. In our control model, the duration of pyloric closure was set to TC/T = 2/3, and the pylorus began to close at t/T = 0.6, which has no delay from the onset of the terminal antral contraction, TD/T = 0. In impaired coordination models, we varied these two parameters (0 ≤ TC/T ≤ 1 and 0 ≤ TD/T ≤ 1). An incompressible Newtonian liquid was considered for gastric contents. At the initial state, 80% of the stomach and the whole duodenum were filled with liquid, while the rest of the stomach and the oesophagus were filled with air. While the density of the liquid was constant ρ = 1.0 × 103 kg m−3, the viscosity of the liquid was varied to examine the effect of viscosity. Viscous traction of air is negligible, and free-surface flow modelling was applied to this problem. The average vertical position of the air–liquid interface was obtained from the emptied volume of the liquid. Assuming that the air–liquid interface remained flat during the process, an external force was applied if the vertical positions differed from the average vertical position. To ignore the effect of hydrostatic pressure difference between the antrum and the duodenum on gastric emptying, hydrostatic equilibrium was also considered. The position and velocity of the gastrointestinal wall were prescribed, and given by moving boundary conditions. We solved the fluid dynamics using the multiple-relaxation-time lattice Boltzmann method [31], with uniform Cartesian grids of 1.5 mm spacing. To implement the moving and curved wall boundary conditions, a bounce-back scheme [32] was applied. Distribution functions at the air–liquid interface were extrapolated [33]. The computation was accelerated using graphics processing unit computing [34]. The emptying rate was computed by the outflow flux at the pylorus. Mixing efficiency was defined as the spatial average of the second invariants of the strain tensor for the cycle of the antral contraction. For more details, see [25,26]. We first examined gastric emptying of a liquid with a low viscosity, μ = 4.2 × 10−3 Pa s, for a control case, where TC/T = 2/3 and TD/T = 0. Figure 2a shows the time history of the instantaneous emptying rate, and the time chart of antral contraction and pyloric closure, where t/T is the time normalized by the period of antral contraction. A peristaltic contraction reached the proximal antrum at t/T = 0, and continued to move towards the pylorus (peristaltic contraction). The pylorus began to open at t/T = 0.4. While the peristaltic contraction still travelled in the proximal antrum, emptying of the liquid contents began (see also snapshots in figures 2b and figure 3a; electronic supplementary material, video 1). At t/T = 0.6, the terminal antral contraction began, and the pylorus began to close. The instantaneous emptying rate then sharply decreased. The terminal antral contraction finished at around t/T = 0.8, and the terminal antrum was relaxed until t/T = 1.0 (antral relaxation). Figure 2. Gastric emptying and mixing for a control case. (a) The time history of the instantaneous emptying rate and the time chart of antral contraction and pyloric closure, where t is the time and T is the period of antral contractions. Content viscosity was 4.2 mPa s. (b) Corresponding snapshots where white and red contents were initially located in the stomach and blue contents were initially located in the duodenum. (c) Time-averaged emptying rate and (d) mixing efficiency, as a function of liquid viscosity. (Online version in colour.)
Figure 3. Location of gastric contents after 1 min, where white and red contents were initially located in the stomach and blue contents were initially located in the duodenum. (a) Control case (TC/T = 2/3, TD/T = 0). (b) The pylorus was unable to close (TC/T = 0, TD/T = 0). (c) Impaired coordination (TC/T = 2/3, TD/T = 1/8). (d) Impaired coordination (TC/T = 2/3, TD/T = 3/8). (Online version in colour.)
To investigate the effect of liquid viscosity on gastric emptying, the viscosity was varied from 4.2 × 10−3 to 4.2 Pa s. Examples of liquid foods with a viscosity of O (10−3) Pa s include water, milk and beer, and those with a viscosity of O(1) Pa s include molasses and honey. Figure 2c shows the time-averaged emptying rate, as a function of the liquid viscosity. The emptying rate decreased as the content viscosity is increased, but the emptying rate became nearly constant for viscosities higher than μ ∼ 0.1 Pa s. The emptying rate ranged from 3 to 8 ml min−1 for the control case. We also calculated mixing efficiency and found that the mixing efficiency had similar trends to the emptying rate, as shown in figure 2d. We next investigated the effects of the duration of pyloric closure on gastric mixing and emptying. To check the relationship between the emptying rate and the pressure difference between the antrum and duodenum, we first considered a case where the pylorus was unable to close (i.e. the pylorus remained open; see also figure 3b; electronic supplementary material, video 2). We calculated the pressure difference between two locations 5 mm either side of the pylorus (i.e. in the distal antrum and proximal duodenum). The instantaneous emptying rate and the pressure difference are shown for low-viscosity liquid (μ = 4.2 × 10−3 Pa s) in figure 4a,b. The pressure difference between the antrum and duodenum promoted a near-constant emptying rate during the peristaltic contraction. The terminal antral contraction increased the pressure in the terminal antrum, and high-velocity flows appeared from there towards the proximal antrum and the duodenum (figure 4c). A large pressure difference during the terminal antral contraction resulted in an increase in the instantaneous emptying rate. When the antral relaxation began, however, the pressure difference became negative, i.e. pressure at the duodenum was higher than that at the antrum. Retrograde flow from the duodenum to the antrum then occurred (figure 4d), and the instantaneous emptying rate was negative. In the case of low-viscosity contents, negative emptying continued to the beginning of the antral contraction due to inertial effects. Figure 4. Effects of pyloric closure duration. (a) Instantaneous emptying rate and (b) pressure difference between the antrum and duodenum for TC/T = 0 (the pylorus is unable to close). (c) Velocity vectors during terminal antral contraction and (d) those during antral relaxation. (e) Time-averaged emptying rate and (f) mixing efficiency, as a function of TC/T for two values of content viscosity. (Online version in colour.)
We then varied the duration of pyloric closure from TC/T = 0 (always open) to TC/T = 1 (always closed), while we fixed the timing of the onset of pyloric closure to be the same as the control case. The time-averaged emptying rate is shown in figure 4e for two types of viscosities. As expected, the emptying rate tended to decrease when the duration of pyloric closure increased. The emptying rate was maximum at TC/T = 0: 10 ml min−1 for high-viscosity contents (μ = 1.3 Pa s), and 27 ml min−1 for lower viscosity contents (μ = 4.2 × 10−3 Pa s). The emptying rate for TC/T = 1/6 was slightly lower than TC/T = 1/3 for the low-viscosity liquid. In the case of TC/T = 1/6, pyloric opening began at the end of the terminal antral contraction. The pylorus was fully open during antral relaxation, and the time-averaged emptying rate was decreased by retrograde flow from the duodenum. In contrast with gastric emptying, the duration of pyloric closure had a minor effect on gastric mixing. The mixing efficiency remained nearly constant even when the duration changed from TC/T = 0 to TC/T = 1, as shown in figure 4f. We then examined impaired coordination of pyloric closure with the antral contraction. We gave a delay, TD, in the onset of pyloric closure from that of the terminal antral contraction, whereas we fixed the duration of pyloric closure to be TC/T = 2/3 (the same as the control case). The time-averaged emptying rate and the mixing efficiency are shown in figure 5 as a function of the delay of pyloric closure. The delay in pyloric closure drastically altered the time-averaged emptying rate. A delay of TD/T = 1/8 increased the emptying rate two- to threefold for both low- and high-viscosity contents (see also figure 3c; electronic supplementary material, video 3). A delay of TD/T = 3/8, however, resulted in a negative value in the emptying rate, in particular for the low-viscosity contents (see also figure 3d; electronic supplementary material, video 4). We compare the instantaneous emptying rates between these cases in figure 5c,d. In the case of TD/T = 1/8, the pylorus was opened during the terminal antral contraction, resulting in rapid emptying. On the other hand, when the delay was TD/T = 3/8, the pylorus was only open during antral relaxation. Flow through the pylorus was always retrograde from the duodenum. The time-averaged emptying rate was thus a negative value. Note that the mixing efficiency was altered for the low-viscosity content by the delay in pyloric closure but only slightly. Figure 5. Effects of the delay of pyloric closure. (a) Time-averaged emptying rate and (b) mixing efficiency, as a function of TD/T for two values of content viscosity. (c) Instantaneous emptying rate for TD/T = 1/8, and (d) that for TD/T = 3/8. (Online version in colour.)
To gain a more comprehensive understanding of the effects of impaired coordination, we varied both the duration and delay of pyloric closure, and, hereafter, we present our results in a non-dimensional form. Gastric flow was characterized by two dimensionless numbers in fluid mechanics, the Strouhal number, St = D/VT, and the Reynolds number, Re = ρVD/μ. In this study, the Strouhal number was fixed to St = 1, and the Reynolds number depended on the liquid viscosity; for example, Re = 0.1 for μ = 1.3 Pa s, and Re = 30 for μ = 4.2 × 10−3 Pa s. When the time-averaged emptying rate is denoted by Q, the emptying rate can be written as Q/D2V in a non-dimensional form. Total gastric emptying is a consequence of anterograde flow from the antrum to the duodenum, and, if present, retrograde flow from the duodenum to the antrum. The retrograde component, Q−/D2V, is shown for Re = 0.1 in figure 6a as a function of TC/T and TD/T, where the region on the left of the dashed line indicates cases when the pylorus is open during antral relaxation (longer than half the relaxation period). This region corresponds well to large values in the retrograde component. This result suggests that the pylorus must be closed during antral relaxation to prevent retrograde flow through the pylorus. The anterograde component, Q+/D2V, is also shown in figure 6b. When the pylorus was open during the terminal antral contraction (region left of the solid line), the anterograde component was a large value, resulting in rapid emptying, as shown in figure 6c. The duration of pyloric closure and its coordination with the terminal antral contraction had a minor effect on gastric mixing over the cases examined in this study. The difference between the maximum and minimum values in mixing efficiency was approximately 10% (figure 6d). For Re = 30, we found the same tendencies in the emptying rate and mixing efficiency as for Re = 0.1, with a small difference due to inertia effects (figure 6e,f). Figure 6. (a) Retrograde component, (b) anterograde component, (c) time-averaged emptying rate and (d) mixing efficiency as functions of TC/T and TD/T for Re = 0.1. (e) Time-averaged emptying rate and (f) mixing efficiency for Re = 30. (Online version in colour.)
Finally, we investigated the region where gastric contents are first emptied. We initially positioned 480 000 tracer particles randomly in the stomach, and simulated the movement of these particles for 10 min. The duration and delay of pyloric closure were set to the control values of TC/T = 2/3 and TD/T = 0. We divided the whole stomach into 13 regions based on the distance from the pylorus, and calculated ‘emptying probability’ as the percentage of emptied particles in each region for a given time. Figure 7a shows the emptying probability for content with a high viscosity. More than 60% of the high-viscosity content initially located at the terminal antrum was emptied within 3 min, with approximately 20% remaining in the stomach even after 10 min. Figure 7b shows the initial position of the tracer particles which had emptied within 10 min. For high-viscosity contents, the content along the greater curvature emptied in preference to content located along the lesser curvature. However, for low-viscosity contents, such a tendency was diminished. Low-viscosity contents were mixed homogeneously in the antrum and corpus within a few minutes, and, thus, the content was uniformly emptied from these regions, as shown in figure 7c,d. Figure 7. (a,c) Emptying probability and (b,d) initial position of tracer particles which have emptied within 10 min for (a,b) Re = 0.1 and (c,d) Re = 30. (Online version in colour.)
In this study, we investigated the mechanism of gastric emptying using an anatomically realistic geometry of the stomach and duodenum. The results showed that peristaltic contractions at the proximal antrum produced gastric emptying with an emptying rate of 3 ml min−1 for highly viscous liquids (μ > 0.1 Pa s) under hydrostatic equilibrium. The emptying rate was increased when the viscosity was decreased. The emptying rate ranged from 3 to 8 ml min−1 for liquid contents (0.001 Pa s < μ < 0.1 Pa s). Marciani et al. [35] investigated the effect of meal viscosity and nutrients on emptying rate in healthy subjects using MRI. The emptying rates of low-viscosity (0.06 Pa s) and high-viscosity (29.5 Pa s) meals were 6.1 and 4.7 ml min−1 for non-nutrient meals, respectively, and 4.1 and 3.3 ml min−1 for nutrient meals, respectively. Our results correspond well to these values. When the pylorus was unable to close, our model demonstrated that the emptying rate increased to 10–30 ml min−1, and instantaneous retrograde flow from the duodenum to the antrum occurred during antral relaxation. This situation may be related clinically to pyloromyotomy, pyloroplasty or pyloric botox procedures, which are performed for selected indications including post-surgical gastric drainage, strictures and gastroparesis. Surgical or endoscopic opening of the pylorus may result in dumping syndrome, in which the gastric contents are rapidly emptied [36]. A pylorus that remains open can also lead to bile reflux, whereby bile flows backward from the duodenum to the antrum. Bile is an irritant to the stomach, and, if prolonged, may induce intestinal metaplasia and potentially progress to gastric cancer [37]. We showed that, when the pylorus was open during antral relaxation, retrograde flow from the duodenum to the antrum occurred as a result of the negative pressure gradient generated by recoil of the antral wall. This novel finding may offer a simple mechanism for promoting bile reflux. Notably, the occurrence of dumping and bile reflux was not restricted to the case where the pylorus remained continuously open, as both pathophysiological events could also theoretically occur due to antropyloric discoordination. Dumping syndrome was evidenced when the pylorus opened during the terminal antral contraction phase, leading to a transient major increase in the emptying rate, and bile reflux could occur when the pylorus opened during the antral relaxation phase. Although retrograde flow has been reported even in normal subjects [12,38], the coordination between the pylorus and antrum may have been incomplete for these subjects. A two-dimensional numerical study by Pal et al. [23] suggested the presence of a gastric emptying road, ‘Magenstrasse’, from the pylorus to the fundus on the lesser curvature side. The gastric contents inside Magenstrasse were emptied rapidly compared with other regions of the stomach. However, our three-dimensional model predicted a different behaviour. In the case of highly viscous contents, a ‘road-like’ region appears from the pylorus to the proximal antrum, but on the greater curvature side. In the case of low-viscosity contents, the road-like region disappears, because the gastric contents are mixed homogeneously within a few minutes. While Pal et al. [23] prescribed the time history of the volume of gastric contents, i.e. the emptying rate, we evaluated the gastric emptying produced by peristaltic contractions, which are known to work in combination with pressure gradients to generate gastric outflow [11]. In addition, Pal et al. [23] only considered a high-viscosity content (μ = 1.0 Pa s). Magenstrasse would appear if gastric emptying is dominated by tonic contractions, and the viscosity of gastric contents is high enough. For the present computational modelling and simulation, we assumed some idealized conditions, and they are the limitations of this study. For example, we assumed hydrostatic equilibrium to ignore any hydrostatic pressure difference across the pylorus. In reality, a hydrostatic pressure difference may be present, particularity when the proximal duodenum is empty, driving flow through the pylorus. Thus, our simulation may underestimate the emptying rate. The depth of the contraction can increase for low-viscosity contents [4], and, thus, the propulsion and retropulsion for low-viscosity contents may also be underestimated. We also assumed that gastric contents did not contain any solid components. It is well known that solid particles are not emptied until their sizes become smaller than a few millimetres. If gastric contents contain solid particles, the overall emptying would be slower because of the size effect of the particles [39]. Fluid–structure interaction modelling is necessary for the simulation of emptying of solid food particles. Computational modelling of solid foods will provide detailed information on mechanical variables, such as stress distribution, which are difficult to obtain from clinical data, and will enable us to quantitatively understand solid food emptying in disease states. However, owing to such idealized conditions, we have successfully quantified the isolated effect of peristaltic contractions on gastric emptying. In future, this advance could be a basis for further investigating the combined effects of gastrointestinal motor functions, such as tonic contractions of the stomach, and duodenal motor functions. Furthermore, while some modifications might be necessary, our computational model would be applicable to other digestive organs, such as the oesophagus and intestine. The ability to predict in silico pathophysiological features that are difficult to isolate experimentally is an important outcome of gastrointestinal computational models [40], as shown here by the quantification of the effect of impaired coordination between the pyloric and antral motor functions. Although numerical simulation of the digestive system is still only an emerging field, it could therefore become an effective methodology for delivering a more comprehensive understanding of gastrointestinal physiology and related digestive diseases, in order to guide new diagnostics and therapies. This article has no additional data. No conflicts of interest, financial or otherwise, are declared by the authors. This research was supported by the Japan Society for the Promotion of Science KAKENHI grants nos. 16H03187, 17H02075 and 18K18817, the Riddet Institute, Centre of Research Excellence, New Zealand and the Health Research Council of New Zealand. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4576454. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 10Cystic fibrosis (CF) is a genetic disease caused by mutations in a chloride channel: the CF transmembrane conductance regulator (CFTR). Only individuals with mutations in both alleles of the CFTR gene present the disease. Individuals with one mutated copy do not exhibit CF, but we refer here to healthy lungs (HL) in cases of individuals possessing no CF causing mutations. A mutated CFTR may not function or it may fail to reach the plasma membrane, not only affecting the lungs but also many other organs, such as the pancreas. In the lungs, CF leads to gradual tissue damage, due to repeated inflammation and infections [1,2]. The root cause of CF has been known for 30 years, but an effective treatment for the disease has eluded the best efforts of the biomedical community, although substantial gains in life expectancy and quality of life have been achieved by targeted management of symptoms. Good results were recently obtained with CFTR modulator drugs lumacaftor and ivacaftor. The more common mutations in CF lead to CFTR retention in the endoplasmic reticulum (ER) and posterior degradation. Lumocaftor increases the probability that mutated CFTR passes ER quality control and reaches the plasma membrane. Ivacaftor increases the open probability of the mutated CFTR, which is also affected in several CF mutations. Unfortunately, some mutations remain untreatable [3]. The epithelium sodium channel (ENaC) is also often affected in CF [2,4–6]. The upregulation of this channel function is thought to contribute to accumulation of thick dehydrated mucus, one of the hallmarks of CF and a frequent source of consequent lung problems [2]. Inadequately hydrated mucus and/or mucus over-production also cause a number of other diseases, including primary ciliary dyskinesia, chronic obstructive pulmonary disease, bronchiectasis and asthma [1]. ENaC is furthermore important in non-mucus-related conditions like blood pressure control problems, oedema [7] and heart disease [8]. Given the important roles of ENaC in such a range of conditions, it is imperative to improve our fundamental understanding of this channel in health, disease and with respect to novel treatments. In this work, we focus on the cross-talk between two crucial regulators of ENaC: the extracellular short palate lung and nasal epithelial clone 1 (SPLUNC1) protein and phosphatidylinositol 4,5-biphosphate (PI(4,5)P2). Distinct parallel mechanisms of ENaC regulation have been investigated [9,10], and several models of airway surface liquid (ASL) and ENaC regulation and their impact on ion-driven water fluxes have been proposed [11–16]. However, these models do not account for the synergistic regulation of ENaC and ASL by SPLUNC1 and PI(4,5)P2. Our overall goal is to deepen our understanding of this synergism between ENaC, ASL, SPLUNC1 and PI(4,5)P2 with a simple, targeted mathematical model, allowing the exploration of specific roles and interactions of these regulators. ENaC is a membrane-bound ion channel that transports Na+ ions according to an electrochemical gradient. Functional ENaC is a trimeric structure where one of the subunits is either of type α or δ, a second subunit of type β and an third subunit of type γ. The different subunit types present common tertiary structures and similar amino acid sequences. The most common channel is an αβγ trimer [17]. The role of ENaC in CF is still unclear [18–20]. Most researchers, however, suppose that ENaC function is upregulated in CF. Indeed, several hypotheses explaining this upregulation have been proffered [2,4,7,21–23]. Among them is the postulate that the absence of functional CFTR in the cell membrane causes the acidification of the ASL, which lines and protects the lungs [24–26]. This acidification, in turn, is suspected to inactivate SPLUNC1, a protein involved in protecting ENaC from proteolysis, among other roles. SPLUNC1 also promotes ENaC ubiquitination and disassembly by removing the α and γ subunits from the plasma membrane [6]. Taken together, the lack of CFTR activity is hypothesized to lead to SPLUNC1 loss and elevated ENaC activity or an increased number of channels. We analyse this hypothesis here, because it could lead to truly novel therapeutic applications [27], if it can be confirmed. The dynamics of ENAC is represented graphically in figure 1. Figure 1. ENaC regulation by proteases, SPLUNC1, PI(4,5)P2 and NEDD4-2. When ENaC reaches the cell membrane, it has low activity. Full activation is contingent on cleavage of its extracellular loops by proteases. To expose the loops, ENaC's N-termini need to interact with PI(4,5)P2 in the cell membrane. Extracellular protein SPLUNC1 induces a conformational shift in ENaC, exposing intracellular ubiquitination sites of α and γ subunits. These sites are ubiquitinated by the neural precursor cell expressed developmentally downregulated protein 4-2 (NEDD4-2), which causes channel disassembly and removal of the ubiquitinated subunit from the cell membrane. In CF, ASL is more acidic due to lack of active CFTR. Acidity inactivates SPLUNC1, which in turn reduces the ubiquitination of ENaC by NEDD4-2 and consequently decreases removal of ENaC. (Online version in colour.)
Phosphoinositides are lipids in the cell membranes that have numerous functions [28]. One is the regulation of plasma membrane proteins. Several studies have shown that some phosphoinositides, specifically PI(4,5)P2 and phosphatidylinositol 3,4,5-triphosphate (PI(3,4,5)P3), influence ENaC [28–32]. This control mechanism of ENaC was advanced by Kota et al. [33], who found that when the intracellular N-terminal of γ ENaC binds to these phosphoinositides, a conformational change occurs that exposes the extracellular loops to proteases. Once these loops are cleaved, the channel-open probability (Po) is increased. While ENaC is initially cleaved by furin in the Golgi, further protease action can greatly increase Po [34]. Therefore, if levels of PI(4,5)P2 or PI(3,4,5)P3 are decreased, proteases do not cut the extracellular loops of ENaC and channel activity is reduced. The N-terminal can also be ubiquitinated, which marks ENaC for internalization and degradation. The fact that ENaC is sensitive to anionic phospholipids and ubiquitinated in the same region raises the question whether PI(4,5)P2 or PI(3,4,5)P3 could protect ENaC from ubiquitination. Kota et al. [33] observed that the γ N-terminal is structurally more compact in the presence of PI(4,5)P2, which indicates that connecting to PI(4,5)P2 or PI(3,4,5)P3 might protect these sites from ubiquitination. Here, we investigate the competition between phosphoinositide and NEDD4-2 for the ENaC N-terminus by comparing two models: one where PI(4,5)P2 does not protect ENaC from degradation and another where it does. While these chains of possible causes and effects appear to make a priori sense, it is known that intuition regarding the consequences of changes of any of the components in a complex nonlinear system can be treacherous and often unreliable. We therefore incorporate all pertinent information regarding the regulation of ENaC and ASL by SPLUNC1 and PI(4,5)P2 into a dynamic mathematical model, which permits explorations of every aspect of the system in a rigorous, quantitative manner. In particular, this model allows us to test if the dual effect of PI(4,5)P2, namely increasing the Po of ENaC and decreasing ENaC degradation, is compatible with experimental observations. More generally, the model enables us to investigate quantitatively whether PI(4,5)P2 has a significant impact on ASL thickness, which ultimately determines the severity of the CF lung phenotype. Two slightly different implementations of the model of ENaC and ASL regulation by SPLUNC1 and PI(4,5)P2 are diagrammed in figure 2. Each contains two dependent variables, ENaC and ASL, and accounts for channel production and degradation, as well as ASL influx and efflux. SPLUNC1 and PI(4,5)P2 are independent variables with regulatory roles in the system. Figure 2. Diagrams of the two model variants tested. (a) Model A: PI(4,5)P2 influences only ENaC's Po. (b) Model B: PI(4,5)P2 influences ENaC's Po and protects the channel from degradation. V1, ENaC production; V2, ENaC degradation; V3, ENaC degradation (via SPLUNC1); V4, ASL influx; V5, CFTR-dependent ASL influx; V6, ENaC-dependent ASL efflux; V7, ASL efflux. Solid arrows represent fluxes and dashed arrows regulatory processes. (Online version in colour.)
We simulate 1 µm2 membrane patch, so that changes in ASL thickness are equivalent to changes in ASL volume. Details of model design, calibration, diagnosis and validation are presented in electronic supplementary material. ASL variable represents thickness or height of the liquid, expressed in micrometres, above the apical plasma membrane. ASL has two influxes: V5 is the influx of material that depends on CFTR activity, whereas V4 is a CFTR-independent influx. ENaC activity induces water absorption and reduces ASL thickness through flux V6. V7 is an ENaC-independent ASL efflux, ensuring that a steady state is reached if ENaC activity is blocked. CFTR and ENaC are not main water transporters but regulate ionic species concentration gradients which are driving forces for water transport. Aquaporins and tight junctions allow material to pass through according to such driving forces. To study the hypothesis that PI(4,5)P2 additionally protects ENaC from ubiquitination, we created two model variants: in model A, PI(4,5)P2 does not protect ENaC from ubiquitination (figure 2a), whereas in model B, PI(4,5)P2 inhibits both degradation fluxes of ENaC (figure 2b). Differential equations for both models are presented in table 1.
Model parameters were optimized (see electronic supplementary material), so that model behaviour was consistent with experimental observations retrieved from the literature (table 2). The model was adjusted to replicate steady states in HL (figure 3a), the known half-life of ENaC and SPLUNC1 dose–response (figure 4). To simulate CF phenotype, SPLUNC1 and γ5 were set to zero. Diagnostics suggest that the model is identifiable. Figure 3. Model results corresponding to ENaC activity and ASL time courses in HL and CF. Plots display model A results (lines) superimposed on time course data of ASL thickness (symbols) after perturbations under HL (a) and CF (b) conditions [25,35–37]. The plots corresponding to model B are displayed in electronic supplementary material, figure S3. Results suggest no notable differences between outputs of the two models. (Online version in colour.)
Figure 4. Effects of SPLUNC1 levels in (a) ASL and (b) ENaC in model A. Choi et al. [38] experimentally increased ASL thickness to 25 µm with different SPLUNC1 concentrations and measured ASL thickness after 4 h (black symbols). Blue line represents model predictions for ASL thickness after 4 h that correspond to Choi et al.'s data. The cyan line shows model predictions for ASL thickness when the model reaches steady state. The green curve in (b) represents model simulations for ENaC numbers at steady state. Results from model B are shown in electronic supplementary material, figure S4. (Online version in colour.)
To estimate parameters γ5 and γ6, the model results were compared to data of ASL thickness dynamics in HL airway epithelial cultures after an artificial increase to 25 or 30 µm (figure 3a; electronic supplementary material, figure S3A). Choi et al. [38] conducted a dose–response analysis of SPLUNC1 on ASL 4 h after a 25 µm increase in ASL thickness. We explored different values for SPLUNC1, altered initial ASL thickness to 25 µm and ran simulations for 5000 min, registering ASL thickness at 4 h and at the end of simulations with optimized parameter values for γ5 and γ6. Comparisons between data and model results are displayed in figure 4a. We also registered levels of ENaC at the end of simulations (figure 4b). We implemented the model without saturating Michaelis–Menten or Hill functions, and the simpler power-law representation is sufficient to capture the saturation of ASL thickness and ENaC numbers relative to SPLUNC1 concentrations. To validate the two models, we compared our predictions with experimental observations that were not used to define parameter values. For this purpose, we used data from: (i) ASL thickness time courses above CF airway epithelial cultures; (ii) ENaC activity changes after phospholipase C (PLC) activation; (iii) ASL thickness time courses in human bronchial epithelial cells (HBECs) with different SPLUNC1 concentrations; (iv) ASL thickness with simulated denufosol addition; (v) ASL thickness in HL and CF cells with or without adenosine triphosphate (ATP) addition; and (vi) several drug effects on ASL thickness as documented in Wu et al. [16]. These auxiliary data are presented with more detail in table 2. Both model variants successfully replicated the dynamics of ASL thickness in CF airway epithelial cultures after an initial artificial increase to 25 or 30 µm (figure 3b; electronic supplementary material, figure S3B). ENaC activity was measured under HL and CF conditions with and without activation of PLC [46]. PLC is not a variable in our models, but PLC activity is known to reduce PI(4,5)P2 levels. Thus, we estimated the degree of PI(4,5)P2 reduction due to PLC activation by determining the decrease in PI(4,5)P2 that predicts, for an HL system, a reduction in ENaC activity similar to the experimental observation. We found that reductions of 10% and 9% in PI(4,5)P2 levels, for models A and B, respectively, reproduce the PLC activation effect. This reduction is mild when compared with reports of 50% reduction after PLC activation [45]. However, this numerical discrepancy might be attributable to differences in cell type. We are simulating human airway epithelial cells, whereas Pochynyuk et al. used immortalized mouse renal cells from the collecting duct. Different experimental procedures or activation protocols could also contribute to the discrepancy. Using the estimated reductions in PI(4,5)P2 for both models, ENaC activities for CF without PLC activation are within the reported interval (figure 5a; electronic supplementary material, figure S5A). Although the model predicts a large drop of ENaC activity in CF with PLC activation, results are above the experimentally observed interval. This difference suggests that the PI(4,5)P2 drop may be underestimated. Using a 15% reduction in basal PI(4,5)P2 in model A and 14.5% in model B, all predictions fall within the experimental intervals of data published by Almaça et al. (figure 5b). Figure 5. Consequences of alternative model A parametrization. Both plots exhibit a comparison between model results (bars) and Almaça et al.'s data [46] of ENaC activity for different PI(4,5)P2 levels (means and confidence intervals). (a) PI(4,5)P2 levels were set in order to match the first two bars (HL) with data. Validation correspond to the CF situation (two rightmost bars). (b) Reduction by 15% of basal PI(4,5)P2 levels causes the predictions of the two models to fall inside confidence intervals reported by Almaça et al. [46]. Results from model B are presented in electronic supplementary material, figure S5. (Online version in colour.)
Choi et al. [38] produced a time course for ASL absorption and a dose–response curve after treatment with recombinant SPLUNC1 on HBECs. They first washed each culture apically prior to loading of ASL with varying concentrations of SPLUNC1 (10, 25 and 100 µM). Over the next 8 h, excess ASL was absorbed due to ENaC-mediated ion transport. Comparison between data and simulations is shown in figure 6. Figure 6. ASL thickness with varying SPLUNC1. For each plot, SPLUNC1 concentration is altered to the indicated value and the initial ASL thickness matches experimental values. Dots are data from Choi et al. [38]. Model A simulations are represented as blue lines. Results from model B are shown in electronic supplementary material, figure S6. (Online version in colour.)
Sandefur et al. proposed a model [14] where ATP activates P2Y2 receptors that in turn decrease PI(4,5)P2 concentration via PLC and activate CFTR-dependent and independent secretion. They also simulated addition of denufosol, which is an agonist of P2Y2 subtype purinergic receptors. The addition of denufosol activates P2Y2 receptors, decreasing PI(4,5)P2 and increasing CFTR-dependent and independent secretions. We supposed that PI(4,5)P2 and CFTR-dependent ASL influx will change in the same proportion as the P2Y2 agonist. CFTR-independent ASL influx will rise and fall with the agonist but with only half the magnitude. These settings gave a good agreement between model predictions and data; simulation results are presented in figure 7. The HL data points were used to fine tune perturbations and the CF case is the actual validation. Figure 7. Simulation of denufosol addition to ASL with model A. Denufosol is an agonist of P2Y2 subtype purinergic receptors. (a) Rate of denufosol addition to ASL, where 1 represents the initial amount. (b) Simulation results. Blue and red lines represent model A results in HL and CF, while the light blue and pink symbols show results from a model proposed by Sandefur et al. [14]. Results from model B are shown in electronic supplementary material, figure S7. (Online version in colour.)
Choi et al. [38] measured ASL thickness in normal and CF HBECs, with and without 300 µM of ATP. We simulated addition of ATP as we did before for Sandefur et al.'s data, by decreasing PI(4,5)P2 by 50%, increasing CFTR-dependent ASL influx by 50% and increasing CFTR-independent ASL influx by 37.5%. Choi et al. found ASL thickness in HL tissue to be higher (8.85 µm) than the values used to set up our model (7 µm). Comparisons between data and model simulations can be seen in figure 8. Figure 8. ASL thickness in the absence and presence of 300 μM of ATP on normal and CF HBECs. Bars represent the model A simulation results, intervals are data from Choi et al. [38]. Results from model B are presented in electronic supplementary material, figure S8.
Wu et al. [16] tested their model against data characterizing the effects of several drugs and mechanical stress on ASL thickness. Even though our model is minimal, with only two independent variables, and our objective was never to simulate effects of drugs and mechanical stress, we tested our models against these data. The results are detailed in electronic supplementary material. Although they do not always agree with Wu et al.'s data, the results are encouraging, considering that our models were not developed to simulate the effect of drugs or mechanical stress. After parametrization and validation of both model variants, we assessed if changes in PI(4,5)P2 have a significant impact on ASL. Towards this end, we analysed the sensitivities of model variables relative to small changes in independent variables and parameters at the steady state (table 3).
Almost all relative sensitivities to changes in parameter values are smaller than 1 (in absolute value) in the HL scenario, which means that perturbations are attenuated, which is desirable. Under CF conditions, only the sensitivity of ASL with respect to γ4 barely exceeds 1. These low values show that the model is very robust to reasonable changes in model parameters. Considering independent variables, sensitivities are sometimes called gains. In the model, only one gain stands out in both model variants, namely the gain of ASL with respect to changes in PI(4,5)P2; in CF, this gain is roughly 5.5 in magnitude (table 3), but it is more than doubled in CF versus HL conditions. The fact that ASL has a relatively higher sensitivity to PI(4,5)P2, especially in CF, supports the hypothesis that targeted alterations of PI(4,5)P2 could affect ASL thickness with possible therapeutic benefit for CF patients. The increase in ASL sensitivity to PI(4,5)P2 from HL to CF conditions is associated with a change in flux distribution between the respective steady states (table 4). These distributions are equal for both model variants, A and B, because adjusted rate constants compensate for inclusion of PI(4,5)P2 in fluxes V2 and V3. In HL, CFTR is only responsible for 21% of ASL influx. ENaC activity is also a minor contributor (29%) to the ASL efflux in HL. In CF, V3 and V5 are set to zero to simulate disease conditions. These alterations increase the contribution of ENaC in ASL efflux to 76%. This CF-specific increased role of ENaC in ASL efflux may be responsible for the greater sensitivity of ASL to changes in PI(4,5)P2. Reports in the literature suggest that the increase in ENaC activity in CF could be even higher, with increased ratios of 4.5 within a model of epithelial ion and water transport [14], 3 in human colon cells [51] and 2.2 in bronchial epithelial cells [46].
With respect to ENaC degradation, 56% is due to SPLUNC1 activity in HL. In CF, this flux is set to zero due to inactivation of SPLUNC1 by the acidification of the extracellular space. The sensitivity of ENaC with respect to PI(4,5)P2 constitutes the main difference between model variants A and B. In variant A of HL, where PI(4,5)P2 only influences ENaC's Po, a small increase in PI(4,5)P2 leads to a decrease of almost the same magnitude in the number of ENaC molecules at the cell membrane (table 3). This is an indirect effect, mediated by SPUNC1/ASL feedback regulation of ENaC activity. In CF, where SPLUNC1 is absent, by contrast, this sensitivity is 0. Regarding HL in model variant B, a small increase in PI(4,5)P2 keeps ENaC almost unchanged. In alternative parameter sets that still match other data well, this PI(4,5)P2 perturbation may even lead to a very small increase in the value of ENaC. This shielding of ENaC from changes in PI(4,5)P2 is due to the PI(4,5)P2-mediated protection of ENaC from degradation. Unfortunately, this difference does not allow us to judge which model is biologically more appropriate. On the one hand, if it is better for the cell to change ASL through greater fluctuations in ENaC numbers due to PI(4,5)P2 levels, model A is to be preferred. On the other hand, if it is preferable to have a mechanism that ensures essential constancy of ENaC numbers, while ASL is sensitive to PI(4,5)P2 changes, then model B may be more appropriate. To the best of our knowledge, no experimental evidence can presently elucidate this point. Independent of this lack of decidability, the study of model B delivered an interesting result. Namely, this model variant suggests a qualitative difference in the response of ENaC to PI(4,5)P2 between HL and CF. This observation is interesting because it points to an inversion of a regulatory process in the disease. We will elaborate more on this subject in the discussion. Finally, we assessed how ASL changes with larger perturbations in PI(4,5)P2 and the number of ENaC molecules (figure 9). These results confirm the sensitivity analysis, as ASL changes significantly when PI(4,5)P2 is altered. Again, an increase in ASL is greater under CF conditions. Manipulation of the number of ENaC molecules is also able to influence ASL thickness, but with smaller efficiency. Figure 9. Sensitivity of ASL with respect to PI(4,5)P2 and ENaC in model A. (a) Effects of different PI(4,5)P2 levels on ASL thickness for HL and CF in model A. (b) Effects of different numbers of ENaC channels on ASL thickness, also in HL and CF in model A. In (b), ENaC numbers were changed by varying the ENaC influx in the system, V1. ENaC numbers were artificially inflated in order to reach stabilization of ASL thickness values; thus, the trends for high numbers are very unlikely to occur in vivo. Results from model B are presented in electronic supplementary material, figure S9. (Online version in colour.)
In this work, we propose a model of ASL regulation in the human lung. Available data allow for the determination of almost all parameters and permit the testing of different hypotheses of PI(4,5)P2 control over ENaC. The model analysis reveals that synergism between ENaC, SPLUNC1 and phosphoinositides is an important mechanism for controlling ASL in HL and that this system is compromised in CF. In previous models [11–16], purine metabolism was portrayed as the main controller of PI(4,5)P2, as it is believed to influence many ion channels. Here, we decided to simplify this aspect of the system for five reasons. First, in contrast with these earlier studies, our primary interest is not cyclic shear stress, compressive stress, cilial strain or any mechanical stress to the lung tissue. Second, practical applications based on purine metabolism did not yield positive results. For instance, the P2Y2 receptor agonist denufosol failed to show replicable results in phase III clinical trials [52], and to the best of our knowledge, there are, at the moment, no chemical compounds in clinical trials that target purine metabolism. Third, compounds associated with SPLUNC1, while in the early stages of clinical tests, have been producing positive results [27,53,54]; correspondingly, we focused on the relationships between ENaC, ASL, SPLUNC1 and PI(4,5)P2. Fourth, our results indicate that the components used in our model are sufficient to capture the phenomena of interest here. Thus, we assume, as is implicitly done in essentially all modelling studies, that physiological systems outside the boundaries of our model, in this case purine metabolism, are functioning properly. Finally, it appears that implementing purine metabolism would unduly increase the complexity of the model and cloud possibly the action of the components of interest. Taken together, we consider a somewhat simplified PI(4,5)P2 regulation mechanism that does not depend on purine metabolism. The change in ASL pH, due to CFTR loss of function, has been addressed in the literature with opposing [55–57] and supporting [40,58–62] experimental evidence, and alternative mechanisms to inactivate SPLUNC1 in CF were proposed [63]. Despite these opposing views, our model results remain valid if there is an increase in ENaC activity in CF conditions. Even if the latter is not true, the high sensitivity of ASL to PI(4,5)P2 still supports its use to modulate ASL in CF. One should note that ENaC is subject to multiple additional types of regulation [64], and it could be interesting in the future to expand our model to include effects of protein kinase C and cyclic adenosine monophosphate, purine metabolism, as well as more detailed regulation by NEDD4-2. Extracellular proteases known to activate ENaC [33,34,63] could also be included in the model, since SPLUNC1 protects ENaC from these proteases and their activity could change in CF conditions. More complex models of epithelial ion and water transport are available, but do not facilitate the exploration of the role of SPLUNC1 and PI(4,5)P2 in the regulation of ENaC and, consequently, of ASL. However, we found that these models were not suitable for our needs. For instance, Sandefur et al. [14] and Wu et al. [16] developed detailed, sophisticated models, in which they modulated CFTR and ENaC interactions by activating CFTR and inhibiting ENaC action with increased adenosine concentration, which is insufficient for studying the action of SPLUNC1 on ENaC that is of interest here. They also implemented purine metabolism and the activation of P2Y2 receptors by ATP as the main controller for PI(4,5)P2 levels. We decided to simplify this aspect by assuming that purine metabolism operates normally and therefore making PI(4,5)P2 an independent variable, which allows a more convenient exploration of different values of this phospholipid. It would seem to be mandatory to consider purine metabolism if our focus were on mechanical stresses to the lung tissue, but this is not our objective. Even so, our model turned out to be able to replicate many of the results of the model proposed by Sandefur et al. (figure 7). The simplicity of the model proposed here permits the direct determination of almost all parameter values from limited experimental data. Only γ5 and γ6 had to be estimated through optimization. Indeed, a very appealing aspect of our model is that we did not need to estimate the model parameters of HL and CF conditions separately, but that one parameter set satisfies both conditions by setting two quantities to zero, namely the CFTR-dependent ASL influx and the amount of SPLUNC1 protein. Given this direct derivation of parameter values and the good agreement between model predictions and validation data, we cautiously conclude that the model represents the ASL and ENaC regulation by SPLUNC1 and PI(4,5)P2 quite well. The sensitivity analysis of model B yielded an interesting result. Namely, this model variant predicts a qualitative difference of ENaC response to PI(4,5)P2 from HL to CF. In model variant A, an increase in PI(4,5)P2 leads to reduced ENaC numbers in HL, while ENaC in CF is insensitive to PI(4,5)P2, due to the inhibition of SPLUNC1, that will shut down the V3 flux. By contrast, in model B, an increase in PI(4,5)P2 will cause ENaC to increase in CF, while it decreases in HL. This difference points to the inversion of a regulatory process in the disease. These findings suggest a new hypothesis to explain ENaC upregulation in CF, namely: ENaC upregulation in CF may be caused, at least in part, by an inverted regulation of this channel by phosphoinositides. This finding is interesting because shedding light onto the specific mechanism and role of ENaC regulation in CF might lead to a distinctly new explanation with subsequent applications in CF treatment. Even though CF has been known for a long time, and enormous efforts have been devoted to this disease, a cure for all CF causing mutations still eludes us. This fact may be interpreted as an indication that some mutations may trigger several simultaneous dysregulations. This proposal is original but not mutually exclusive with other ENaC upregulation hypotheses, and it may be a critical piece of the puzzle. Furthermore, while advancements in drug therapies for CF have produced compounds that are effective against some mutations, including the common F508Δ, many other mutations are still not treatable, and a mutation agnostic treatment is still relevant. Strategies moderating ENaC have the potential to fill this role. As our model considers PI(4,5)P2 as an independent variable, it can be easily coupled with models of phosphoinositide pathway that are now starting to emerge [65,66]. This coupling will allow a deeper exploration of potential manipulations of phosphoinositide metabolism leading to a more efficient recovery of ASL in CF patients. Two model variants were explored to test the hypothesis whether PI(4,5)P2, besides its influence on ENaC's Po, may also protect ENaC from ubiquitination. We did not find any notable differences between the two variants when comparing model predictions with validation data. Both were equally successful in reproducing the systems at the HL and CF steady states, as well as the observed ASL dynamics in HL. Thus, the simpler model, where PI(4,5)P2 only regulates ENaC open probability, is sufficient to explain system behaviour. However, this sufficiency does not rule out the protective role of PI(4,5)P2 in ENaC ubiquitination, and if future experimental evidence supports this role of PI(4,5)P2, the more complex model variant may be deemed more appropriate. This secondary role of PI(4,5)P2 could be of practical interest, because the dual regulatory function would make ASL more sensitive to PI(4,5)P2 changes, thereby improving the possible success of PI(4,5)P2 manipulations as a therapeutic approach. This article has no additional data. D.V.O., L.L.F., E.O.V. and F.R.P. designed the project and drafted the manuscript. D.V.O. performed the literature review, retrieved data, created the ASL/ENaC model and performed the analysis. L.L.F. helped in the creation of the ASL/ENaC model. All authors reviewed the manuscript. The authors declare no competing financial interests. This work was supported in part by US grants from NSF (MCB-1517588 (PI: E.O.V.) and MCB 1411672 (PI: Diana Downs)) and NIH (2P30ES019776-05 (PI: Carmen Marsit)) and by an UID/MULTI/04046/2019 Research Unit grant from FCT, Portugal (to BioISI). The funding agencies are not responsible for the content of this article. D.V.O. is the recipient of a fellowship from BioSys PhD programme PD65-2012 (Ref SFRH/BD/52486/2014) from FCT (Portugal). We would like to acknowledge Drs Margarida D. Amaral and Maria Margarida Ramos for helpful discussions. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608434. References
Page 11Leishmaniasis is a family of diseases caused by an intracellular protozoan parasite (genus Leishmania) transmitted by the bite of a female phlebotomine sand fly [1]. Leishmaniasis may be primarily categorized based on three main types of clinical symptoms (cutaneous, muco-cutaneous and visceral). The type of the disease is a result of which species of sand flies, species of parasite, and hosts are present in a region. Cutaneous leishmaniasis (CL), common in Latin American countries, is transmitted by sand flies of the subfamily Phlebotominae of the genus Lutzomyia [2]. The sand flies are infected with various species of Leishmania parasite (e.g. two dominant species in Ecuador are L. (V.) guyanensis and L. (V.) braziliensis), when they bite a natural reservoir including humans [1,3]. Existence of different hosts complicates the transmission cycles as it could change the efficiency of transmission by less competent hosts in the transmission cycle [4,5]. The transmission occurs in natural sylvatic, rural and peri-urban regions, seldom emerging in urban zones because of human movements [6]. There are many Leishmania reservoirs including wild vertebrates like mammals, marsupials and potentially birds [6,7], and domestic animals such as dogs [8,9]. The sand flies also feed on hens as chicken coops are often their shelters but these poultry are not considered as reservoirs of Leishmania [7,10,11]. However, the birds Anser anser and Phasianus colchicus have been mentioned as putative hosts of L. infantum [7]. In the neotropics, the Leishmania spp. sand fly vectors belong to genus Lutzomyia and in a natural state these sand flies inhabit humid forests with soils rich in decaying organic matter, tree holes, fallen logs and burrows of wild mammals [12]. The population density of Lutzomyia species increases in the rainy season. Its flight activity is usually performed at dawn and dusk, which corresponds to the feeding period; it can fly a few metres from the ground and up to 200 metres away [12]. In Ecuador, out of 81 phlebotomine species reported [13] 15 species have been reported as anthropophilic and considered to be potential vectors of human leishmaniasis [14]. According to Kato et al. [3] and Hashiguchi et al. [13], Leishmania guyanensis, L. braziliensis, L. naiffi, L. lainsoni, L. panamensis, L. amazonensis, L. mexicana and Leishmaniamajor-like have been isolated from human samples from tropical and subtropical areas of Ecuador. Ecuador is a country located in northwest of South America that is extremely ecologically diverse and leishmaniasis in it is characterized by both diversity and complexity. Ecuador’s ecology ranges from dry forest in the coast, cloudy forest in the Andes to tropical rain forest in the Amazon lowlands [15], which are typical environments for abundance of different species of sand flies. In Ecuador, the first case of CL was reported in 1920 from Esmeraldas province close to the border of Colombia [13]. Since then the country has reported many outbreaks of CL including the recent one in the small rural town of Valle Hermoso of Santo Domingo de los Tsachilas province in 2010 (figure 1). During the last decade (2001–2011), more than 20 000 human CL cases were recorded through surveillance in Ecuador, ranging yearly from around 800–2000 [13]. There are 22 provinces out of 24 that report leishmaniasis cases in Ecuador. The highest percentage of cases were reported in Pichincha, Esmeraldas and Santo Domingo de los Tsachilas [16]. The most affected province in this period was Pichincha (20% of the total in the country), followed by Esmeraldas (16.4%) and Santo Domingo de los Tsachilas (12%) [13]. In 2014, a total of 1183 cases were reported to Ecuadorian Ministry of Public Health (MSP in Spanish), with 262 cases (22.1%) from Pichincha province, 148 cases (12.5%) from Santo Domingo de los Tsachilas and 136 cases (11.5%) from Esmeraldas (Departamento de Epidemiologia, MSP, 2014). Currently, clinical diagnosis of CL in Ecuador remains the only method to confirm cases and to get officially reported. Hence, these reported numbers are likely to underestimate the incidence because the disease is principally found in the populations living in the remote, rural and forested areas of Ecuador, where transportation and medical care systems are very poor, making it difficult for patients to access limited health clinics, especially in the Amazonian and western, northernmost provinces. In this study, one of the objectives is to estimate case underreporting levels of CL using the 2010 outbreak in the town of Valle Hermoso, which lies in the Santo Domingo de los Tsachilas province of Ecuador. Figure 1. The study site, the town of Valle Hermoso, on the map of Santo Domingo de Los Tsachilas province of Ecuador. (Online version in colour.)
CL remains an important public health problem in Ecuador with cases getting reported regularly from 23 out of the 24 provinces of the country (except in the Galapagos Islands) [13] in spite of enhanced control programmes. Limited studies have been conducted to determine possible reservoir hosts of CL. Some studies from Latin America have identified three new mammalian species with the parasite including the sloth Choloepus hoffmanni didactylus, the squirrel Sciurus granatensis, and the kinkajou Potos flavus [17–22]. Serological studies in the Pacific and Andean region determined CL infection in dogs with the same human strain isolated in the respective region [15,23,24]. Studies have used novel molecular techniques which allow estimation of the biting rate of vectors on various types of hosts, thereby making it possible to determine the host preferences of vectors that influence transmission of the parasite [25]. In CL hyper-endemic area of the coastal region of Ecuador, like Valle Hermoso (Santo Domingo de los Tsachilas province), birds have been found as the main blood meal source of sand flies via molecular technique [26,27]. However, the presence of alternative hosts (potentially birds) has also been shown as a critical risk factor of the CL in the region [28]. Hence, it seems that the presence of avian hosts may be linked with outbreaks in Ecuador. Lutzomyia longipalpis/L. mexicana model has shown that chickens may be able to host Lu. longipalpis parasite population; however, there is no conclusive information as to whether chicken blood is likely to support the development of transmissible Leishmania infections in Lu. longipalpis [29]. Mathematical models have been used to study various aspects of transmission dynamics of leishmaniasis [30]. In particular, models can be used to design control strategies, via analysing a critical threshold quantity, model’s basic reproduction number (R0), interpreted as the average number of secondary cases of infection as a result of the introduction of a primary infection into a completely susceptible population [31–33]. For example, Gorahava et al. [34] developed an optimization model for anthroponotic visceral leishmaniasis control in India and used it to identify an optimal allocation strategy of choosing and distributing insecticide based on the number of human and cattle populations in each district of the affected region. Studies have also shown that the culling of seropositive dogs, the use of insecticide-impregnated dog collars, and the vaccination of dogs significantly contribute to reducing the prevalence of zoonotic visceral leishmaniasis infection in both canines and humans [35,36]. Besides, models have been used to quantify incidence underreporting levels and to study its impact on visceral leishmaniasis transmission dynamics [32]. This last study showed that reported data highly misinterpreted the true incidence levels in districts of Indian state of Bihar. Studies have linked population life cycle with CL transmission dynamics and have attempted to understand patterns of infection. They have used models incorporating vectors, humans, reservoirs, and/or environmental factors [31] to analyse the relationship between deforestation and movement of individuals [37], to illustrate understanding of the life cycle of the Leishmania parasite [38], to study the role of different types of hosts [39] and to estimate CL reproduction number (R0) in ecologically different localities [40]. However, these studies have been either theoretical or have used data which highly underestimate true burden. This is because of two reasons: (a) available data related to CL are limited and (b) reported data are underreported. In the present research, we link theoretically derived quantities with empirical information and quantify true incidence of CL in Ecuador. The manual of procedures for disease control of the MSP-Ecuador recommends determining possible foci of the disease once a case of leishmaniasis has been reported from the region, spraying of dwellings of infected patients and identifying possible reservoirs and its management [41]. However, the basis of these controls need to be throughly and systematically analysed. On the other hand, MSP-Ecuador aims to understand the impact of ongoing interventions in the face of limited surveillance. The goal of this study is to assist and extend the ongoing government efforts in understanding complex cycles of leishmaniasis and estimating its true burden. Specifically, this study uses surveillance and entomological data together for the first time from Ecuador and (i) quantifies infection rates in potential alternative hosts, (ii) attempts to understand the impact of presence of alternative hosts and vector feeding preferences on the transmission dynamics of CL, (iii) estimates case-underreporting levels and (iv) suggests effective control policies for this resource-limited region. The goal is achieved via the development and analysis of a mathematical model, which captures the transmission dynamics of CL infections in the presence of primary (humans) and alternative hosts and limited reporting of cases through surveillance. This study is expected to assist in providing specific actions on vector control and host management interventions as a response to control of CL in Ecuador. We conducted monitoring of sand flies in the town of Valle Hermoso which lies in the Santo Domingo de Los Tsachilas province of Ecuador. Valle Hermoso is a hyper-endemic area for leishmaniasis (figure 1). Phlebotominos were collected during the dry season, in July 2013 and during the rainy season, in March 2014. The samples were captured with the Centers for Disease Control and Prevention (CDC) miniature light traps (John W. Hock, Gainesville, FL). Four traps were set from 18.00 to 06.00 during two consecutive nights in the rainy and dry seasons. The nearest trap was placed 150 m from inhabited houses (peri-domiciliary area) and outward into the forest with a distance of 150 m between them, and the last light trap was placed 600 m from inhabited houses. Specimens collected were killed and stored at −20°C and transported to the laboratory. Specimens were identified, counted and classified into three groups: blood-fed, unfed and gravid females. Sand fly species taxa were morphologically identified using keys from Galati [42] and Young & Duncan [2]. Females with blood meals were easily recognized by the presence of engorged abdomens. Female abdomens were dissected for DNA analyses. DNA was extracted, amplified and sequenced to identify the potential food source and identify parasitic infection in each sand fly. PCR amplifications were carried out first for vertebrate-specific primers (cytochrome B), and then positive samples were amplified for vertebrate prepronociceptin gene (PNOC) and avian DNA (cytochrome B). Positive samples for mammalian DNA were subjected to a primer specific multiplex PCR to identify cytochrome B DNA from humans, dogs, cows, and pigs. Positive samples for avian DNA were tested with PCR specific amplification for domestic chicken cytochrome B. Blood from chickens, humans, dogs, cows, and pigs was used as positive controls [26]. The data were collected from different chicken farms in the rural regions of our study site. Epidemiological case surveillance data were also obtained from Valle Hermoso, a rural region of the province of Santo Domingo de los Tsachilas located in the northwest of Ecuador, which has approximately 10 000 inhabitants. The town is located at an altitude of 307 m.a.s.l. and the average temperature is 25°C. The data consist of incidence from 2009 to 2011 passively collected and registered at the MSP (shown in figure 2) and were used to estimate the model parameters. There were 318 total number of cases reported in these 3 years from Valle Hermoso. Since CL is non-fatal disease, the population from rural and distant areas never comes for treatment and hence CL in this region is highly under-diagnosed. In general, CL cases were reported by the government local healthcare unit. The epidemiological surveillance of CL and data collection were carried out by the Surveillance, Epidemiology Unit of the Ministry of Public Health and the reports are made accessible through the technological data platform, Sistema Integrado de Vigilancia Epidemiologica (SIVE in Spanish) [43]. The detailed historical data were difficult to gather because of many reasons including the lack of notification of resources (both personnel and equipment), the tendency to report only the most serious cases and lack of information from private health centres among others [44]. In Panama, significant underreporting of CL was estimated and it was believed to be attributed to the lack of diagnostic methods and low levels of access to healthcare services [45]. In Ecuador, underreporting is considered between 2.8- and 4.6-fold, based on comparative data with Argentina [46]. Figure 2. Cutaneous leishmaniasis incidence by epidemiological week in the town of Valle Hermoso of the Santo Domingo de Los Tsachilas province from 2009 to 2011. Data from Ecuador’s Ministry of Public Health. (Online version in colour.)
A compartmental type epidemiological framework was proposed to model anthropozoonotic parasite transmission in a community of two hosts (birds as potential host and humans as an alternative host) and a vector species (similar models are developed and analysed in [30,34,47,48]). The model includes an additional compartment that corresponds to the unreported cases, considering that in rural areas there is no access to conventional treatment, and lacks traditional and ancestral knowledge of the disease [15,32,49]. Such models are not only good to capture dynamics of vector-borne diseases (VBDs) but also have a capability of tracking missing information from empirical studies such as underreported cases. The human population was divided into susceptible (Sh1), infected unreported (Ih1), infected reported (Ph1) and recovered (Rh1) individuals, where total human population was Nh1=Sh1+Ih1+Ph1+Rh1. The vector and alternative host (potentially bird) populations were both divided into susceptible (S) and infected (I) subcategories with corresponding total population as N* = S* + I*. The variables related to vector populations were indicated with subscript v and alternative hosts (birds) with subscript h2. Human population: The sand flies bite humans at a constant rate b (defined as average number of mosquito bites received by a host in a unit time, which is assumed constant over seasons). Infected individuals can recover from CL and become susceptible at the per capita rate δ. Not all the infected individuals are identified and reported to the surveillance system. Infected people do not die from CL. Population is assumed to remain constant over time. Reported infected individuals receive effective treatment and recover faster than unreported infected individuals. Vector population: Sand flies bite humans and alternative hosts (birds) at different rates based on the preference for the two hosts. Biting of hosts by infected sand fly might result in successful transmission of the Leishmania parasite. The birth and death rates are assumed to be equal. Susceptible sand flies can also get infected from infectious humans and alternative hosts (birds). Sand flies once infected remain infected throughout their life and also do not die from the infection. Alternative host (potentially bird) population: The preference of sand flies for alternative hosts (birds) relative to humans is defined by parameter αv. Alternative hosts (birds) can get infected due to bites from infected sand flies and infected alternative hosts (birds) can transmit the infection to susceptible sand flies. Additionally, we assume that there is no disease-induced deaths in infected alternative hosts (birds). The population of alternative host (bird) remains constant over time. In order to systematically study each of the components of the modelling framework and develop robust model-based estimates and control implications, the framework was categorized into four different models (Models 1, 2, 3 and 4). Model 1 is described by the complete framework that included two hosts (alternative hosts and humans) and a vector species with underreporting explicitly incorporated in the model (figure 3). Underreporting was not included in Model 2 whereas population of alternative hosts (birds) was not considered in Model 3. Model 4 was the simplest model with neither alternative host (birds) population nor underreporting considered. All the four model systems are explicitly stated in the electronic supplementary material and the parameters are defined in table 1. Figure 3. The flow chart of the modelling framework, that is, Model 1. (Online version in colour.)
Control of CL depends currently on passive case detection and rapid treatment and in some locations vector management. The World Health Organization (WHO) [56] is now suggesting local public health departments to improve collective effectiveness of interventions while using existing resources for the reduction of morbidity. However, to improve existing control of disease, the evidence for the effectiveness of different prevention and intervention strategies is needed. The integrated ecosystem approach to human health is a more comprehensive and coherent approach to controlling CL. In this study, we use an integrated disease control model-based approach that includes improvement in social participation at different levels, vector and reservoir management programmes, and therapeutic interventions. The integrated strategy can allow us to understand the role of lack of knowledge of CL among communities which can be improved via educational programmes, increase participants’ adherence to the intervention, and active participation and surveillance of local healthcare individuals. However, in order to have sustainable long-term interventions, systematic understanding of the prevention and therapeutic programmes is required. Here, we study the effectiveness of various interventions using a CL transmission dynamics mathematical model for the case of Ecuador. Educational programmes may be effective in controlling CL because such programmes can educate individuals when, how and where to seek medical assistance and consequently reduce further transmission of infection and even improve underreporting [57]. It is believed that underreporting for CL in Latin America may range from 2.8 to 4.6 fold of reported cases [46]; however, there is no systematic study from Ecuador that measures the level of underreporting in the region. The impact of interventions are captured in our model via various model parameters. For example, a parameter γ2 (per capita rate of case reporting) can be altered to study the role of surveillance efforts via educational programmes (or active reporting) on patterns of CL. The impact of insecticide spraying and larvae management programmes can be studied through our model via changing estimates of the parameters μv (mortality of the vector population) and Nv (total vector population), respectively. The increases in insecticide spraying rate in a community can result in higher death rates of sand flies (μv) whereas larvae control programmes can limit the development rates of sand flies from larvae to adult stage and thus can reduce the total population size of adult vectors (Nv), which are only modelled in the equations. On the other hand, the use of insecticide-treated bed nets by individuals can reduce average transmission rate of a VBD. This is because insecticide kills the vectors and hence the number of vector-bites received by those individuals becomes zero. In our model, the use of impregnated bed nets by population is captured via transmission efficiency parameter β1. Host management programmes (such as culling and treatment of alternative hosts and reservoirs) can also be used for controlling CL and the role of such programmes is incorporated in the model via the parameter Nh2 (total alternative host (bird) population). The models are evaluated here via thorough mathematical analysis, estimations of model parameters using empirical data, and parameter sensitivity and uncertainty analysis. The details of analyses are given in electronic supplementary material. Model 1 equilibrium analysis resulted in computation of two equilibria (disease free and endemic equilibrium) and a threshold quantity, basic reproduction number, R0. The disease free state and the basic reproduction number of the model (system of equations (A.1) in electronic supplementary material) are given, respectively, as: E0=(Nh1,0,0,0,Nv,0,Nh2,0) and R02=(bβ1μv)(Nh1αvNh2+Nh1)⏞sandfly−human interaction(bβ2γ1+γ2+μh1)(Nv(αvNh2+Nh1))⏞human−sandfly interaction+ (bβ1~αvμv)(Nh2αvNh2+Nh1)⏟sandfly−alternative host interaction(bβ2αvμh2)(NvαvNh2+Nh1)⏟alternative host−sandfly interaction2.1 The expression of R0 indicates that it depends on parameters related to human interventions (through γ2 and β1), alternative host and reservoir management (through Nh2), and vector control (through Nv and μv). See previous section for more details on implementation of interventions in the model. The analysis suggests that if R0 > 1 the CL will become endemic whereas if R0 < 1, CL can be controlled. The electronic supplementary material provides some mathematical details related to the endemic states. Some parameters of the model are taken directly from the literature. Estimates of the rest of the model parameters such as the probability of successful transmission of infection from vector to human host given a bite, β1, the probability of effective transmission from human host to a vector given a bite, β2, and the per capita reporting rate, γ2, were unknown for Ecuador and thus were estimated indirectly by fitting the four models separately (see table in figure 3) to the 2009–2011 CL cumulative incidence surveillance data (see data in figure 2) via the WLS procedure (WLS; see electronic supplementary material for details). Parameter sensitivity analysis (SA) is used to quantify the effects of variation in uncertain model input parameters on the model outputs [58,59]. SA allows for prioritization of the most influential parameters on the model output, to the least important parameters, and quantifies those intervention strategies that influence the system most. Here, we also quantify uncertainty generated in the output variables as a result of measurement errors in data via the parameter estimation procedure using observational data and the model. In order to analyse the behaviour of the model, we first estimated parameters and then used parameter estimates to simulate various scenarios of dynamics of CL infection. The parameters are estimated using two different procedures: (1) finding estimates in the literature and fixing them to the obtained value for further model analysis and (2) parameters for which estimates cannot be found from the literature were estimated using WLS method and cumulative incidence data. The data used in the estimation include our collected entomological data and information gathered from similar studies from Ecuador and other leishmaniasis affected countries. Since most parameters were estimated using data from Ecuador, the model results were considered as a representative for the whole of Ecuador. The estimates of parameters are collected in table 1. Epidemiological parameters related to human host: It is assumed that all the populations (human, vectors and alternative hosts) are constant and were estimated using the census data and surveys. According to the survey carried out by the National Institute of Statistics and Census, INEC (for its acronym in Spanish), the estimated human population size of the town of Valle Hermoso in 2010 was around 10 000 inhabitants. Hence, Nh was taken as 10 000. Since the data corresponding to CL from Ecuador were limited, we used a prospective longitudinal survey of CL from Peru to estimate per capita rate of loss in immunity (δ), and it was found to be as 0.0033 per day [54]. The manifestations of CL include the presence of lesions, which may later ulcerate. Lesions may appear in humans about 7 days after receiving a bite from an infectious sand fly and since time to access treatment was unknown, using expert opinion from the region, we assumed that the average time to access treatment after clinical manifestation was around 8 days. Hence, we used an average time to reporting among individuals receiving treatment as around 15 days (i.e. 1/γ2 = 15 days). Often patients do not approach public healthcare facilities and take much longer time to recover via natural spontaneous resolution of infection. It is found that such patients recover in an average of 15 months, hence, 1/γ1 = 15 × 30 = 450 days [55]. Pentavalent antimony derivatives such as metglumine antimoniate and antimony sodium stibogluconate, recommended by WHO, are used for treatment of CL in Ecuador. Treatment may be repeated three times at intervals of 15 days [55]. The doses applied to children and adults may vary and are based on reference to an individual’s body weight. Hence, we estimated recovery rate of individuals reporting and receiving treatment as σ = 1/(15 × 3) = 1/45 per day. The biting rate of sand flies is assumed to be equal to the number of sand fly bites received by an individual per day and is estimated as 0.2856 per day [51]. Parameters related to alternative hosts (birds): The alternative hosts in our model are assumed to be chickens. According to the poultry breeding manuals [60,61], the time that a hen remains in hatcheries is from 60 to 70 days; poultry that are raised in country houses can live for 90 days on average. Since we collected data from different chicken farms in the rural regions of our study site, we assumed that the life span of a chicken(1/μh2) is 90 days. Parameters related to sand flies: A study was conducted in the Valle Hermoso town of the Santo Domingo de Los Tsachilas province of Ecuador to determine the sources of blood meal for phlebotomine sand flies (figure 4a,b). A total of 442 female sand flies were collected and classified as non-engorged and engorged. The 106 engorged females were identified morphologically, and selected for blood meal identification by PCR technique. A total of 84 sand flies of these were positive for blood meals from birds, primarily chickens. Since humans and chickens were the most preferred hosts for sand fly species in our samples, we assumed that these sand flies prefer to bite only humans and birds (chickens). Hence, we estimated the feeding preference for the bird hosts is αv ≈ (84/22) = 3.8182. Figure 4. Percentage of fed sand fly by each host. (a) Sand flies collected in Valle Hermoso. (b) Type of sand fly blood meals. (Online version in colour.)
In our dataset, we found that out of the 106 samples of engorged females, 42 were positive for leishmaniasis and 22 were positive for blood meals from mammals. Since there were no time-dependent data available for sand fly feeding and preference behaviours, we took initial estimate of probability of transmission of parasite from a vector to a human host as β1 ≈ (22/106) × (42/106) = 0.08223. However, the parameter was also formally but indirectly estimated via fitting model to the incidence data as explained in §3.1.2. Using the sand fly data, out of the 106 engorged females, 84 were positive for blood meals from birds and 33 of these (who fed on birds) were also positive for leishmaniasis. Hence, in the absence of detailed data from the region, we took estimate of the probability of transmission of parasite from a vector to a bird host as β1~≈(33/106)=0.3139. Since some estimates were less precise, we also carried out parameter uncertainty and SA on some model outputs. The entomological laboratory observations were used to estimate the daily mortality rate of an adult sand fly and it was taken to be around 1/14 per day [52]. We used WLS procedure to estimate parameters of the four different models (see figure F.1 for Model 1, F.2 for Model 5 and table 2). The four models are:
The model (out of these four models) that best fitted the surveillance incidence data was identified via a metric, selection score, defined as the Euclidean norm of a vector whose entries are the coefficient of variation of parameters that are estimated [62]. The selection scores for the models were computed and the best fit model was identified based on the lowest selection score value (see last column of table 2). Model 4 followed by Model 2 were found to be the best models using the reported incidence data from Valle Hermoso. In the models, we obtained point estimates, standard error, and confidence interval of γ2 (only for Models 1–3), β1, and β2 (table 2). The distribution of R0 was also estimated using uncertainty quantification and fitting procedure (figure 5). Since Model 1 was the most comprehensive among the four models, it was used to report the mean estimates of R0 for CL in Ecuador and we estimated mean (R0) = 3.9. The uncertainty of R0 was obtained via parameter uncertainty analysis. Each parameter in the analysis was sampled 10 000 times from its respective distribution and R0 values were computed (figure 5a). This procedure was repeated for 1000 iterations to estimate robustness in the probability that R0 is greater than certain value (figure 5b). We also found that it takes on average 14.5 days (i.e. 1/γ2 = 1/0.0689) for a symptomatic case to be reported in Ecuador presently and approximately 75% (≈γ2/(γ1 + γ2) = 0.0689/(0.0689 + 0.0222)) of the symptomatic cases are eventually reported. Using Model 4 (the best fit model), the transmission probabilities from alternative hosts (birds) to sand flies, humans to sand flies, sand flies to birds and sand flies to humans are estimated as β2αv = 0.07 × 3.8 = 0.27, β2 = 0.07, β~1=0.3, and β1 = 0.03, respectively (table 2). Reporting level using Model 5, i.e. under assumption that alternative hosts are dead end hosts, we estimated its value as approximately 62% (≈γ2/(γ1 + γ2) = 0.0371/(0.0371 + 0.0222); hence, expected underreporting is around 38% with 95% CI=(29%, 47%); figure 6). Figure 5. Distribution of R0 from uncertainty analysis for Model 1. The estimated R0 value is 3.9. (Online version in colour.)
Figure 6. Distribution of percentage of underreporting of cases using Model 1 and Model 5. (Online version in colour.)
In this section, we provide implications from model-based interventions (details are given in §2.2.3). Simulations are carried out to study the role of feeding preference (αv) in patterns of CL in humans for low, medium and high regions of transmission (that is, for low, medium and high values of probability of transmission from a vector to a human host, β1, or to an alternative host (bird), β1~). Under certain conditions, when preference for alternative host (bird) relative to human host (αv) is increased, the equilibrium prevalence of CL in humans decreases; however, the rate of decrease depends on the level of endemicity of the region and transmission probability to host (that is, decrease is different for β1 and β1~ with much larger variations for β1~; figures A.3(a) and A.3(b) in the electronic supplementary material). As expected, increasing the value of β1 increases the number of human cases overall. However, increasing the transmission probability to alternative hosts (that is, increasing β1~) decreases the number of human cases overall because of dilution effect (electronic supplementary material, figure A.3(b)). Moreover, in low transmission areas, increases in sand fly preference for alternative hosts (birds) result in slower rate of decrease in human CL prevalence but faster rate of decrease for prevalence in alternative hosts (birds). As expected, prevalence in humans drastically decreases as reporting of cases improves (that is, (γ2/γ2 + γ1) increases); however, this decrease in prevalence could be significantly enhanced with the implementation of vector control programmes (through increases in μv) up to a critical value of increase in the reporting (electronic supplementary material, figure A.1). Increases in reporting (that is, increases in γ2; because of improvement in diagnosis, treatment, or surveillance) and decreases in alternative host density (Nh2; via host and reservoir management programmes) result in decreases in reproduction number, R0 (figure 7). In other words, improved reporting can eliminate the disease locally (via R0reduction); however, the rate of decrease depends on alternative host (potentially bird) density in the region. Similar trends are observed when β1 (probability of transmission from a sand fly to human) and Nv (density of vectors) are both varied to see their impact on R0. That is, control programmes that reduce effective contacts of sand flies with humans such as distribution of impregnated bed nets, can reduce R0significantly; however, rate of reduction depends on the density of the sand flies in the region (electronic supplementary material, figure A.2). Figure 7. Variation in the mean value of R0 (for Model 1) when γ2 and Nv are varied. (Online version in colour.)
A global SA was performed to identify the parameters with the greatest influence on the model output, R0. In this study, we use partial rank correlation coefficient (PRCC) as a standard measure of global sensitivity [63]. We first verified assumptions of PRCC SA including confirming that R0 varies monotonically with respect to each of the model parameters. A PRCC was obtained for 13 model parameters to understand the sensitivity of each of the parameters on R0. The SA showed that the model parameters such as human mortality rate (μh1), infectious period (1/γ1), rate of reporting (γ2), alternative host (bird) mortality rate (μh2) and sand fly mortality rate (μv) statistically have a significant influence on estimating R0 with each of them being negatively correlated (negative PRCC between them) (see figure 8). Model parameters such as μh1, σ, δ, μh1 and γ1 are statistically insignificant to the estimation of R0. Moreover, parameters b, β1, αv, β1~, β2 are also statistically significant and are found to have a positive correlation with R0. Sand fly related parameters are most influential in estimating R0, followed by parameters for alternative hosts and parameters related to humans. In order to eliminate CL, these results strongly support the need to improve the monitoring and controlling of the sand flies, and the alternative host (potentially birds) population. In other words, this SA provides a specific intervention/monitoring strategy for controlling leishmaniasis in endemic areas in Ecuador. The analysis showed a strong negative correlation between μv and R0 and a strong positive correlation of β1, β2 and β1~ with R0 (figure 8). The local analytical computation of sensitivity indexes (collected in the electronic supplementary material) further validates numerical global sensitivity results. Figure 8. Partial rank correlation coefficient (PRCC) indexes from parameter sensitivity analysis of R0 (for Model 1). (Online version in colour.)
Phlebotomine sand flies transmit Leishmania that affects humans, animals, and potentially bird alternative hosts worldwide in many tropical countries including in Latin America. In this study, we developed a data-driven modelling framework to study the role of sand fly feeding behaviours and host preferences in the transmission dynamics of leishmaniasis in Ecuador, when existing surveillance system is passive. Historically, cases of CL in Ecuador have fluctuated over years, with some years showing major epidemics in different regions of the country. However, some of the epidemics never came to light in time probably in part due to considerable underreporting. Moreover, the Leishmania infection has neither been systematically monitored in humans nor in the other hosts across Ecuador. Recently, some researchers in Latin America have suggested a rising trend in the number of CL cases in Ecuador, and attributed this to a lack of access to medical treatment, increased human migration into leishmaniasis endemic areas, and/or to ecological changes triggering vector adaptation to different alternative hosts. There are many Leishmania natural hosts and reservoirs all over the world. In particular, it is known that the transmission of parasites of the genus Leishmania involves a large diversity of mammalian hosts. The identification of Leishmania reservoirs and hosts is usually carried out by blood meal analysis in the insect vectors. In Latin America, Brazil leads the effort in conducting studies on blood meal preferences among sand fly species and found that Lutzomyia longipalpis (the vector for transmission of leishmaniasis) also has a preference for birds [64]. We carried out a study in a coastal region of Ecuador that showed that birds, primarily chickens, are the preferred blood source for several species of sand flies [26]. The study by Anaguano et al. [26] further serves as an evidence of feeding preference of sand flies on birds during the dry and wet seasons. A modelling framework is therefore developed to understand the dynamics of CL transmission between local sand fly species and its natural and preferred hosts [31]. The research goals of this study are to understand the role of alternative hosts (potentially birds), existing case surveillance system, and the current control programmes in the transmission dynamics of CL using epidemiological and entomological data from Ecuador. The modelling framework was classified into five different models in order to systematically study the research goals. The models presented in this study describe the interactions between the sand fly species and its two main hosts: humans and alternative hosts (birds). We collected two types of datasets for carrying out model parametrization: (i) incidence data from 2009 to 2011 outbreak in the town of Valle Hermoso obtained through the Ecuador public health surveillance system and (ii) feeding behaviours of sand flies to estimate host preference of vectors, via sand fly data collected by our research group. Our results suggest that there is a wide gap between the reported and the total infected cases (around 40% of cases were unreported), confirming our hypothesis that there is huge underreporting of the disease in the region. Underreporting is a result of multiple factors including difficult accessibility to endemic areas, registration system failures, and ineffective diagnosis and treatment of patients in private medical centres [65]. The mathematical analysis resulted in computation of the threshold quantity, the basic reproduction number (R0), a useful number for understanding the transmissibility of CL and designing of various intervention strategies. This threshold quantity is often used to describe the condition for the existence of an outbreak. If R0 is less than 1, the disease will decline and eventually die out, and if R0 is more than 1, there may be an outbreak or epidemic. For our models, it is found that estimates of R0 depend on multiple parameters including density of alternative hosts (birds), feeding preference of sand flies, mortality rate of sand flies, density of sand flies, and coverage of surveillance system. Using data from Valle Hermoso, we estimated R0 to be around 3.9, suggesting potential for regular outbreaks in future and need for significant improvements in existing intervention programmes. Our estimate of R0 is higher than estimate reported from neighbouring countries Colombia (R0 = 1.3 [48]) and Peru (R0 = 1.9, if domestic dogs are primary reservoirs [66]), which could be due to a difference in the reporting system and/or ecology of the subregions. Nevertheless, the results suggest that improved reporting and early treatment of cases can control the disease drastically; however, the rates of decrease in cases will depend on intensity to control alternative host (potentially bird) density in the region. We observed a direct relationship between the increase in feeding frequency for potential alternative hosts (birds) (or the decrease in time to reporting of a case to surveillance system) and decreases in the CL prevalence in human population. The parameter SA of R0 showed that vector control programmes are the most effective interventions for CL elimination when compared with the control programmes that focus on alternative host management or are directly related to humans. The analysis on equilibrium prevalence levels was also performed. It showed that the alternative hosts (birds) can play a significant role in the dynamics of the transmission of CL in Ecuador. These results further confirm the necessity of improving the monitoring and controlling of the sand flies as well as the alternative hosts. Our modelling and data analysis provide a novel approach for investigating the transmission and control of CL in Latin America. We have (a) estimated for the first time transmission rates between different hosts (preferred and alternative) and vector species in the presence and absence of alternative reservoirs, (b) estimated underreporting levels in the presence and absence of alternative reservoirs (estimates comparable to those reported in the literature [67]), (c) estimated CL reproduction in Ecuador, and (d) suggested that control of alternative reservoir hosts is critical to achieving optimal results of the vector control programme. The study also provides new dimensions to understanding the dynamics of VBDs in general: (a) it gives a method to estimate underreporting levels of a VBDs based on current incidence from surveillance, and data on vector feeding preferences and host competence, (b) it provides a procedure to link host-related empirical information to a dynamical model, and (c) it suggests how qualitatively characteristics of VBD prevalence can be studied when data are scarce and how uncertainty in prevalence can be quantified. In conclusion, we present the first model-based estimates of CL underreporting and infection rates for potential alternative hosts (birds) in Ecuador. We collected some entomological data and used novel data-driven approach to parametrize the model. Our estimates constitute useful procedure for decision making and prioritization of CL control interventions in the endemic areas of Ecuador. Our research clearly shows that there is a need for improvement in data collection on different avian species and for implementation of active surveillance system to thoroughly evaluate long-term CL patterns in Ecuador. Vector density is critical to the establishment of CL in new susceptible regions. We anticipate that in near future surveillance programmes will integrate these methods and results in their systems. The current methodology should be further developed to address its limitations and provide more accurate estimates but this is dependent on the collection of detailed data. In the future, we would like to collect and use surveillance data from other provinces of Ecuador in order to comprehensively validate model results for the whole country, identify and suggest effective sampling techniques to collect fine-grained sand fly related data in an effort to determine feeding preferences, and develop mechanisms for sampling birds to obtain data for understanding CL transmission efficiencies among alternative host species. An approval for conducting the sampling study for collecting blood meal analysis on sand flies was obtained from Institutional Review Board (IRB) at the Instituto Nacional de Investigacion en Salud Publica (INSPI) of Ecuador in 2013. The case surveillance data was obtained from Ministry of Public Health, Ecuador. The details of the data are presented within the study. Additional materials are collected in electronic supplementary material. All authors contributed in the development of study idea and writing of the initial manuscript. D.M.V., M.P., E.J.M.-B., M.C. and A.M. developed and analysed the mathematical models. D.M.V., V.C., P.P. and A.M. collected the data and verified it. M.P., E.J.M.-B., M.C. and A.M. fitted the data to the model. L.A. and A.M. performed sensitivity analysis on model outputs. M.P., A.M. verified the empirical and analytical methods. A.M. supervised and finalized the findings of this work. All authors discussed the results and contributed to the final manuscript. We declare we have no competing interests. This project has been partially supported by A.M.’s grants from the National Science Foundation (NSF, grant no. DMPS-0838705, and grant no. ACI 1525012). Partial funding is also from SENESCYT-PIC grant no. 0014. This research was initiated at the Mathematical and Theoretical Biology Institute (MTBI) of Arizona State University, Tempe. The authors wish to thank MTBI’s sponsors (NSF, NSA, Sloan Foundation and ASU), the INSPI-Ecuador, and to express their appreciation to the MTBI staff and fellow researchers, especially Prof. Carlos Castillo-Chavez, whose suggestions improved the manuscript. FootnotesElectronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.4616081. References
Page 12Cell proliferation is essential for a range of normal and pathological processes. Many different mathematical models of proliferation have been proposed [1–7]. It is often assumed that cells proliferate exponentially dM(t)dt=λM(t),M(t)=M(0) eλt,1.1 where M(t) is the number of cells at time t and λ > 0 is the proliferation rate.The eukaryotic cell cycle consists of four phases in sequence, namely gap 1 (G1), synthesis (S), gap 2 (G2) and mitosis (M) (figure 1a). A key assumption implicit in equation (1.1) is that the cell population is asynchronous, meaning that the cells are distributed randomly among the cell cycle phases (figure 1b), yielding a constant per capita growth rate, (1/M(t)) dM(t)/dt = λ. By contrast, a population of cells is synchronous if the cells are in the same cell cycle phase (figure 1c), or partially synchronous if only a subpopulation of cells is synchronous (figure 1d). In this case, the synchronous cells divide as a cohort in discrete stages, producing a variable per capita growth rate. In addition to the implicit assumption of asynchronicity, classical exponential growth models and generalizations thereof [8] do not account for subpopulations, and predict monotonic population growth. Figure 1. C8161 experimental data and multi-stage model solution. (a) The cell cycle, indicating the colour of FUCCI in each phase. (b–d) Asynchronous, synchronous and partially synchronous cells. (e–h) Images of a proliferation assay with FUCCI-C8161 cells. Scale bar, 200 µm. (i) M(t). Linear regression of ln M(t) versus t gives R2 = 0.99. (j) R(t), Y(t) and G(t). (k) Q(t). Experimental data are shown as discs and the model solutions as curves. (Online version in colour.)
Here we provide new experimental data from two-dimensional cell proliferation assays in which the cell growth appears exponential as in equation (1.1). Unexpectedly, however, we observe oscillatory subpopulations arising from a phenomenon we refer to as inherent synchronization. We reveal the normally hidden inherent synchronization by identifying subpopulations based on cell cycle phase, employing fluorescent ubiquitination-based cell cycle indicator (FUCCI) [9]. FUCCI enables visualization of the cell cycle of individual live cells via two sensors: when the cell is in G1 the nucleus fluoresces red, and when the cell is in S/G2/M the nucleus fluoresces green. During the G1/S transition, called early S (eS), both sensors fluoresce and the nucleus appears yellow (figure 1a). We explain these seemingly inconsistent observations by applying a multi-stage mathematical model for cell proliferation. Previous studies of cell synchronization using FUCCI induce the synchronization using methods including serum starvation, cell cycle-inhibiting drugs, environmental pH or contact inhibition [10–15]. Our assays are prepared using a standard method [10] normally thought to produce asynchronous populations, and we take utmost care to ensure that there is no induced synchronization in our cell cultures due to serum starvation, low pH or contact inhibition (electronic supplementary material, S1). Over three cell lines and four independent experiments, however, we consistently observe inherent synchronization. Neglecting synchronous subpopulations can have important implications for experiment reproducibility. For example, the accurate experimental evaluation of cell cycle-inhibiting drugs is highly dependent on the cell cycle distribution of the cell population [10,16]. In a partially synchronous population, the drug may have a delayed or advanced effect compared with an asynchronous population, depending on the cell cycle position of the synchronous cells. Generally, the presence of synchronization may affect the reproducibility of experiments that investigate cell cycle-dependent mechanisms, such as changes in migration and drug response. Revealing any synchronization with quantitative techniques like FUCCI will lead to a better understanding of these mechanisms. Our experimental data are time-series images from two-dimensional proliferation assays using three melanoma cell lines, C8161, WM983C and 1205Lu [15,17,18], which have mean cell cycle durations of approximately 18, 27 and 36 h, respectively [15]. Four independent experiments are performed for each cell line. Live-cell images are acquired at 15 min intervals over 48 h. Images from one position in a single well of a FUCCI-C8161 proliferation assay at 7, 16, 25 and 34 h show red, yellow or green nuclei corresponding to the phases G1, eS or S/G2/M (figure 1e–h). We quantify the population growth by counting the total number of cells in each image (electronic supplementary material, S1) to give M(t) at time t (figure 1i). The total number of cells appears to grow exponentially over 48 h, supported by the best fit of equation (1.1) (electronic supplementary material, S1) since we have R2 = 0.99 from the linear regression of ln M(t) versus t. The temporal variations in the numbers of cells in the subpopulations R(t), Y(t) and G(t) with red, yellow or green nuclei (figure 1j), respectively, where M(t) = R(t) + Y(t) + G(t), are oscillatory. In an asynchronous population, the subpopulations would exhibit monotone growth. The oscillations we observe, however, reveal that the cells are partially synchronous. To explore the inherent synchronization further, we group cells in eS and S/G2/M together, since eS is part of S, and consider the ratio Q(t) = R(t)/(Y(t) + G(t)) (figure 1k). Synchronization is clearly evident in the oscillatory nature of Q(t). Note that the troughs at 7 and 25 h and the peaks at 16 and 34 h are separated by 18 h, which is the approximate cell cycle time for C8161. We can visualize the oscillations in these two subpopulations (figure 1e–h), where the ratio of the number of red cells to the number of yellow and green cells is lower at 7 and 25 h and higher at 16 and 34 h. Equation (1.1) and related generalizations [8] cannot account for the oscillations in these subpopulations. Similar observations are made for further examples of this cell line, and the two additional cell lines (electronic supplementary material, S1). We quantitatively confirm the presence of oscillations in Q(t), arising from inherent synchronization, for all 90 datasets by calculating the discrete Fourier transform of the Q(t) signal, and identifying the dominant frequencies (electronic supplementary material, S1). These results confirm that all 90 experimental replicates display oscillatory subpopulations that are inconsistent with traditional exponential and logistic growth models. We employ a multi-stage model of cell proliferation [19] which can describe synchronous populations. The model assumes that the cell cycle durations follow a hypoexponential distribution, which consists of a series of independent exponential distributions with different rates. To apply this model, we partition the cell cycle into k stages, Pi for i = 1, … , k, where the duration of each Pi is exponentially distributed with mean μi. If T is the mean cell cycle time then ∑i=1kμi=T. The stages Pi do not necessarily correspond to phases of the cell cycle, but instead are a mathematical device which allows control over the variance of cell cycle phase durations in the multi-stage model, whereby more stages correspond to less variance in the phase durations for a cell population. If we let the transition rates be λi = 1/μi and consider the partitioned cell cycle P1⟶λ1P2⟶λ2⋯⟶λk−1Pk⟶λk2P1, we arrive at a system of differential equations describing the mean population Mi(t) in each stage [19], dMi(t)dt=2λkMk(t)−λ1M1(t),for i=1,λi−1Mi−1(t)−λiMi(t),for i=2,… ,k.2.1 Note that M(t)=∑i=1kMi(t). If k = 1, equation (2.1) simplifies to equation (1.1). Within the 48 h duration of our experiments, none of the cell lines exhibits contact inhibition of proliferation, consistent with the typical loss of contact inhibition in cancer cells [20]. Consequently, a carrying capacity is not incorporated into the model.We solve equation (2.1) numerically with the forward Euler method, and estimate the parameters by fitting the solution to our experimental data (electronic supplementary material, S1). Using 18 stages for each of the three cell cycle phases described by FUCCI, giving k = 54, we obtain M(t) (figure 1i), R(t), Y(t), G(t) (figure 1j) and Q(t) (figure 1k), which all correspond well with the experimental data. In particular, the multi-stage model replicates the oscillations in R(t), Y(t), G(t) and Q(t), a feature that is not possible with traditional exponential models. While the multi-stage model can replicate the oscillatory subpopulations, the model is unable to predict all features of the inherent synchronization in a cell proliferation experiment due to variable initial conditions, as the inherent synchronization is a stochastic phenomenon which likely arises from cell division and intercellular interactions. The model can, however, be used to predict general features of the inherent synchronization of each cell line. Our new experimental data demonstrate that cell populations may appear to grow exponentially despite subpopulations exhibiting oscillatory growth arising from normally hidden inherent synchronization. We use standard experimental methods thought to produce asynchronous populations; however, all of our proliferation assays exhibit inherent synchronization. We use FUCCI to track cell cycle progression, which is necessary to confirm cell synchronization. As the standard exponential growth model cannot account for subpopulations with oscillating growth, we use a multi-stage mathematical model of cell proliferation to replicate oscillations in population growth. Our results are important because revealing any synchronization will help to better understand cell cycle-dependent mechanisms, such as changes in migration and drug response. Without quantitative techniques like FUCCI to probe the cell cycle, synchronization and its effects on experimental outcomes and reproducibility may remain hidden. All experimental data are available in the electronic supplementary material documents. All algorithms required to replicate this work are available on GitHub at https://github.com/ProfMJSimpson/Vittadello2019. All authors designed the research. S.T.V. performed the research. All authors contributed analytic tools and analysed the data. S.T.V. wrote the manuscript, and all authors approved the final version of the manuscript. N.K.H. and M.J.S. contributed equally. We declare we have no competing interests. N.K.H. is a Cameron fellow of the Melanoma and Skin Cancer Research Institute and is supported by the NHMRC (APP1084893). M.J.S. is supported by the ARC (DP170100474). We thank the editor and four anonymous referees for helpful comments. Footnotes†These authors contributed equally to the study. Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608440. References
Page 13Global incidence of symptomatic dengue is estimated at 50–100 million cases annually, with burden concentrated in low- and middle-income countries [1,2]. While many organizations advocate various dengue control measures, these settings have limited resources and many competing options to improve quality of life; thus, potential control efforts must be prioritized. One option, the only currently licensed dengue vaccine, CYD-TDV (commercially: Dengvaxia), presents a complicated assessment: there is potential benefit, but also safety risks and associated mitigation costs. Dengue infections elicit complex immune responses, particularly in regions where people typically experience multiple infections. There are four known dengue serotypes; infection by one confers apparently lifelong immunity to it, and temporary immunity to others. Disease threat varies substantially by infection number: primary infections are generally asymptomatic, and when symptomatic are rarely severe; secondary infections are more often symptomatic, and more often severe; and post-secondary infections are almost always asymptomatic. This pathogenicity pattern is caused by antibody-dependent enhancement [3,4]. To avoid enhancement, vaccine development has focused on products effective against all serotypes. Though CYD-TDV initially appeared to achieve this goal [5,6], subsequent work concluded that the vaccine acted more like a silent natural dengue infection [7–11]: enhancing disease risk in seronegative (i.e. no prior dengue infection) recipients while being efficacious for previously infected recipients [12]. These disparate outcomes pose ethical challenges. A multi-model comparison study estimated that using CYD-TDV in high-burden settings would reduce both moderate and severe cases overall [13]. These findings informed initial recommendations by the World Health Organization (WHO) to consider CYD-TDV for settings with high seroprevalence in the target age for routine vaccination, with a minimum target age of 9 years old to increase the likelihood of past infection [14]. Continuing observation in trial populations confirmed increased risk of severe outcomes in seronegative recipients [15]. The Strategic Advisory Group of Experts on Immunization (SAGE) suggested avoiding this risk by verifying prior infection with serological testing [16] and WHO revised its recommendations accordingly [17,18]. Practically, the revised guidance necessitates a point-of-care rapid diagnostic test (RDT); as of July 2019, no such test exists, precluding test-then-vaccinate strategies. At US$78 per vaccinated individual, CYD-TDV is a marginal investment in many settings [13], so adding costs would seemingly decrease its attractiveness. But screening can plausibly optimize use, limiting vaccination to the individuals likely to benefit. To provide a decision tool for such investment, we developed a model of the relationship between three pertinent costs: secondary infections, vaccination and testing. Local decision-makers can use this approach for their specific circumstances to determine if CYD-TDV is worth further consideration. The model has deliberately generous assumptions, providing a simple way to reject CYD-TDV for a region, but additional work, using more realistic assumptions, is required to determine whether CYD-TDV is sufficiently beneficial. Using this model, we found that test-then-vaccinate strategies generally provide health benefits compared with both non-vaccination and vaccination without testing, but not necessarily outweighing the additional costs. We evaluate the balance of benefits and costs using return-on-investment (hereafter ROI; net benefit per unit cost), which also enables comparison against other development options. We found that test-then-vaccinate strategies typically yield their highest ROI when testing starts younger than the currently recommended 9 years. We also found ROI improves with periodic re-testing of initially seronegative individuals. We derive ROIs using two limiting assumptions. We ignore transmission, and thus indirect vaccination benefits, and assume interventions work deterministically. Treating dengue incidence like an environmental risk is justified in endemic settings, given the predicted limited impact on transmission from CYD-TDV [13]. Assuming that the intervention is deterministic provides upper benefit limits. Because more realistic assumptions will reduce benefits, local authorities may reject CYD-TDV (e.g. for insufficient ROI) with this framework, but would need a more detailed model (e.g. incorporating test sensitivity and specificity based on real trials, incomplete vaccine efficacy in seropositive recipients) to justify positive ROI estimates. The electronic supplementary material provides derivations; the R package, denvax, implements the analyses [19]. We represent dengue disease with three infection outcomes—primary like, secondary like and post-secondary like—each with cost reflecting their respective disease risk and severity. Note that these average costs include all outcomes, from asymptomatic infections to death. We assume secondary infections have the highest cost, and post-secondary infections have zero cost. We assume CYD-TDV acts like a silent natural infection, preventing one of these outcomes, and testing reveals an individual’s infection history. Figure 1 shows life trajectories under different interventions. Figure 1. Lifetime outcomes by intervention. Trajectories shown for no vaccination, vaccination without testing and vaccination with multiple testing. Each path represents a possible life history, resulting in health outcome and intervention costs weighted by share of population following that path. For detailed branching probabilities, see electronic supplementary material, figures S2–S4 and S7–S8. (Online version in colour.)
We calculate individual lifetime costs, using average primary and secondary infection costs (F, S) weighted by relevant life history probabilities denoted NPX{A} (probability of N out of X lifetime infections by testing age A). Costs also include testing and, potentially, vaccination (T, V) during the intervention window. Strategies are defined by initial testing age and maximum allowed tests (L) at a rate of one per year. The model denotes the seroconversion probability between age A and A + 1 as C{A}, which determines the average test count. We estimate these probabilities with the exposure model described in the next section. We considered two test mechanisms: binary, detecting only the presence of past infection, and ordinal, detecting the number of past infections. Because the ordinal test also eliminates vaccination costs for people with multiple past infections, it is always more effective (ignoring cost). Though ordinal tests are theoretically better, we expect binary tests are more realistic, and assumed them for our general results. Using these assumptions, we derived equation (2.1). The left side has intervention costs: weighted average number of tests administered, 〈n(A, L)〉 (equation (2.2)), and probability of vaccination, PV†{A+L−1} (equation (2.3)), for a particular strategy. The right side is the lifetime difference in health costs. For all interventions with testing, first infection cost, F, cancels. Therefore, we can generalize across settings by expressing intervention costs relative to secondary infection costs: ν = V/S and τ = T/S. ⟨n(A,L)⟩τ†−PV†{A+L−1}ν≤(P2+− 0P2+{A+L−1}+ 2+P∀), 2.1 ⟨n(A,L)⟩=1+ 0P∀{A+L−1}(L−1)+∑i=0L−2C{A+i}0P1+{A+i}(i+1)2.2 andPV†{A+L−1}=1− 0P∀{A+L−1}.2.3 ROI is the net benefits (difference in health outcome costs minus the intervention cost) per intervention cost, P2+− 2+P∀− 0P2{A+L−1}⟨n(A,L)⟩τ†−PV†{A+L−1}ν−1≥0,2.4 for positive returns.To identify circumstances where adding testing increases intervention benefits, we compared vaccination with and without testing, which produces a similar equation, but which depends on F, ⟨n(A,L)⟩τ∗+(1−PV∗{A+L−1})ν≤( 0P1+− 0P2+{A+L−1})− 0P1+FS.2.5 We model dengue exposure in annual increments: each year, an individual is potentially exposed. We divide the population into risk groups, low and high. Thus, the exposure model has three parameters: pH, population fraction at high risk; and the annual avoidance probability for low (sL) and high (sH) risk groups. The probability of being seropositive at age A is thus P+{A}=ρH(1−sHA)+(1−ρH)(1−sLA).2.6 We use a maximum likelihood approach to fit this model to age-seroprevalence data, then use the parameters to simulate exposure histories. For each life-year, we uniformly select one of the four serotypes to cause exposures, then probabilistically expose individuals based on their risk. Exposure leads to infection, unless the individual was (i) previously infected by this serotype or (ii) infected in the previous year. We estimate lifetime outcome probabilities by aggregating many simulated individual histories. In these sections, we demonstrate the types of analyses available to the decision-makers using the models described above. Because each region will differ, in costs, epidemiology and decision criteria, we cannot present a singular recommendation. We illustrate detailed application of the framework with two examples including particular epidemiology and costs, but these are demonstrations, not conclusive results. We applied our model across a range of epidemiological settings and potential relative costs. Figure 2 shows test-then-vaccinate strategies that start after age 4, testing annually until a recipient is seropositive and thus vaccinated or reaches a maximum age (up to 20 in these results). For each epidemiological setting and relative cost, we report the lifetime value for the optimal strategy (i.e. the pair of initial and maximum testing ages that maximize ROI). We can view the same results from a different angle to understand trends in initial testing age and maximum number of tests. Figure 3 highlights the general trend that testing earlier is better, but this effect depends on epidemiological setting and costs. Some regions have a minimum number of tests for positive ROI; others, a maximum. We found trend reversions in several cases; while atypical, they highlight the need for context-specific analyses. Figure 2. Lifetime ROI surfaces across settings. Increasing transmission (columns left to right) generally increases ROI, but only for strategies where testing starts young enough. Disparity (rows) combines the high-risk population size and how much additional exposure they endure. By the middle disparity setting, the high-risk individuals are effectively exposed every year, so shrinking that population in higher disparity settings must be balanced by increasing low-risk exposure probability to maintain the seropositivity rate; more low-risk exposure increases ROI opportunity. (Online version in colour.)
Figure 3. Sensitivity to number of tests. This area of epidemiological and economic parameters illustrates how there may be higher return with more tests (left-most), a minimum number required for positive return (middle) or a maximum (right-most), depending on test cost. (Online version in colour.)
While the general results suggest trends, they also identify the need for region-specific analyses. To that end, we implemented the model in an R package, denvax, which can be used with local epidemiological and economic data. We demonstrate applying the approach with two practical examples for Malaysia and Peru, using epidemiological data from CYD14 [20] and a long-term study of Iquitos, Peru [21], and crude economic assumptions from previous studies [13]. We use the serological data to fit the two-risk, constant-FOI model, and then estimate the coefficients for equation (2.4). For this example, we assume disease and intervention costs, but decision-makers would estimate these from their regional data. Given local policy, disease cost might be based on a variety of measures, such as hospital costs, lost productivity or willingness to pay. Vaccination cost estimates could be based on quoted prices, and delivery costs derived from existing programmes with similar delivery strategies, e.g. human papillomavirus vaccine. Though costs are not yet available for an improved RDT, decision-makers could use the model to understand the maximum affordable test cost given other assumptions. Here we assume that vaccination (i.e. full dose regimen and service delivery) costs US$78 per fully vaccinated child in both locations [13]. We assume testing costs of US$5 per recipient per year of testing (likewise full testing and service delivery). Finally, we assume the regions differ by secondary infection cost, using the South East Asian and Latin American societal costs from [13], US$86 and US$223, respectively. Thus for equation (2.4), the test and vaccine cost fractions are {τ, ν} = {0.06, 0.91} for Malaysia and {0.02, 0.35} for Peru. Since we are assuming costs, we focus on ROI for varying initial and maximum testing ages in figure 4. The settings produce very different surfaces, and highlight the importance of initial testing age when evaluating strategies. Given the demonstration data, no strategy produced a positive ROI in Malaysia, even with the ordinal test, and more testing only decreases ROI. Peru presents a different story: test-then-vaccinate strategies are beneficial if started young enough, and repeat testing improves ROI for all ages in the range considered, even for initial ages where single testing has a negative ROI. Figure 4. Practical comparison. ROI trends for two settings with seroprevalence between 70% and 80%. Using the assumed vaccination and testing costs, we find that a low secondary infection cost, S, and high exposure disparity (as assumed for Malaysia) results in negative ROI. However, in settings with high S and low exposure disparity (as assumed for Peru), ROI is positive when vaccination starts young enough. We show results for both binary and ordinal tests. While the more optimistic ordinal test can be substantially better (as shown for Peru-like results), that advantage may not be enough to make the intervention worthwhile (as shown in the Malaysia-like results). (Online version in colour.)
We demonstrated an approach that identifies CYD-TDV vaccination scenarios worthy of further investigation, and implemented a framework for local authorities to assess their potential return-on-investment using region-specific epidemiology and costs. It is impossible to provide universal answers to how or whether CYD-TDV should be used, but it is possible to describe a limiting relationship among a small set of factors and ascertain general trends. We identified two such trends that should inform work on test-then-vaccinate strategies. First, repeat testing may improve cost-effectiveness over single testing; for some settings, this occurs even with relatively high testing costs. Second, we found routine test-then-vaccinate programmes often provide better ROI when targeting younger recipients, including those below the currently recommended 9 years old. Since our model uses several optimistic assumptions, it is only appropriate for ruling out CYD-TDV in unfavourable settings, not for conclusively supporting its use. Justifying CYD-TDV requires more model realism. Future work should refine vaccine performance, which we know to be imperfect even in seropositive recipients and which has uncertain durability. Likewise, even the gold-standard tests for detecting prior dengue infection provide imperfect classification, particularly in the presence of other circulating arboviruses. Should a point-of-care RDT be licensed, it is unlikely to have better performance than the current gold standard [22]. Including these details will thus necessarily lead to lower ROIs than those estimated by our model. However, incorporating insights from our model may help recover benefits. For example, we found annually repeating testing generally has superior performance, and under our assumptions biennial testing would be even better. This suggests repeat testing with lower frequency may be more cost-effective in more realistic models as well. Similarly, the substantial ROI improvement for younger intervention ages suggests that the current age guidelines for vaccination should be revisited if safety can be guaranteed by a highly specific test. Despite current licensing, CYD-TDV trials included younger participants and, after controlling for seropositivity, found safe and efficacious outcomes [15]; both the original and re-issued WHO guidance indicate that the 9-year-old limit is to ensure sufficiently high seroprevalence in the target population. Accounting for these potential advantages properly will require more detailed data. Repeat testing may enable economies of scale (e.g. cheaper per unit tests), but will impose additional costs (e.g. record-keeping and service logistics). A model with repeat tests will also need to represent how individual test results are correlated. A lower routine intervention age will also affect test performance and vaccine efficacy. These concerns should be addressed in test development, so that data are available to inform future modelling work. Multiple dengue vaccine candidates are currently in trials [23,24]. While they may prove durably tetravalent, given the complex dengue immunology and vaccine development history, there is a risk that these new vaccines will also have quirks. CYD-TDV, the only currently licensed vaccine, is flawed but reasonably understood and available now. We provide a clear tool to determine where it is not useful, but we also show that following the WHO-recommended test-then-vaccinate strategy for CYD-TDV can improve cost-effectiveness in many settings while satisfying clinical safety and ethical requirements, if a low-cost, suitable performance test can be developed. Whether that improvement is sufficient to warrant deploying the intervention will require additional work and detailed local analyses. All relevant source code is available from https://gitlab.com/cabp_LSHTM/denvax/. C.A.B.P. conceptualized and implemented the research and analyses. C.A.B.P. developed the visualizations and formal derivations, with contributions from S.C. and T.J.H. C.A.B.P. conducted the economic analysis, with contributions from K.M.A. and S.F. C.A.B.P. drafted the manuscript, with all authors contributing to revisions. Aside from C.A.B.P., author order is alphabetical by last name. We declare we have no competing interests. C.A.B.P. is supported by EBOVAC3. S.F. and S.C. are supported through a Sir Henry Dale Fellowship jointly funded by the Wellcome Trust and the Royal Society (grant no. 208812). K.M.A. is supported by NIH/NIGMS R01 GM109718. T.J.H. is supported by NIH/NIGMS U54 GM111274. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608953. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 14Biological systems exhibit a variety of unexpected, sometimes paradoxical, behaviour. Examples include catastrophic shifts [1] at tipping points in ecosystems [2], phase changes in polyphenic insects [3] and hormesis in toxicology, whereby a cell or organism exhibits a biphasic response when exposed to increasing amounts of a substance or external conditions [4–6]. Ecological paradoxes involving interactions between two or more species include coexistence of competing species [7] and the ‘paradox of the plankton' [8,9]. Other examples arise from repeated applications of pesticides that have unexpected consequences because of Volterra's principle, when an intervention in a predator–prey system that removes predator and prey in proportion to their population sizes increases the prey population [10]. In addition, a food type may become rarer in a diet even when it becomes more abundant [11] and, in single species systems as discussed here, incorrect use of pesticides may not control pests effectively and can also lead to rapid increases in the number of pests, thus inducing bigger outbreaks [12]. Hormetic (paradoxical) effects (HPEs) also pose significant challenges for decision-making in treatments of cancer [13–15] and neurodegenerative diseases [16], and in the management of nutrition [17] and ecotoxicology [18]. Strategic applications of hormetic effects have shown promise in optimizing the management of agro-ecological systems with pesticides [19,20] or harvesting [21–23] in relation to variations in endogenous regulatory mechanisms, intervention timing and dose–response specifics. However, there is a gap separating experimental designs and field observations from simple mathematical models that can be parametrized by experiments and which, when applied, can also exhibit all of the above paradoxical behaviour. Here, we fill this gap by deriving a simple unifying three-parameter model for discrete generation single species populations. It is based on the Ricker equation [24] for continuous population dynamics between observation intervals and is interrupted by an external dose at a particular time between the observations. The model has three critical parameters: the intrinsic growth rate, the dose–response and the dose timing of interventions [25], from which we can see that the time factor plays a fundamental role in designing proper experiments, and in understanding hormetic effects. We hypothesized that the dose and dose timing can interact in our new model, which would thus be able to reproduce a wide range of hormetic phenomena, and our analytical results did indeed reveal hormetic biphasic dose and dose timing responses either in a J-shape or an inverted U-shape, yielding a homeostatic change or a catastrophic shift. A multi-parameter bifurcation analysis showed that hormetic effects occur in many parameter regions due to different interactions between intrinsic growth dynamics and the strength and timing of external stimulations; hence many more hormetic effects in nature than those currently recognized should be expected. We also use our model to show that hormetic effects can be significantly enhanced under multiple low-level stimulations [26] within one generation, and this enhanced effect may reduce threshold levels, including for the hormetic zone, as well as reducing both the maximum response and inhibition. This shows that multiple and repeated low-level stimulations may not only provide insights into observations of hormetic effects more clearly and quickly, but also reduce associated risks such as those involved in dosage decisions for medical treatments. Data fitting of our model to laboratory and field data demonstrates the effectiveness of our framework for using experimental data to inform intervention strategies for species with discrete generations under multiple impulsive interventions. Consider discrete generations of a single species population, modelled by Nn+1=f(Nn,r,p,θ),2.1 where the population size at the (n + 1)th generation is determined from its previous generation, subject to internal continuous dynamics with the intrinsic growth rate r. The parameter q=1−p with p∈(0, 1] (survival rate) is characterized by pesticide or drug dose or effectiveness and describes the pesticide or drug efficacy and is closely related to (but not equivalent to) the applied dosage of pesticides or drugs at n + θ, θ∈[0, 1]. Thus, for convenience, we call the parameter q the dose–response, and the parameter θ is accordingly called the dose timing response. Developing a realistic formulation of f to describe the combined effects of internal regulation and the dose–response and dose timing response of external simulations on hormesis has long been regarded as a challenging task [18]. Here, we addressed this challenge by using analytical and piecewise methods [27–30] (electronic supplementary material, §1) to derive a formulation of f by incorporating impulsive external stimulation into the Ricker model [24]. Our formulationf(N, r, p, θ)=pNexp[r(1−NK(θ+(1−θ)pexp(r(1−NK)θ)))],2.2 henceforth referred to as the hormesis Ricker model (HRM), with K being the carrying capacity, provides a closed-form description of how the dose–response and dose timing response of external stimulations affect the reproductive capacity. The HRM clearly shows how intraspecific competition affects population growth after the pesticide application at time θ, which is markedly different from the models previously proposed [22,23]. From those publications, we can see that a linear combination relation between the growth function before a pesticide application and the growth function after an application has been assumed. This could largely reduce the effects of the intraspecific density regulation on the reproductive capacity once the external stimulation occurs at time n + θ. Moreover, this simplifying assumption cannot really explain the essence of hormetic and paradoxical effects, which can be verified based on the theoretical analyses shown in the electronic supplementary material.The two special cases are θ = 0 and θ = 1, which result in the following two discrete models: Nn+1=pNnexp[r(1− pNnK)] andNn+1=pNnexp[r(1−NnK)]. These two formulations correspond to control measures being applied at the beginning and end of a generation, respectively, and, obviously, these two special models could have the same dynamics as those of the classic Ricker model [27–29]. In particular, the above two special discrete models describe the iterative relationship between the number in a population at two successive generations, but do not reflect the population dynamics within the two generations. However, the practical problem is that the external stimulation is often applied within the two generations, rather than at the start or end of the two generations. This is precisely what our new model (2.2) focuses on, i.e. the impact of the external stimulation at specific times within the two generations on paradoxical and hormetic effects. Therefore, the dose–response and dose timing response must act together, which means that in order to reveal the important factors affecting the occurrence of hormetic effects it is necessary to consider the two factors simultaneously. The importance of the dose timing response (i.e. θ∈(0,1)) has been addressed theoretically in the electronic supplementary material.We assume that model (2.1) has a stable equilibrium N∗(r,q,θ) at which the homeostatic state is normalized to unity when q=0 and θ=0, that is, N0∗(r,0,0)=1. The existence and stability of all possible equilibria can be found in the electronic supplementary material. Hormesis related to the dose–response, characterized by low-dose stimulation and high-dose inhibition, is described by the following paradoxical effect: ∂N∗(r, q, θ)/∂q>0 for small q, ∂N∗(r, q, θ)/∂q<0 for large q, and N∗(r,q,θ)<1 when q is large enough. A field experiment was conducted at the Seven Mile Camp experimental base of Yuanyang County of the Henan Academy of Agricultural Sciences in 2012. There were seven experimental fields arranged and numbered sequentially, each with three replicates giving a total of 21 plots. Each plot was 20 m long and 10 m wide with seven replicates such that the total area involved was about 4200 m2. The cotton planting array pitch was 1 m between plants with rows spaced 0.28 m apart. Each plot was separated from its neighbour by 2 m, with corn planted in the gaps for quarantine. Different thresholds for spraying were used in each field according to previous control experience. The gradient of the action threshold was 1, 2, 5, 10, 20, 40 and 60 infested heads/100 plants for invoking spraying in fields 1–7, respectively. The numbers of Apolygus lucorum (Hemiptera: Miridae) bugs were recorded and damage estimated every 4–6 days from 28 May to 5 September 2012. When the pest numbers exceeded the given action threshold in each field, the pesticide was applied before the next day (every unit of 2.5% permethrin was dissolved in 1500 units of fogging liquid). In table 1, South 1–7 represent the seven fields. Based on the above action thresholds, the timings of pesticide applications for the seven fields are italics in table 1.
We emphasize that our experimental field dataset described above is not a hormetic dataset. It is used only for confirming that our model can be fitted successfully to data with single or multiple pulsed external stimulations within each pest generation. In order to confirm that the model can be fitted to hormetic data, we analysed two published datasets. The first concerned effects of applications of the herbicide glyphosate on the growth and yields of chickpea Cicer arietinum (fig. 1b in [31]). The second involved applications of deltamethrin to an insecticide-resistant strain of the maize weevil Sitophilus zeamais (fig. 4 in [32]). First, we assumed that the growth rate of the pests in our field experiment depended on the number of cotton plant squares [33], such that r(t) = r¯f(t + τ), where τ represents the phase difference between the numbers of squares and pests, f denotes a function for the number of squares with respect to time t, and r¯ is the growth rate coefficient. The number of squares with respect to the days after planting is based on data in [33]. Hence, the data for squares can be extracted (as shown in figure 1a), and the function can be obtained by using the B spline method. As the earliest date for the pesticide spraying was 17 July, we let the initial time of our model be 17 July. The initial number of pests was the same as for our data. The least-squares method was used to estimate the unknown parameters p; K; r¯; τ, and the estimations and fitting results for all seven fields are given in table 2 and figure 1b–h. Figure 1. Case study. (a) The function f for determining the growth rate of the pest population during the plant period. (b–h) Results of data fitting for fields South 1–7. (i) Effects of dose–response on the pest population number with parameter values K=125.6677,r¯=0.006,τ=19.874, determined by field South 3. Red circles in (b–h) represent points when the pesticide was not applied. (Online version in colour.)
Moreover, the published hormetic datasets could be fitted by using similar methods. To do this, we chose the dose–response function p=e−ρ×Ds to estimate the unknown parameter ρ rather than p, where Ds denotes the glyphosate dose in figure 2a and deltamethrin dose in figure 2b. Note that the carrying capacity K for both hormetic datasets is determined by the first data point (i.e. Ds = 0). The dose timing θ was determined by the experimental design, for example, the various glyphosate doses were applied four weeks after chick pea emergence, and then the data points were collected after 21 days of glyphosate applications. Therefore, the dose timing θ for glyphosate hormesis is about 4/7, which is confirmed by our data fitting and parameter estimation in figure 2a. Similarly, the dose timing θ for deltamethrin hormesis is confirmed in figure 2b, and it is in good agreement with the experimental design. Figure 2. Hormetic data fitting. (a) Hormetic effect of glyphosate on growth of chick pea after 21 days spraying, measured by root length. The estimated parameter values are r=4, θ=0.53, ρ=0.09. (b) Hormetic effect of deltamethrin on predicted population size of maize weevil. The estimated parameter values are r=4.999, θ=0.1066, ρ=8.08.
In this section, we show how the hypothesis raised in the introduction that the HRM can reproduce a wide range of hormetic phenomena with general applicability is realized. Homeostatic changes and catastrophic shifts due to low-level stimulations can be evinced by compensation as a new equilibrium (figure 3a) is established or by switching from one stable state to another larger stable state (figure 3b), where at a relatively large population size, the reproductive capacity is effectively enhanced when a low-dose external stimulation is applied. The population's intrinsic reproductive capacity is not fully expressed under natural conditions, but a low-dose external perturbation may result in hormetic effects such that the population size is pushed beyond its previous homeostatic state to a new larger equilibrium. The establishment of a higher level equilibrium, as shown in figure 3b, is achieved under the dual effects of catastrophic bifurcation and the higher equilibrium stability induced by the low-dose stimulus. Figure 3. Homeostatic changes and catastrophic shifts. The homeostatic state reaches an altered equilibrium (a) and experiences a catastrophic shift to a larger steady state (b) when the population is challenged by a hormetic stressor for the HRM. (c–f) A sample of single-parameter catastrophic bifurcation diagrams, where a blue/red line represents a stable/unstable equilibrium. The saddle-node or flip bifurcations with respect to dose timing response (θ), dose–response (q) and growth rate (r) show the occurrence of hormetic effects for a wide range of parameter space. (g–n) Two-parameter equilibrium bifurcations. White areas denote where no positive equilibrium exists; grey areas indicate where unstable unique positive equilibria exist; green areas indicate single stable equilibria; magenta areas show bistable areas with three equilibria; blue areas represent where three equilibria exist of which only the smallest one is stable, and yellow areas show where three equilibria exist of which only the largest is stable. (Online version in colour.)
The catastrophic bifurcation diagram figure 3c reveals the role of the dose timing response θ, in the occurrence of hormetic effects. For low-level stressors, a slight incremental change to θ may induce a catastrophic transition to a larger, alternative, stable state. The saddle-node or flip bifurcations with respect to one of the three parameters clearly show the occurrence of hormetic effects for a wide range of parameter values (figure 3c). We have also conducted a two-parameter equilibrium bifurcation analysis to examine the synergistic interaction of internal regulation, dose–response and dose timing response. The analysis showed that bi-stability occurs only when q is relatively small (the magenta areas in figure 3g–i)). Further illustrations based on three-parameter bifurcation analyses are reported in the electronic supplementary material. Note that parallel analyses based on the Beverton–Holt model [34] cannot produce hormetic effects, confirming the importance of the internal regulation mechanism for hormesis to take place (electronic supplementary material, §S1). When the carrying capacity is normalized to one (K = 1), a necessary condition for hormetic effects to occur is f(1, r, q, θ)>1, i.e. the low-level stimulation reduces intraspecific competition to enhance intrinsic reproductive capacity to exceed unity. When this happens, a sufficient condition can be derived (electronic supplementary material, table S1) for the system to shift from a homeostatic state to a higher level equilibrium as r < rc, with the threshold rc being given by θ and q, i.e. rc=1/θ[ln(θ/(1−θ)p)+2]. The stability analysis, summarized in electronic supplementary material, table S2, shows how (under the above-threshold condition), low-level stressors induce homeostatic changes/catastrophic shifts. In particular, bi-stability and catastrophic shifts may occur only when the parameter pair (r, p = 1 − q) is within a certain region in the rp-parameter space that is separated by the curve r+ln(p)=2 into two parts. Within the region 0<r+ln(p)<2, we can observe different homeostatic modulations: if r < 1 (i.e. N0∗(r, 0, 0)=1 is stable), then r+ln(p)<2 holds, and the stable population level can be pushed beyond the normalized value of one to reach a new larger equilibrium N1∗>1 by the application of a low-level stimulation. If r > 2 (i.e. N0∗(r, 0, 0)=1 is unstable), then slightly increasing the stimulation can lead to r+ln(p)<2, inducing the stability of the new larger equilibrium and yielding hormetic effects. Note that, from electronic supplementary material, table S2, we observe that in the region r+ln(p)>2 the system can also be stabilized at the new larger equilibrium N1∗>1 or N3∗>1. The occurrence of paradoxical effects is closely related to the mechanism towards hormesis. The discussion summarized in the electronic supplementary material, tables S3 and S4 for the occurrence of hormetic effects shows that a more refined parameter space is needed, in comparison with those for stability conditions. In particular, electronic supplementary material, table S4 confirms that the dose timing response of external stimulations may have a stronger influence on populations with large intrinsic growth rates. The two-parameter bifurcation diagrams, figure 4a–h, show how r, q and θ together influence the stability of equilibria and their variations (i.e. the sign of ∂N∗/∂q) when the dose changes (electronic supplementary material, table S3). For example, figure 4j shows that increasing the low-level stimulation before reaching the maximum response will result in a stronger paradoxical effect within the hormetic zone. Moreover, the smaller that θ and q are, the larger are the green areas and, for a relatively small intrinsic growth rate r, hormetic effects only occur for extremely small θ and q. However, when the growth rate r is large, the hormetic effects are presented in complex patterns, as illustrated in figure 4d,h. Figure 4. Paradox and hormetic-like biphasic dose–responses. Two-parameter bifurcation spaces for HPEs related to the efficacy of the external stimulation q in (a–h) for the HRM. Green areas represent where there is a unique stable equilibrium with ∂N∗/∂q>0, which indicates that the level of a single stable equilibrium will be increased as q increases for a wide range of parameter space. Magenta areas denote bistable regions with ∂N1∗/∂q>0 and ∂N3∗/∂q>0. Cyan areas indicate bistable regions with ∂N1∗/∂q>0 and ∂N3∗/∂q<0; red denotes ∂N1∗/∂q>0 and N3∗ is unstable; blue denotes ∂N3∗/∂q>0 and N1∗ is unstable. Homeostatic changes and catastrophic shifts for hormetic-like biphasic dose–responses (i.e. inverted U-shape curves or J-shape curves) are shown in (i–p), and the colour bars are consistent with the colour maps in (a–h). (Online version in colour.)
Single-parameter bifurcation analyses, shown in figure 4i–p, reveal one of the most common features of hormesis: hormetic biphasic dose–responses (inverted U-shape or J-shape) of low-dose stimulation and high-dose inhibition. There are two types of inverted U-shape curves: one is a continuous inverted U-shape curve (figure 4i–k), which reveals the homeostatic changes due to external stimulation as a new and large stable equilibrium is established; the other is a piecewise continuous inverted U-shape curve (figure 4l–p), which reveals the importance of catastrophic shifts and the strength of the stability produced by external stimulations in homeostatic changes in generating a hormetic-like biphasic dose–response [2]. The dose timing response, θ, could also be closely related to the hormetic effects. The two-parameter bifurcation diagrams, figure 5a–f, show how r, q and θ influence the stability of equilibria and their variations when the timing changes. This presents an even more complex pattern in comparison with figure 4a–h (electronic supplementary material, table S4). Similarly, a smaller stressor q, a larger growth rate r, and a different dose timing response can result in very complex patterns, as shown in figure 5c–f. Continuous inverted U-shape curves (figure 5g–i) and piecewise continuous inverted U-shape curves (figure 5j–n) can also be generated by the dose timing response, resulting in hormetic-like biphasic dose timing responses, with earlier dose timing response stimulation, and later dose timing response inhibition. Figure 5. Paradox and hormetic-like biphasic dose timing responses. Two-parameter bifurcation spaces for HPEs related to the timing of the external stimulation θ in (a–f) for the HRM. The signs of the derivative N* with respect to θ have been marked in different colours. Homeostatic changes and catastrophic shifts for hormetic-like biphasic dose timing responses (i.e. inverted U-shape curves) are shown in (g–n), and the colour bars are consistent with the colour maps in (a–f). The dose timing responses generate inverted U-shaped curves similar to those for dose–responses, indicating that the timing could also be an important factor for HPEs. (Online version in colour.)
The hormetic-like biphasic dose–responses and dose timing responses can be significantly modulated and enhanced by multiple applications of low-dose stimulations within each generation at different dose–response times (figure 6). Cumulative effects of multiple low-level stimulations result in a faster increasing of N* for very small values of q, and the larger the number of external stimulations, the faster N* increases (figure 6a–d). Moreover, the hormetic zones, maximum responses and threshold levels at which the control is effective can be substantially reduced as the number of external stimulations increases. As the dose timing response and dose–response alter, different patterns of equilibrium variation emerge: the spectral ranges for occurrence of hormetic effects significantly change with increasing dose–responses and growth rates (figure 6e–p). This reveals a complication of hormetic effects due to the integration of multiple parameters, which poses a serious challenge for evaluating and avoiding the risks produced by hormesis. Nevertheless, there is also a threshold number of dose–responses above which the hormetic effects never occur. Figure 6. Cumulative effects. Results of applying multiple low-level stimulations at different dose timing responses θi with the same dose–response q defined in the model (electronic supplementary material, S5.28). (a–d) Three hormetic dose–response curves with one to three dose–response stimulations at times θi (i = 1, 2, 3): blue (one dose), magenta (two doses) and red (three doses). Three-dimensional plots for single dose–responses (e–h), two dose–responses (i–l) and three dose–responses (m–p), with different intrinsic growth rates r. The baseline parameter values are shown at the top for all subfigures; only the stable equilibria are shown in these subplots. (Online version in colour.)
We successfully fitted the model using data from the field micro-plot experiment. The intrinsic growth rate of the pest population was determined by proxy from the growth curve of the cotton squares (figure 1a). Other unknown parameters of the HRM were estimated, and the best data fits for different experimental plots are shown in figure 1b–h. The results clearly show the effects of dose–responses (figure 1i): when the low dose is applied, the control is effective at the early stage, but the pest population can increase later, after a few days, to exceed the number when no pesticide is applied. However, a high dose can successfully suppress the pest population to a low level, using the action threshold specified in the field (South 3). The results confirm that the HRM depicted the field data with multiple pulse control measures well. Note that if we fixed the intrinsic growth rate r as a constant and estimated the different survival rates p at each observation, we obtained similar results, but this could lead to difficulties for the parameter estimations. The green line in figure 1i shows the simulation results without pesticide spraying. The red and blue lines present the simulations when 5% and 10% of the pest are killed after pesticide applications. The effectiveness of the pesticide in the early stages of low-dose applications and loss of the effectiveness of the pesticide in the late stages confirm that our new model can successfully reveal the occurrence of paradoxical and hormetic effects, predict the potential risks, and provide guidance for designing new experiments on hormetic effects. To confirm these conclusions, we analysed the two hormetic datasets shown in figure 2 to estimate the unknown parameters r, θ, ρ. The results confirmed that the HRM depicted the hormetic data well, and revealed that the larger the intrinsic growth rate, the more likely is the occurrence of hormetic effects. Moreover, a comparison of the last data point and its simulated value in figure 2a, shows that the two large doses lead to redundancy. In order to formulate our novel discrete single species model with perturbation within each generation presented here, analytical and piecewise constant methods were used to extend the classic discrete Beverton–Holt and Ricker models, referred to as EBHM and HRM, respectively (see the electronic supplementary material). Based on the HRM, we revealed complex three-parameter spaces including: (1) the existence and stability of equilibria, shown in electronic supplementary material, tables S1 and S2 and (2) the occurrences of hormetic and paradoxical effects with respect to control parameters θ and q, shown in the electronic supplementary material, table S3. Moreover, the hormetic and paradoxical effects can be enhanced by multiple low-level stimulations within each generation, as shown in figure 6, i.e. the cumulative effects can significantly reduce the threshold levels, maximum responses and inhibition. Theoretical analyses and numerical investigations confirm that hormesis is difficult to investigate, as it requires that three factors (growth, dose–response and dose timing response) can act together in a complex parameter space. The novel model that we have devised is capable of describing such hormesis and the phenomenon of ecological paradoxes. The results derived from the model show how interactions between intervention dose and dose timing and between dose–responses and intrinsic factors can model hormesis in toxicological experiments and ecological paradoxes, providing insights into their complex dynamics and a methodology for improved design and analysis of experiments, with wide-reaching implications for understanding hormetic effects. Our study reveals two basic situations with hormetic effects and thus two routes to hormesis. One is the dual role of bioregulatory and compensation mechanisms in inducing new homeostatic states once the external stimulation occurs at the right time, even though external stimulation reduces the intrinsic growth rate (for example, when r < 2, i.e. the previous homeostatic state is stable). However, due to the variation in intraspecific regulatory factors and compensation mechanisms, the intrinsic reproductive capacity can be enhanced when the population density reaches a certain threshold level (i.e. the rapidly increasing f(N), as shown in figure 3a), to achieve a new high-level equilibrium that is much greater than the previous one, occurring with hormetic effects (figure 4i–k). The other situation results from the interaction between the intraspecific regulation factor, compensation mechanism and induced stability (the previous homeostatic state is unstable when r > 2) when catastrophic shifts may be realized and alternative high homeostatic states reached, also occurring with hormetic effects (figure 4l–p). For ecotoxicological applications, our model helps to determine whether a low-dose results in hormetic effects, and to evaluate the effectiveness of a high-dose pesticide. Moreover, it provides an important cue for determining action thresholds and pest control strategies [35,36]. Recognition of hormetic-like biphasic dose–responses and dose timing responses and how they are produced are crucial for elucidating bioregulatory actions and their biomedical implications [18,37–40]. The main features shown in figures 4–6 and the complex parameter spaces accounting for the occurrence of the features listed in the electronic supplementary material, tables S1–S4 reveal why HPE phenomena are generalizable. Also, they explain why most toxicological experiments lack the capacity to assess possible hormetic dose–responses [4]. Our main results reveal that not only dose factors, but also the dose timing responses of interventions and population growth patterns are important determinants of hormetic effects. Therefore, the theory predicts that there may be as yet undetected hormetic effects in many practical circumstances, in addition to those involved in pest control and disease treatments. The theoretical analyses revealed the close relationships between the internal growth process of the pest population and the dosage of pesticide application, as well as the dose timing response of the applications, i.e. the occurrence of hormetic and paradoxical effects are determined by the complex three-parameter spaces including the intrinsic growth rate, the dose–response and the dose timing response of pesticide implementations. Nevertheless, we found that the three parameters give the dynamic behaviour of the system in the form of a regular combination. The threshold values related to the combination rθ give the key conditions for the existence of equilibria, as shown in electronic supplementary material, table S1. However, the stability properties of the equilibria and occurrences of hormetic and paradoxical effects are determined by the threshold values of the combination per or r+lnp, as shown in electronic supplementary material, tables S2–S4. It is interesting that in a certain parameter space, the hormetic and paradoxical effects could occur whether the equilibrium is stable or unstable. Moreover, the critical values for the occurrence of hormetic and paradoxical effects related to the q and θ are quite different, i.e. for the fixed dose–response q the critical intrinsic growth is divided into four different intervals shown in electronic supplementary material, table S3, while only two critical intervals are separated for the θ shown in electronic supplementary material, table S4 which requires a large intrinsic growth rate. All these important relations among three crucial parameters can help us to distinguish the occurrence of negative effects of pesticide applications, and then help us to recognize hormesis in toxicological experiments and ecological paradoxes. Importantly, the models and analytical techniques presented here not only provide a possibility for assessing the effectiveness of control interventions within two generations of discrete populations, but they can also be employed to fit field data with multiple pulse perturbations, as shown in figure 1, and to hormetic data, as shown in figure 2. Thus, given the growth characteristics of plant leaves, we fitted the growth rates of pests, and then were able to fit the field data from different regions. Moreover, such data fitting and parameter estimations can help to evaluate the parameter space related to the potential occurrence of hormetic and paradoxical effects. In summary, in this paper, a novel re-formulation of the Ricker population equation showed how interactions between dose–responses, dose timing responses and intrinsic factors can model hormesis in toxicological experiments and ecological paradoxes, providing insights into their complex dynamics and a methodology for improved design and analysis of experiments, with implications for understanding hormetic effects in general. For ecological applications, the data fitting demonstrated the effectiveness of our framework to inform intervention strategies for species with discrete generations under multiple impulsive interventions. We only focused on a single species model to reveal the effects of multiple factors on hormesis, and note that a key problem of how to construct a discrete model for multi-population interactions, based on methods similar to those developed here, remains a key theoretical challenge. Moreover, many factors such as environmental temperature, diet, pesticide or drug resistance, and random perturbation may also play important roles in inducing hormesis. Therefore, our next goal is to develop more practical mathematical models and carry out systematic research related to paradoxical and hormetic effects. The field dataset are freely available from S.Y. All authors designed and conducted the research. S.T. and J.L. did the analytical calculations. S.T. and C.X. did the numerical calculations. G.L. provided the field datasets. S.T., Y.X. and R.A.C. were the lead writers of the manuscript. We declare that we have no competing interests. This work was partially supported by the National Natural Science Foundation of China (NSFCs: 11631012, 61772017 and 11401360), and by the Fundamental Research Funds for the Central Universities (GK201901008). J.W.U. would like to acknowledge support from the Natural Sciences and Engineering Research Council of Canada, and the Canada Research Chairs programme. R.A.C. is grateful to the University of Greenwich for research funds that contributed to this work. We thank the anonymous referees for their very helpful comments, which led to significant improvements to the manuscript. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608443. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 15Prostate cancer (PCa) is a major health burden among ageing men worldwide [1]. External beam radiation therapy (EBRT) is a feasible treatment for patients of all ages and PCa risk groups [1–3]. In EBRT, radiation is delivered from an outside beam aiming at disrupting the DNA in the tumour cells’ nuclei, which forces them to undergo programmed cell death due to excessive DNA damage accumulated from both radiation and the previous genetic alterations that generate and support PCa [4]. EBRT requires a precise planning of the radiation dose quantity, distribution over the prostate organ and temporal delivery [1,2]. Classical EBRT plans deliver a total dose of 74–80 Gy in 2 Gy fractions. Moderately hypofractionated plans (60–66 Gy delivered in 3 Gy fractions) are also used after recent clinical trials that have shown that they are non-inferior to conventional EBRT [1]. Neoadjuvant and adjuvant androgen deprivation therapy (ADT) may improve EBRT performance, but can also provoke bothersome side effects (e.g. low libido, impotence, anaemia, osteoporosis, depression). Hence, combination of EBRT with ADT is only recommended for intermediate-risk PCa (four to six months) and mandatory for high-risk tumours (two to three years) [1]. Local recurrence after EBRT can be managed with radical prostatectomy, cryoablation, brachytherapy and high-intensity focused ultrasound, while patients with advanced PCa are usually prescribed ADT, chemotherapy or a combination of both [1,2]. Patient monitoring after conclusion of EBRT largely relies on prostate-specific antigen (PSA) levels [1,2], which is a common biomarker whose levels in blood tend to rise during PCa [1,2]. Radiation-induced tumour cell death causes PSA to decrease after EBRT, so a continued rise in PSA may be indicative of PCa recurrence due to thriving cancerous cells surviving radiation therapy. However, PSA may also be affected by natural background fluctuations (e.g. diet, lifestyle), a continuous smooth increase due to prostate enlargement caused by benign prostatic hyperplasia (BPH), and sudden rises due to ceasing ADT or to the so-called PSA bounce (a transient rise of at least 0.1 to 0.5 ng ml−1 usually within 24 months after EBRT [5,6]). Therefore, physicians require robust criteria to identify when a rise in PSA corresponds to a PCa recurrence. Initially, biochemical relapse after EBRT was defined as three consecutive rises of PSA after the minimum post-EBRT PSA value registered for a given patient (PSA nadir) [1,2]. Currently, a superior criterion defines biochemical relapse as an increase larger than 2 ng ml−1 over PSA nadir [1,2], which correlates better with clinical recurrence and patient survival. The former three-point rule is still used as a warning sign in patient monitoring. However, these criteria of biochemical relapse detection require PSA to reach a minima and start increasing, which may result in delays in the application of further treatments. Also, this relapse measure does not inform about the expected PCa prognosis. The definition of early markers of PCa recurrence and malignancy would enable physicians to successfully control the disease with an appropriate salvage treatment. This is the purpose of multiple studies aimed at analysing PSA dynamics after EBRT. A high value of PSA nadir, a short time to reach PSA nadir after EBRT termination and short PSA doubling time (or high PSA velocity) during biochemical relapse have been correlated with metastatic disease and reduced patient survival [6–10]. While these studies focus on long-term PSA dynamics, only a few investigations have focused on analysing the PSA evolution shortly after EBRT conclusion. A rising PSA trend, high PSA levels or a rapid PSA decline shortly after EBRT have been linked with poorer prognosis and patient survival [11–13]. To gain further insight, post-EBRT PSA dynamics has also been quantitatively described by fitting mathematical formulae to PSA longitudinal data in different patient cohorts [14–18]. PSA decline after EBRT in cured patients is usually described with an exponential decay (possibly added to a constant or a slowly increasing linear term accounting for benign growth), while a bi-exponential formula best represents the PSA decrease and posterior rise in biochemically relapsing patients [14–16]. This bi-exponential formula has also been leveraged in all cases, such that parametrization using PSA data for cured patients will cause the rising branch to vanish [17,18]. Still, the choice of the mathematical formula in the vast majority of quantitative studies on PSA dynamics only relies on the empirical observation of PSA temporal trends following EBRT and does not account for the underlying tumour dynamics, which is ultimately regulating the PCa recurrence. Here, we present a patient-specific mathematical formulation of PSA dynamics based on biological mechanisms describing tumour response to radiation. Mechanistic mathematical modelling of cancer and response to treatments have improved the understanding of tumour growth and can assist physicians in clinical decision-making on a personalized basis [19–23]. Some mechanistic modelling studies have explored the connection between tumour and PSA dynamics in untreated PCa growth [24–27], under hormonal therapy [28–32], and after radical prostatectomy [33,34]. Radiation effects is a rich topic in the literature of computational modelling of cancer [20–23,35–39]. Several mathematical models have been proposed to describe radiation effects on tumour cells (e.g. cytotoxic action, cell-cycle arrest, promotion of apoptosis). The linear-quadratic model is arguably the most widely used formulation [20–22,35–37,40,41]. However, the linear-quadratic model inherently assumes a relatively fast response to radiotherapy and hence this paradigm works better in rapidly growing tumours (e.g. glioblastoma multiforme). For slowly growing tumours, such as low-grade glioma or PCa, the late response to radiation requires to account for repopulation of tumour cells, i.e. the underlying tumour dynamics [23,42,43]. Although previous modelling efforts have explored alternative formulations of radiation effects on PCa [42–45], mechanistic mathematical descriptions of the complete evolution of prostatic tumour growth and PSA after the delivery of radiotherapy are lacking. Our mathematical formulation addresses this challenge with minimal assumptions on radiation effects. Anonymized patient data were obtained from Centro Oncológico de Galicia (COG, A Coruña, Spain). Ethics approval was obtained from Comité Autonómico de Ética da Investigación de Galicia (Santiago de Compostela, Spain). Informed consent was not required for the patient data used in this study. A total of 1588 men diagnosed with localized PCa confirmed at COG (stage T1 to T2, Gleason score less than 8) and treated with EBRT in this institution between 2009 and 2015 were considered for inclusion in the study. Inclusion criteria were first-line treatment of EBRT delivered only at COG and more than 2 years of PSA monitoring with at least 5 PSA values after conclusion of EBRT. Exclusion criteria were a previous neoplastic disease prior to PCa, any other treatment for PCa (e.g. ADT, radical prostatectomy, radiotherapy, chemotherapy) and EBRT without radical intent. A total of 71 patients satisfied the inclusion criteria and did not qualify for any of the exclusion criteria. EBRT was either conventional (64 patients, 2 Gy/dose) or hypofractionated (seven patients, 3 Gy/dose). In both cases, the original EBRT plan consisted of series of five daily doses delivered on weekdays followed by two days of rest during the weekend. For simplicity, in this preliminary study, we pooled all patients together without differentiating radiation plans. Seven patients experienced biochemical relapse (either three consecutive increasing values of PSA or an increase of more than 2 ng ml−1 over PSA nadir), of which four had reported evidence of PCa recurrence. We will refer to those patients who did not show biochemical recurrence after EBRT as cured patients. Table 1 summarises the characteristics of the patient cohort. Additionally, 43 cured patients and three biochemically relapsing patients had T1 cancer, whereas 21 cured patients and four biochemically relapsing patients had T2 cancer.
Serum PSA P(t) is generally assumed to be proportional to the prostatic tumour mass and it is known to approximately follow an exponential trend in time [1,2,24–26]. Hence, if we denote the number of tumour cells by N(t), then P(t)=ρN(t)=ρN0 etτn=P0 etτn,2.1 where ρ is a proportionality constant, τn is the characteristic time of net proliferation, and N0 = N(t0) and P0 = P(t0) are the population of tumour cells and serum PSA at a time t0, respectively.EBRT for PCa consists of nd radiation doses delivered at times {ti}i=1,…,nd. We will assume that all doses are equal, which applies to our patient cohort. After the delivery of the k-th radiation dose at time tk, we assume that a fraction of tumour cells D~k(t) is irreversibly damaged and undergoes cell death after a characteristic time τd, while the remaining fraction of tumour cells Sk(t) survives and continues to grow with a characteristic time of net proliferation τs. The dynamics of D~k(t) and Sk(t) are given by the following set of ordinary differential equations, dSkdt=Skτs,Sk(tk)=RdSk−1(tk)2.2a dD~kdt=−D~kτd,D~k(tk)=(1−Rd)Sk−1(tk),2.2b for each interval tk ≤ t < tk+1 and where S0(t1)=N(t1)=N0 et1/τn, D~0(t1)=0, and Rd is the dose-dependent fraction of surviving cells after the delivery of the k-th radiation. We do not assume any specific formulation for Rd, such as in most literature of computational modelling of radiation effects [20–22,35–37,40,41]. Instead, we directly compute Rd from PSA data, making the model more flexible and easier to parametrize. As each patient always receives the same dose per session, it suffices to compute one value of Rd per patient. The solutions to equations (2.2) areSk(t)=RdSk−1(tk) et−tkτs2.3a andD~k(t)=(1−Rd)Sk−1(tk) e−t−tkτd,2.3b for tk ≤ t < tk+1.Let Dk(t) be the accumulated population of irreversibly damaged tumour cells due to the radiation doses already delivered for tk ≤ t < tk+1. Its dynamics satisfies the equation Dk(t)=Dk−1(t)+D~k(t),2.4 where D0(t) = 0. Then, the population of total cancerous cells after the k-th radiation dose Nk(t) and the corresponding serum PSA concentration Pk(t) can be computed asNk(t)=Sk(t)+Dk(t)2.5a andPk(t)=ρNk(t)=ρ(Sk(t)+Dk(t)), 2.5b whereSk(t)=RdSk−1(tk) et−tkτs2.6a andDk(t)=Dk−1(t)+(1−Rd)Sk−1(tk) e−t−tkτd, 2.6b for tk ≤ t < tk+1.Using equations (2.6) recursively stepwise from the first radiation dose, we obtain the following explicit formulas for the population of proliferative and damaged tumour cells Sk(t)=RdkN0θ1 etτs2.7a andDk(t)=(1−Rd)(∑i=1kRdi−1 e(ti−t1)(1τs+1τd))N0θ1θ2 e−tτd,2.7b for tk ≤ t < tk+1 and where θ1=et1(1τn−1τs) and θ2=et1(1τs+1τd). Hence,Pk(t)=P0θ1[Rdk etτs+(1−Rd)(∑i=1kRdi−1 e(ti−t1)(1τs+1τd))θ2 e−tτd],2.8 for tk ≤ t < tk+1 and where we have used that P0 = ρN0.In the particular case in which the radiation doses are equispaced in time, tk = t1 + (k − 1)τr. Then, equation (2.7b) simplifies to Dk(t)=(1−Rd)1−Rdk ekτr(1τs+1τd)1−Rd eτr(1τs+1τd)N0θ1θ2 e−tτd,2.9 and hence we may rewrite equation (2.8) asPk(t)=P0θ1[Rdk etτs+(1−Rd)1−Rdk ekτr(1τs+1τd)1−Rd eτr(1τs+1τd)θ2 e−tτd].2.10 for tk ≤ t < tk+1 (see details in electronic supplementary material, annex S2).Alternatively, we may assume that the whole radiation treatment is delivered at a certain time tD. Then, S(t) and D(t) are given by S(t)=RDN0θ1 etτs2.11a andD(t)=(1−RD)N0θ1θ2 e−tτd, 2.11b where RD is the fraction of surviving cells after the total treatment dose, θ1= etD(1τn−1τs) and θ2= etD(1τs+1τd). By using equations (2.5) we getP(t)=P0θ1[RD etτs+(1−RD)θ2 e−tτd].2.12 After the completion of radiotherapy, i.e. for t>tnd, the evolution of PSA will be given by equation (2.8), which for simplicity we will denote by P(t): P(t)=P0θ1[Rdnd etτs+(1−Rd)(∑i=1ndRdi−1 e(ti−t1)(1τs+1τd))θ2 e−tτd].2.13 Let us define the non-dimensional counterparts of P and time t, respectively, as P^=P/(P0Rdndθ1) and t^=t/τd. Then, we may rewrite equation (2.13) in non-dimensional form asP^(t^)= eτdτst^+(1−Rd)Rdnd(∑i=1ndRdi−1 e(ti^−t1^)(1τs+1τd))θ2 e−t^= eβt^+αθ2 e−t^,2.14 where we have introduced two non-dimensional parametersα=(1−Rd)Rdnd(∑i=1ndRdi−1 e(ti^−t1^)(1τs+1τd))2.15a andβ=τdτs.2.15b While α may represent the efficacy of the radiation plan, β controls the dynamics of the tumour cell populations and PSA after radiation (see §4.1). Thus, these parameters may hold predictive value, which will assess in this work.Following a similar procedure, we may also obtain the expressions of α and β for both the periodic-dose model α=(1−Rd)Rdnd1−Rdnd endτr(1τs+1τd)1−Rd eτr(1τs+1τd)andβ=τdτs,2.16 and the single-dose modelα=(1−RD)RDandβ=τdτs.2.17 Additionally, the derivative of equation (2.14) with respect to t^ provides the non-dimensional PSA velocity vP^(t^)= dP^(t^)dt^=β eβt^−αθ2 e−t^.2.18 According to their definition α, θ2 and β, are positive quantities. When αθ2/β > 1, then P^ decreases for at least some time after radiotherapy. Then, we can compute the time to PSA nadir, tn, by solving vP^(t^n)=0 for t^n and substituting the definition of θ2 (see §2.2.1), yieldingtn=t1+τdln(α/β)1+β.2.19 Hence, the time to PSA nadir Pn since the completion of EBRT at time tnd is given by Δtn=tn−tnd. Radiation plans may experience delays due to treatment side-effects, holidays, machine routine maintenance or machine failures. The reported values of EBRT duration in table 1 suggest that these interruptions were common in our patient cohort. In addition, the information about EBRT in our patient dataset consists of the dates of treatment initiation and termination, the radiation dose, and the number of doses. This input information is not compatible with an accurate use of our general model (§2.2.1), which would require the exact dates of EBRT sessions. Thus, in this work, we will focus our analysis on the periodic-dose model (§2.2.2) and the single-dose model (§2.2.3). The possible difference in results between both models, if any, would be related to treatment duration effects. Electronic supplementary material, table S1 summarizes the main quantities in all models. Electronic supplementary material, annex S1, table S2 and figure S1 show that the periodic-dose model is virtually equivalent to the general formulation, and we will analyse the single-dose model as a feasible simplification of both the general and periodic-dose models. We will further assume that EBRT does not change the proliferation rate of surviving cells, so that τn = τs and θ1 = 1. This assumption is common in the literature [21–23], contributes to the simplicity of our models, and facilitates parametrization, especially in those patients with a limited number of PSA values before EBRT. Additionally, we choose t0 = 0 and we will assume that tD is the date of EBRT initiation in the single-dose model. We leveraged nonlinear least squares using the trust-region method to estimate the parameters of our models in a patient-specific manner. Table 2 shows the initial values for the algorithm, the lower bounds and the upper bounds used to fit the single and the periodic-dose models for each patient. We assessed the goodness of fit with the sum of squared errors (SSE), the R2, the adjusted R2 with respect to the degrees-of-freedom in error (R^2) and the root mean squared error (RMSE).
We used the Wilcoxon rank-sum test (WRST) to identify potential markers of biochemical relapse by analysing whether model parameters, non-dimensional parameters, PSA nadir and time to PSA nadir since EBRT completion differed between cured and biochemically relapsing patients. We also tested the goodness-of-fit statistics to analyse whether the estimation of PSA dynamics was more accurate in either patient subgroup. Additionally, we compared the values of model parameters and model-derived quantities obtained with each PSA dynamics formulation. We defined R=Rdnd in the periodic-dose model and R = RD in the single-dose model to compare the values of Rd and RD, respectively. This study was performed both globally by using the WRST and patient-wise by using the Wilcoxon signed-rank test. We used the same tests to compare the goodness-of-fit statistics produced by each mathematical model, and hence to determine whether one of them provided a superior fit. The level of significance was set at 5% for all statistical tests. We constructed the receiver operating characteristic (ROC) curves of the quantities that changed significantly between cured and biochemically relapsing patients to assess their ability to classify patients in either group. We iteratively varied a threshold for each of these quantities independently across the whole range of values provided by each model. Threshold stepping was determined as the difference between the maximum and the minimum value divided by 1000. For each threshold, we computed sensitivity, specificity and accuracy. We also computed the area under the ROC curve (AUC) by using the trapezoidal rule and the optimal performance point by using Youden’s index. Calculations were performed in Matlab (Release R2017b, The Mathworks, Inc., Natick, MA, USA). Parameter estimation was performed with the Curve Fitting Toolbox. Statistical tests were run with the Statistics and Machine Learning Toolbox. We also computed the 95% confidence bounds for each model fit with the Curve Fitting Toolbox. The periodic-dose model and the single-dose model succeeded in fitting individual patient PSA data. Figure 1 portrays the results for both models corresponding to two cured patients and two patients with biochemical relapse. Table 3 shows that model fitting was extraordinarily precise with both PSA dynamics models for the vast majority of patients. We observed that superior fitting results were obtained when several PSA data were distributed in an approximately even manner right before and after EBRT (see electronic supplementary material, figure S2). Conversely, few pre-EBRT PSA values or few post-EBRT PSA measurements close to treatment termination could hinder the accurate reproduction of PSA dynamics (see electronic supplementary material, figure S3). High fluctuations in PSA data always worsened the goodness of fit (see electronic supplementary material, figure S4). Figure 1. Curve fitting results for two cured patients (a,b) and two patients with biochemical recurrence (c,d) using both the periodic-dose model (solid green line) and the single-dose model (dashed blue line). For each patient, each row shows, respectively, the fit provided by the periodic-dose model, the fit obtained with the single-dose model, and a comparison of the fits computed with either model. The shaded areas along the model fits in the first two subfigures of each row depict the corresponding 95% confidence interval of the model fit. PSA values are depicted as red bullets and the duration of EBRT is shaded in light grey. (Online version in colour.)
We did not identify any significant difference between the goodness-of-fit statistics for cured and biochemically relapsing patients with any of the two PSA dynamics models in two-sided WRSTs (table 3). Nevertheless, we observed that our models tended to reproduce PSA dynamics with slightly superior accuracy for the cured patients of this cohort (table 3). The goodness-of-fit statistics of each model did not globally differ neither in the whole cohort nor in any patient subgroup according to two-sided WRSTs (table 4). However, two-sided Wilcoxon signed-rank tests identified significant differences in the accuracy of the fit obtained with each model for each patient (table 4). Corresponding one-sided Wilcoxon signed rank tests showed that the single-dose model produced lower SSE (p = 1.62 × 10−4) and RMSE (p = 2.20 × 10−4), as well as higher R2 (p = 1.08 × 10−4) and R^2 (p = 1.19 × 10−4). We observed the same results for the subgroup of cured patients (table 4), where one-sided tests also demonstrated that the single-dose model rendered lower SSE (p = 3.62 × 10−4) and RMSE (p = 4.72 × 10−4) as well as higher R2 (p = 2.32 × 10−4) and R^2 (p = 2.38 × 10−4). No model was found to provide a significantly superior accuracy in the subgroup of biochemically relapsing patients (table 4).
The values of the parameters P0, Rd or RD, τs and τd obtained with the periodic-dose model and the single-dose model are summarized in table 5. We also used them to compute each model’s non-dimensional parameters (α, β), PSA nadir (Pn), and time to PSA nadir since EBRT termination (Δtn) for each patient, also reported in table 5. P0 was typically close to the first PSA value available for each patient, but it was not necessarily coincident (figure 1). The estimation of Rd, RD and τd provided values inside the parametric domain defined in table 2 for the vast majority of patients. However, we obtained τs ≈ 500 (upper bound) for many patients, especially with the periodic-dose model. This situation only occurred for cured patients, for whom larger values of τs are expected. Indeed, both large τs and very small remnant proliferative tumour cell populations after EBRT lead to similar results, i.e. no tumour regrowth for the time scales studied leads to some uncertainty in the parameter values. However, this fact did not compromise the accuracy of the model fitting to the data (see table 3; electronic supplementary material, figure S5). For the periodic-dose model, two-sided WRSTs identified τs, β and Δtn to be significantly different between cured and biochemically relapsing patients (table 5). The matching one-sided tests revealed that biochemically relapsing patients had smaller τs (p = 4.39 × 10−4), larger β (p = 6.17 × 10−4) and shorter Δtn (p = 0.0123). For the single-dose model, we also found τs, β and Δtn to significantly differ between cured and biochemically relapsing patients in two-sided WRSTs (table 5). Again, the corresponding one-sided tests showed that biochemically relapsing patients exhibited shorter τs (p = 4.70 × 10−4), higher β (p = 8.61 × 10−4) and smaller Δtn (p = 0.0111). Figure 2 depicts the boxplots corresponding to the values of τs, β and Δtn obtained with the periodic-dose model and the single-dose model for the whole cohort, cured patients and biochemically relapsing patients. These boxplots show how τs, β and Δtn cluster around different values in cured and biochemically relapsing patients. Among the other quantities of interest, the non-dimensional parameter α was close to the significance threshold for both models, as well as RD and Pn in the single-dose model. Figure 2. Boxplots of the potential patient classifiers identified in the statistical analysis: (a) the characteristic time of tumour cell proliferation τs, (b) non-dimensional parameter β = τd/τs and (c) the time to PSA nadir since EBRT termination Δtn. The first and the second row correspond to the results obtained with the periodic-dose model (green) and the single-dose model (blue), respectively. Outliers are depicted as hollow grey circles. (Online version in colour.)
Except for P0, the two-sided Wilcoxon signed-rank tests showed that the values of the remainder parameters, the non-dimensional parameters, the PSA nadir, and the time to PSA nadir obtained with either PSA dynamics model for each patient were significantly different within the whole cohort and the subgroup of cured patients (table 6). Corresponding one-sided tests in the whole cohort revealed that the single-dose model provided smaller R (p < 1 × 10−6), τs (p = 5.47 × 10−5), α (p < 1 × 10−6) and Pn (p = 0.004) as well as larger τd (p < 1 × 10−6), β (p < 1 × 10−6) and Δtn (p < 1 × 10−6). Within the subgroup of cured patients, one-sided Wilcoxon signed-rank tests also revealed that the single-dose model produced lower values of R (p < 1 × 10−6), τs (p = 2.23 × 10−5), α (p < 1 × 10−6) and Pn (p = 1.58 × 10−4) as well as larger values of τd (p < 1 × 10−6), β (p < 1 × 10−6) and Δtn (p < 1 × 10−6). Within the subgroup of biochemically relapsing patients, only R, τd, β and α were found to significantly vary with either model for each patient in two-sided Wilcoxon signed-rank tests (table 6). Corresponding one-sided tests showed that the single-dose model produced lower values of R (p = 0.023) and α (p = 0.008) as well as larger values of τd (p = 0.016) and β (p = 0.016). However, the global comparison of the values provided by either model using two-sided WRSTs did not find any significant difference neither within the whole cohort nor within any of the patient subgroups (table 6). Figure 3 shows the ROC curves for the three quantities that were significantly different between the groups of cured and biochemically relapsing patients: τs, β and Δtn. The AUC and optimal performance point obtained for each quantity and model are shown in table 7. The two ROC curves for each classifier were very similar and provided comparable AUC values and optimal points of performance, especially for τs. This suggests the insensitivity in the accuracy of these classifiers with respect to the choice of mathematical model to fit PSA data. Figure 3. ROC curves for the different patient classifiers identified in the statistical analysis: (a) the characteristic time of tumour cell proliferation τs, (b) the non-dimensional parameter β = τd/τs and (c) the time to PSA nadir since EBRT termination Δtn. (Online version in colour.)
The shape of the ROC curves, the AUC, and the balance between optimal sensitivity and specificity showed that τs and β rendered almost equally outstanding results and that both performed better than Δtn, which only showed a fairly satisfactory behaviour. Parameter τs provided the highest AUC and optimal sensitivity. While β provided a slightly lower AUC than τs, it also showed a better trade-off between sensitivity and specificity at optimal performance point. Δtn was found to provide the highest optimal specificity, but the corresponding optimal sensitivity and AUC were remarkably lower with respect to those of τs and β. All potential classifiers showed similar accuracy at optimal performance point. However, as the prevalence of biochemical relapse was low in our cohort (seven out of 71 cases), the accuracy of classifiers was largely driven by the specificity, almost regardless of sensitivity (table 7). Hence, Δtn was also found to provide a slightly higher accuracy at optimal performance point. Our models always lead to an explicit bi-exponential formula of PSA dynamics relying on the coupled dynamics of the radiation-induced irreversibly damaged tumour cell fraction and the surviving tumour cell population. Consequently, we provided a biophysical meaning for the parameters appearing in the empirical biexponential formulations [14–18]. Both models achieved a highly remarkable accuracy in the fitting of patient’s PSA longitudinal data in our cohort (table 3). We observed that even a limited amount of PSA data can provide an excellent fit with both models as long as (i) sufficient PSA values are evenly distributed closely around EBRT and (ii) they do not exhibit large fluctuations (see electronic supplementary material, figures S2–S4). Despite its apparent simplicity, our results show that the single-dose model suffices to accurately describe PSA dynamics before and after EBRT, even providing superior fittings than the more complex periodic-dose model (tables 3 and 4). This also means that the single-dose model is an excellent surrogate for the general model in equation (2.8), which is virtually equivalent to the periodic-dose model (see electronic supplementary material, figure S1 and table S2). This extraordinary balance between simplicity and accuracy is an appealing feature that facilitates forthcoming research on PSA dynamics and its actual clinical use. By formally analysing our models, we found that the evolution of PSA after EBRT is characterised by only two non-dimensional parameters: α and β (see §2.2.4). The non-dimensional parameter α controls the magnitude of PSA decay due to EBRT, i.e. the amount of PSA eliminated due to radiation-induced tumour cell death. Large values of α are related to low values of Rd (equations (2.15a) and (2.16)) or RD (equation (2.17)) what means that radiation successfully eliminates tumour cells. Thus, α accounts for the efficacy of EBRT. As β = τd/τs, this non-dimensional parameter controls the coupled dynamics of the irreversibly damaged and surviving cell fractions that ultimately translates into the observable temporal trends of PSA after EBRT. Because τd < τs (see table 5), larger values of β indicate post-radiation tumour dynamics to be mostly driven by proliferation of the surviving fraction, while lower values of β point out towards a dominance of radiation-induced tumour cell death.
Interestingly, the efficacy of EBRT was better in biochemically relapsing patients, who showed larger α and lower surviving fractions (Rd or RD) than cured patients, even though these observations were statistically not significant (table 5). A dramatic decay of PSA following EBRT has also been linked to PCa recurrence in the literature [13]. We observed that biochemically relapsing patients showed smaller τs (table 5), i.e. tumours proliferated faster, which may explain this counterintuitive phenomenon: programmed cell death is triggered before cell division in case of major genetic damage [4], so fast proliferation accelerates the elimination of tumour cells affected by radiation, which translates in a dramatic decrease in total tumour cell number and thus PSA. This mechanism has also been proposed to explain the poorer prognosis of diffuse low-grade glioma patients who experience a rapid tumour volume decrease following radiotherapy using both a clinical and mathematical approach [23,46]. As α was not significantly different between biochemically relapsing and cured patients, the latter may also experience a steep PSA decay after EBRT. Hence, we require a larger cohort to validate this mechanism in PCa. This study resulted in three classifiers that showed great potential to identify biochemically relapsing patients: a short characteristic time of tumour cell proliferation τs, a large non-dimensional parameter β, and an early time to PSA nadir since EBRT termination Δtn (tables 6 and 7). Indeed, both β and Δtn are inherently controlled by τs. As τd does not vary much between cured and relapsing patients, large β values are also a consequence of a small τs (see §4.1). Then, large β promotes an early PSA nadir (see equation (2.19)), which correlates with PCa recurrence and worse survival rates [8,9]. The additional dependence of Δtn on α, which does not significantly vary between cured and biochemically relapsing patients, might explain the comparatively worse performance of Δtn as a patient classifier in ROC analysis with respect to τs and β.
We believe that τs holds a promising, robust prognostic value for PCa both before and after EBRT. An elevated tumour cell proliferation rate (i.e. short τs) has been correlated with increased aggressiveness of PCa in terms of a high Gleason Score [47], which is a crucial clinical variable in clinical management of PCa that has been linked to a higher probability of PCa local recurrence and distant metastases [2,7,8]. While Gleason Score is normally determined from histopathological assessment of biopsy samples, τs would enable to non-invasively monitor Gleason Score and to justify further biopsies when model estimations suggest a more aggressive cancer than the baseline, diagnostic biopsy. Moreover, the PSA doubling times and velocity on the rising branch in biochemically relapsing patients can be approximated as DT ≈ τs ln2 and vP ≈ (P/τs) for all models in §2.2. Hence, small values of τs would render short doubling times and high velocities of PSA increase, which have been associated to poor prognosis in PCa recurrence [6–8,10]. Our estimation of τs in biochemically relapsing patients (table 5) agrees with previously reported tumour doubling times [48], PSA relapsing doubling times [7,10], and time to PSA nadir since EBRT termination [9]. Parameter τs also enables to estimate pretreatment PSA doubling times and velocity, whose prognostic value is controversial [49]. Our model provides a robust and systematic procedure to estimate these dynamic variables, which may facilitate the assessment of their role as PCa prognostic markers. Our study presents several limitations. The patient cohort featured a limited number of patients experiencing biochemical relapse, which makes it difficult to accurately identify and assess patient classifiers. Our results need to be tested in larger independent cohorts, in which we could also explore the correlations between common PCa clinical characteristics and model parameters, non-dimensional parameters and PSA nadir estimation. While our models were rather robust against PSA fluctuations, a larger cohort would also contribute to reduce their effect on statistical analysis. We could further reduce the impact of these fluctuations by using robust nonlinear least-square methods, which associate a weight to each PSA value that tends to zero as it deviates from the average trend. Furthermore, we are using biochemical relapse as a surrogate for PCa recurrence. Ideally, our PSA dynamics models should be tested to identify clinically confirmed PCa recurrence after EBRT. We plan to specifically update our cohort with patients for whom such evidence is available to conduct further research with our PSA formulations. Hence, we could also characterize local recurrence and distant metastases using model-based markers. Despite our methods could only approximate τs ≈ 500 (upper bound) for some cured patients, we believe that this is a minor limitation for four reasons: (i) large τs is expected in cured patients, so τs = 500 mo might be an acceptable approximation; (ii) model fitting was not compromised (see electronic supplementary material, figure S5); (iii) τs plays a little role in post-EBRT dynamics of cured patients; and (iv) τs = 500 produces small β and large Δtn, contributing to classify the patient as cured. Multiple pre-EBRT PSA values, robust nonlinear least-squares fitting and problem non-dimensionalization could facilitate the accurate estimation of τs with our models. We assumed that the proliferation rate of tumour cells did not vary after EBRT, i.e. τn ≈ τs. While this is a common assumption [21–23], recent studies suggest that radiation may also affect tumour proliferation [22,50]. To explore this phenomenon in PCa patients undergoing EBRT, we would need to estimate both τn and τs, which requires multiple PSA data both before and after EBRT. Additionally, our models do not differentiate between the PSA produced by PCa and BPH. We could add a term to equations (2.1) and (2.5b) to include the BPH contribution PBPH(t) to the tumour-generated PSA levels, i.e. P(t) = ρN(t) + PBPH(t) and P(t) = ρ (S(t) + D(t)) + PBPH(t). For times t < 10 years, we may approximate PBPH(t) with a linear term or another exponential [26,51]. This simple model update would enable a more accurate determination of τs and model fitting. We also plan to explore alternative radiobiological definitions for Rd and RD to refine the modelling of radiation effects [20–22,35–37,40–43]. By introducing an explicit dependence of Rd and RD on the radiation dose one could pursue more sophisticated optimal EBRT plans [52]. PSA is currently the cornerstone of clinical decision-making during follow-up after local radical radiotherapy for PCa, so we focused our models on this biomarker. Emerging urine and blood tests are showing a promising performance in PCa diagnosis (e.g. PCA3, prostate health index, four kallikrein panel) and they may complement or even substitute PSA in the future [1,53]. Circulating tumour cells have also been shown to contribute to the diagnosis of advanced PCa. However, these tests are not recommended yet for routine screening due to the limited and sometimes inconsistent reported data. Once these biomarkers become routine, their corresponding dynamics could be coupled with our model to explore their joint performance to identify biochemically relapsing patients. Personalised volumetric data of prostate and tumour could further refine the estimation of PSA production by both benign and malignant tissue [24,25]. Multiparametric magnetic resonance is an emerging imaging technique that provides a wealth of anatomic data and is increasingly used to diagnose and monitor mild PCa during active surveillance protocols. In this context, the underlying tumour dynamics model could be refined, e.g. by using a phase-field or Fisher–Kolmogorov model and linking the variable identifying tumour growth with PSA production [22,24]. Initial tumour geometry and parameter selection can then be determined by combining longitudinal PSA and imaging data [19–22,24]. However, tumour volume is not measured in routine monitoring of patients after radiotherapy and longitudinal imaging follow-up for each patient would be required besides the standard PSA data. Thus, extending our models to include volumetric data will inevitably require a specific research monitoring protocol featuring an adequate image acquisition plan. Our mathematical models can help in the early identification of biochemically relapsing patients. This requires a good parameter identification, for which we recommend collecting at least three pre-EBRT PSA values and no less than four post-EBRT PSA values. This would translate in measuring PSA every three to six months before and after EBRT, which is compatible with current clinical guidelines. This recommendation stems from the results of this study, but we plan to determine the minimal data that enables an optimal prediction of PSA dynamics with our models in forthcoming studies. Likewise, we also plan to compare observed PSA data with simulated PSA trends corresponding to alternative treatment plans, which may help to determine the window of curability and best timing for EBRT. These initial PSA data would allow a first evaluation of the patient’s risk of relapse. Later, as further PSA data are gathered, the physician can update the prognostic variables to provide more accurate patient-specific predictions. Moreover, the predicted PSA dynamics can suggest an adequate frequency of new PSA tests to accurately parametrize our models, for instance, with shorter time intervals to precisely capture the decay following EBRT, confirm the date of nadir, and characterize a potential rising branch in relapsing patients, or longer time intervals to confirm the plateau or benign linear growth in cured patients. Hence, physicians could design a personalized PSA monitoring plan adapted to the unique PSA dynamics of each patient and informed by the underlying tumour evolution, instead of the fixed conventional recommendations currently provided by clinical guidelines. Anonymised patient data were obtained from Centro Oncológico de Galicia (COG, A Coruña, Spain). Ethical approval was obtained from Comité Autonómico de Ética da Investigación de Galicia (Santiago de Compostela, Spain). Informed consent was not required for the patient data used in this study. This work did not generate data other than those presented in the article and the electronic supplementary material. G.L., V.M.P-.G., L.A.P.-R., A.M., A.R. and H.G. conceived and designed the research, structured and analysed the results, and participated in the preparation and editing of the manuscript. G.L., V.M.P-.G., A.R. and H.G. developed the mathematical models and defined the analytical methods. G.L. performed the computations and created the displays. We have no competing interests. G.L. and H.G. were partially supported by the European Research Councilthrough the FP7 Ideas Starting Grant program (Contract # 307201). G.L. and A.R. were partially supported by Fondazione Cariplo—Regione Lombardia through the project ‘Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica’, within the program RST—rafforzamento. V.M.P-.G. work was partially supported by the Ministerio de Economía y Competitividad/FEDER, Spain (grant no. MTM2015- 71200-R) and Junta de Comunidades de Castilla-La Mancha (grant no. SBPLY/17/180501/000154). FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4603298. References
Page 16Coordination is central to living systems and biological complexity at large, where the whole can be more than and different from the sum of its parts. Rhythmic coordination [1–3] is of particular interest for understanding the formation and change of spatio-temporal patterns in living systems, including, e.g. slime mould [4,5], fireflies [6,7], social groups [8,9] and the brain [10–14]. Theoretical descriptions of biological coordination are often in terms of coupled oscillators, whose behaviour is constrained by their phase relations with each other [2,15–18]. Existing studies of phase coordination often focus on systems of either very few (small-scale, mostly N = 2) [13,19,20], or very many oscillators (large-scale, N → ∞) [21–23]. Here we inquire how the two might be connected and applied to midscale systems with neither too many nor too few components. The present work answers this question by modelling empirically observed coordinative behaviour in midscale systems (N = 8), based on data collected in a specially designed human experiment [24]. The resultant model that captures all key experimental observations happens to also connect previous theories of small- and large-scale biological coordination in a single mathematical formulation. But first, how are small- and large-scale models different? Small-scale models were usually developed to capture empirically observed coordination patterns, as in animal gaits [25–27], bimanual movement coordination [28,29], neuronal coordination [30], interpersonal coordination [31,32], human–animal coordination [33] and human–machine coordination [34,35]. They describe multiple stable coordination patterns (multistability) and the transitions between them (order-to-order transitions), e.g. from a trot to a gallop for a horse [36]. In humans, dyadic coordination patterns like inphase and antiphase (synchronization, syncopation) were found across neural, sensorimotor and social levels (see [13,14] for reviews), well captured by the extended Haken–Kelso–Bunz (HKB) model [29,37,38]. However, the extended HKB was restricted to describing coordination phenomena at N = 2 (i.e. not directly applicable to higher-dimensional coordination phenomena). By contrast, large-scale models are concerned more about statistical features like the overall level of synchrony and disorder-to-order transitions, but not so much about patterns at finer levels. As a representative, the classical Kuramoto model [2] is applicable to describing a wide range of large-scale coordination between, e.g. people [23,39], fish [40] and neural processes [22], and is often studied analytically for its incoherence-to-coherence transition (at the statistical level, for N → ∞; see [41,42] for reviews). Although the extended HKB and the classical Kuramoto model emerged separately, they connect to each other by an interesting difference: the Kuramoto model with N = 2 is almost the extended HKB model except that the former lacks the term responsible for antiphase coordination in the latter (more accurately, the bistability of inphase and antiphase). Bistability of inphase and antiphase coordination, with associated order-to-order transitions and hysteresis, happens to be a key observation in small-scale human experiments [28,43]. This begs the question of whether there is a fundamental difference between large- and small-scale coordination phenomena. Does the existence of antiphase, multistability, and order-to-order transitions depend on scale N? With these questions in mind, we recently conducted a human experiment [24] at an intermediate scale (N = 8), such that the system is large enough for studying its macro-level properties, yet small enough for examining patterns at finer levels, ideal for theories and empirical data to meet at multiple levels of description. In the following sections, we demonstrate how the marriage between the two models (not either one alone) is sufficient for capturing empirical observations at multiple levels of description and we discuss its empirical and theoretical implications for biological coordination. Before getting into the model, we briefly review the mid-scale experiment and key results [24]. In the experiment (dubbed the ‘Human Firefly’ experiment), ensembles of eight people (N = 8, total 120 subjects) spontaneously coordinated rhythmic movements in an all-to-all network (via eight touchpads and eight ring-shaped arrays of eight LEDs as in figure 1; see Material and methods for details), even though they were not explicitly instructed to coordinate with each other. To induce different grouping behaviour, subjects were paced with different metronomes prior to interaction such that each ensemble was split into two frequency groups of equal size with intergroup difference δf = 0, 0.3 or 0.6 Hz (referred to as levels of ‘diversity’), and were asked to maintain that frequency during interaction after the metronome was turned off. Subjects’ instantaneous tapping frequencies from three example trials (figure 2a–c) show intuitively the consequences of frequency manipulations: from (a) to (c) a supergroup of eight gradually split into two frequency groups of four as diversity increased from δf = 0 to 0.6 Hz. Figure 1. Experimental set-up for multiagent coordination. In the Human Firefly experiment [24], eight subjects interacted simultaneously with each other via a set of touch pads and LED arrays. Each subject’s movements were recorded with a dedicated touchpad. Taps of each subject were reflected as the flashes of a corresponding LED on the array presented in front of each subject. In each trial, each subject was paced with a metronome prior to interaction. The metronome assignment split the ensemble of eight into two frequency groups of four (group A and B, coloured red and blue, respectively, for illustrative purposes; the actual LEDs are all white). The frequency difference δf between group A and B was systematically manipulated to induce different grouping behaviour. See text and [24] for details. (Online version in colour.)
Figure 2. Social coordination behaviour observed in the Human Firefly experiment in terms of frequency dynamics and aggregated relative phase distributions. Panels (a–c) show instantaneous frequency (average over four cycles) from three example trials with diversity δf = 0, 0.3, 0.6 Hz, respectively. Viewed from bottom to top, in (c), two frequency groups of four are apparent and isolated due to high intergroup difference (low-frequency group, warm colours, paced with metronome fA = 1.2 Hz; high-frequency group, cold colours, paced with metronome fB = 1.8 Hz). As the two groups get closer (b), more cross-talk occurred between them (note contacting trajectories especially after 30 s). Finally, when the intergroup difference is gone (a), one supergroup of eight formed. Panels (d–f) show relative phase ϕ distributions aggregated from all trials for δf = 0, 0.3, 0.6 Hz, respectively (each distribution was computed from the set of all pair-wise relative phases at all time points in all trials for a given diversity condition; histograms computed in [0, π), plotted in [− 2π, 2π] to reflect the symmetry and periodicity of relative phase distributions). When diversity is low (d), the distribution peaks near inphase (ϕ = 0) and antiphase (ϕ = π), separated by a trough near π/2, with antiphase weaker than inphase. The two peaks are diminished as δf increases (e,f), but the weaker one at antiphase becomes flat first (f). (Online version in colour.)
Key results involve multiple levels of description, in terms of intergroup, intragroup and interpersonal relations. The level of intergroup integration is defined as the relationship between intragroup and intergroup coordination (β1, slope of regression lines in figure 3a. Here intragroup coordination is measured by the average pair-wise phase-locking value over all intragroup dyads, and likewise, intergroup coordination over all intergroup dyads. Phase-locking value per se is a measure of stability of a relative phase pattern within a period of time, which equals to one minus the circular variance. See section ‘Phase-locking value and level of integration’ for technical details). Intuitively, we say that two groups are integrated if intragroup and intergroup coordination facilitate each other (positive relation between respective phase-locking values, β1 > 0), and segregated if intragroup and intergroup coordination undermine each other (negative relation between respective phase-locking values, β1 < 0. We will see later in figure 4 how this measure meaningfully captures coordination dynamics). In the experimental result, two frequency groups were integrated when diversity is low or moderate (δf = 0, 0.3 Hz, blue and red lines, slope β1 > 0) and segregated when diversity is high (δf = 0.6 Hz, yellow line, slope β1 < 0). A critical level of diversity demarcating the regime of intergroup integration and segregation was estimated to be δf* = 0.5 Hz. Within the frequency groups, coordination was also reduced by the presence of intergroup difference (figure 3b, left, red and yellow bars shorter than blue bar). At the interpersonal level, inphase and antiphase were preferred phase relations (inphase much stronger than antiphase; distributions in figure 2d–f), especially when the diversity was very low (figure 2d, peaks around ϕ = 0, π, in radians throughout this paper), but both were weakened by increasing diversity (figure 2e,f; in episodes of strong coordination, antiphase is greatly amplified and much more susceptible to diversity than inphase, see [24]). Notice that subjects did not remain locked into these phase relations but rather engaged and disengaged intermittently (two persons dwell near and escape from preferred phase relations recurrently, a sign of metastability [13]; see figure 6a red trajectory for example), reflected also as ‘kissing’ and ‘splitting’ of frequency trajectories (e.g. in figure 2b). In the following sections, we present a model that captures these key experimental observations at their respective levels of description. Figure 3. Comparison between human and model behaviour at intragroup and intergroup levels. (a) How intragroup coordination relates to intergroup coordination for different levels of diversity (δf, colour-coded) in the ‘Human Firefly’ experiment [24]. Each dot’s x- and y-coordinate reflect the level of intragroup and intergroup coordination, respectively (measured by phase-locking value; see text) for a specific trial. Lines of corresponding colours are regression lines fitted for each diversity condition (slope β1 indicates the level of integration between groups). With low and moderate diversity (blue and red), two frequency groups are integrated (positive slopes); and with high diversity (yellow), two frequency groups are segregated (negative slope). Black line (zero slope) indicates the empirically estimated critical diversity δf*, demarcating the regimes of intergroup integration and segregation. The exact same analyses applied to the simulated data (200 trials per diversity condition) and results are shown in (c), which highly resemble their counterparts in (a). (b) A break-down of the average level of dyadic coordination as a function of diversity (colour) and whether the dyadic relation was intragroup (left) or intergroup (right). Intragroup coordination was reduced by the presence of intergroup diversity (δf ≠ 0; left red, yellow bars shorter than left blue bar); intergroup coordination dropped rapidly with increasing δf (right three bars; error bars reflect standard errors). Results of the same analyses on simulated data are shown in (d), which again highly resemble those of the human data in (b). (Online version in colour.)
Figure 4. Simulated coordination dynamics changes qualitatively and quantitatively with coupling strength and frequency diversity. (a–c) Frequency dynamics of three simulated trials, with increasing coupling strength (a = b = 0.1, 0.2, 0.4, respectively) and all other parameters identical (members of the slower group, in warm colours, spread evenly within the interval 1.2 ± 0.08 Hz, similarly for members of the faster group, in cold colours, in the interval 1.8 ± 0.08 Hz; initial phases are random across oscillators but the same across trials). When the coupling is too strong (c), all oscillators lock to the same steady frequency. When the coupling is moderate (b), oscillators split into two frequency groups, phase-locked within themselves, interacting metastably with each other (dwell when trajectories are close, escape when trajectories are far apart). When the coupling is weak (a), intragroup coordination also becomes metastable seen as episodes of convergence (black triangles) and divergence. (d) Level of intergroup integration quantitatively (β1, colour of each pixel) for each combination of frequency diversity δf and coupling strength a = b. White curve indicates the critical boundary between segregation (blue area on the left, β1 < 0, minβ1 = −0.2) and integration (red and yellow area on the right, β1 > 0). Within the regime of integration, the yellow area indicates complete integration (β1 ≈ 1) where there is a high level of phase locking, and the red area indicates partial integration (0 < β1 ≪ 1) suggesting metastability. Dashed grey lines label δf’s that appeared in the human experiment. Solid grey line labels the empirically estimated critical diversity. (Online version in colour.)
Our model of coordination is based on a family of N oscillators, each represented by a single phase angle φi. We will show that a pair-wise phase coupling [2,25,29] of the form φ˙i=ωi−∑ j=1Naijsin(φi−φ j)−∑ j=1Nbijsin2(φi−φ j),2.1 suffices to model the key features of the experimental data identified above. The left side of this equation is the time derivative of φi, while the constant ωi > 0 on the right is the natural (i.e. uncoupled) frequency of the ith oscillator. The coefficients aij > 0 and bij > 0 are parameters that govern the coupling.Equations (2.1) include a number of well-studied models as special cases. For instance, setting ϕ := φ1 − φ2, δω := ω1 − ω2, a~:=a12+a21 and 2b~:=b12+b21 for N = 2, the difference of the two resulting equations (2.1) yields the relative phase equation ϕ˙=δω−a~sinϕ−2b~sin2ϕ,2.2 of the extended HKB model [37]. The HKB model [29] was originally designed to describe the dynamics of human bimanual coordination, corresponding to equation (2.2) with δω = 0 (i.e. describing the coordination between two identical components). The extended HKB introduces the symmetry breaking term δω to capture empirically observed coordinative behaviour between asymmetric as well as symmetric components (i.e. the HKB model is included in the extended HKB model, which is further included in equations (2.1)). It has since been shown to apply to a broad variety of dyadic coordination phenomena in living systems, e.g. [13,14,19,43,44]. Equations (2.1) can be considered a generalization of the extended HKB model from 2 to N oscillators. It is remarkable that such a direct generalization can reproduce key features of the collective rhythmic coordination in ensembles of human subjects at multiple levels of description.Another well-studied special case of equations (2.1) is the Kuramoto model [2], which has bij = 0 (and typically aij = a, independent of i and j). We will see below, however, that the Kuramoto model cannot exhibit at least one feature of the experimental data. Namely, the data show a secondary peak in the pairwise relative phase of experimental subjects at antiphase (see figure 2d–f). Simulations using the Kuramoto model do not reproduce this effect, while simulations of equations (2.1) model do (compare figure 5d–f and g–i). We give additional analytical support for this point by studying relevant fixed points of both models in the electronic supplementary materials (section ‘Multistability of the present model’). Figure 5. Model simulations of frequency dynamics and aggregated relative phase distributions. (a–c) An example of how intergroup difference may affect intragroup coordination using frequency dynamics of three simulated trials (a = b = 0.105; note that frequency is the time derivative of phase divided by 2π, and consequently the distance between two frequency trajectories reflects the rate of change of the corresponding relative phase, which increases and decreases intermittently during metastable coordination). These three trials share the same initial phases and intragroup frequency dispersion but different intergroup difference i.e. δf = 0, 0.3, 0.6 Hz, respectively. When intergroup differences are introduced (b,c), not only is intergroup interaction altered but intragroup coordination also loses stability and becomes metastable (within-group trajectories converge at black triangles and diverge afterwards). The timescale of metastable coordination also changes with δf, i.e. the inter-convergence interval is shorter for (b) than (c). (d–f) Relative phase distributions, aggregated over all time points in 200 trials (a = b = 0.105) for each diversity condition (δf = 0, 0.3, 0.6, respectively). At low diversity (d), there is a strong inphase peak and a weak antiphase peak, separated by a trough near π/2. Both peaks are diminished by increasing diversity (e,f). These features match qualitatively the human experiment. (g–i) The same distributions as (d–f) but for a = 0.154 and b = 0 (i.e. the classical Kuramoto model). There is a single peak in each distribution at inphase ϕ = 0, and a trough at antiphase ϕ = π. (Online version in colour.)
Given the spatially symmetric set-up of the ‘Human Firefly’ experiment (all-to-all network, visual presentation at equal distance to fixation point), it is reasonable to further simplify equations (2.1) by letting aij = a and bij = b (a, b > 0), φ˙i=ωi−a∑ j=1Nsinϕij−b∑ j=1Nsin2ϕij,2.3 where ϕij = φi − φj is the relative phase between oscillators i and j (henceforth we use the notation ϕij instead of the subtraction, since relative phase is the crucial variable for coordination [28,29]).The behaviour of the model itself clearly depends on the coupling strength (a, b) and frequency diversity (distribution of ωi’s). While the latter was explicitly manipulated in the human experiment [24], the former was unknown. A qualitative look at simulated dynamics (see examples figure 4a–c for δf = 0.6 Hz) indicates that weak coupling better captures human behaviour (members of the same group do not collapse to a single trajectory in figure 4a as in figure 2). By contrast, stronger coupling (figure 4b,c) deprives the system of much of the metastability. Quantitatively, we fitted the coupling strength (assuming a = b) to the human data based on the level of intergroup integration (β1) (see distribution of model β1 in figure 4d) particularly for diversity condition δf = 0.3 Hz (i.e. using only one-third of the data to prevent overfitting). We show below how the model captures human behaviour across all diversity conditions and levels of description under the best-fit coupling strength (a = b = 0.105; see section ‘Choosing the appropriate coupling strength’ in the electronic supplementary materials for more details). At the level of intergroup relations, model behaviour (figure 3c) successfully captures human behaviour (figure 3a) at all levels of diversity. Similar to the human experiment, low diversity (δf = 0 Hz) results in a high level of integration in the model (blue line in figure 3c slope close to 1; β1 = 0.972, t199 = 66.6, p < 0.001); high diversity (δf = 0.6 Hz) comes with segregation (yellow line slope negative; β1 = −0.113, t199 = −3.56, p < 0.001); and in between, moderate diversity (δf = 0.3 Hz) is associated with partial integration (red line positive slope far less than 1; β1 = 0.318, t199 = 4.23, p < 0.001). Here we did not estimate the critical diversity δf* the same way as for the human data (by linear interpolation), since we found theoretically that the level of integration depends nonlinearly on diversity δf, and as a result the theoretical δf* is 0.4 Hz (figure 4d). This prediction can be tested in future experiments by making finer divisions between δf = 0.3 and 0.6 Hz. In the human experiment, not only did we uncover the effect of diversity on intergroup relations, but also, non-trivially, on intragroup coordination (outside affects within, a sign of complexity). Statistically, this is shown in figure 3b (three bars on the left): with the presence of intergroup difference (δf > 0), intragroup coordination was reduced (red, yellow bars significantly shorter than blue bar). This is well captured by the model as shown in figure 3d (two-way ANOVA interaction effect, F2,19194 = 3416, p < 0.001; the simulated data also capture the rapid decline of intergroup coordination with increasing δf in human data, shown in figure 3b,d, right). In addition to capturing this statistical reduction of intragroup coordination due to intergroup difference, the model, more importantly, provides a window to the dynamical mechanism underlying such statistical phenomena. For example, comparing three simulated trials with identical intragroup properties but different levels of intergroup difference (figure 5a–c), we see that the presence of intergroup difference (figure 5b,c for δf = 0.3, 0.6 Hz) dramatically elevates metastability in the system (compare intermittently converging–diverging dynamics in figure 5b,c to the rather constant behaviour in figure 5a for δf = 0). This suggests that the decrease of intragroup coordination in a statistical sense reflects the increase of metastability in a dynamical sense (see section ‘Examples of dynamics with intergroup coupling removed’ in the electronic supplementary materials for baseline dynamics when intergroup coupling is removed). Indeed, if we remove intragroup metastability from all simulations (by reducing intragroup frequency variability), they no longer capture the empirically observed statistical result (see section ‘Effect of reduced intragroup variability in natural frequency’ in the electronic supplementary materials). At the interpersonal level, human subjects tended to coordinate with each other around inphase and antiphase, especially when the diversity is low (δf = 0 Hz; figure 2d, peaks around ϕ = 0, π separated by a trough near ϕ = π/2); and the preference for inphase and antiphase both diminishes as diversity increases (δf = 0.3, 0.6, figure 2e,f ). Both aspects are well reproduced in simulations of the model (figure 5d–f). Note that these model-based distributions are overall less dispersed than the more variable human-produced distributions (figure 2d–f), likely due to the deterministic nature of the model (i.e. no stochastic terms). Yet as demonstrated above, a deterministic model is sufficient for capturing key empirical results at all three levels of description, i.e. coexistence of inphase and antiphase tendencies and their reduction with diversity, reduction of intragroup coordination with the presence of intergroup difference, and intergroup integration∼segregation at different levels of diversity. Thus, the deterministic version of the model is preferred for simplicity. Equation (2.3) becomes the classical Kuramoto model [2] when b = 0. We follow the same analyses as in the previous section but now for a = 0.154 and b = 0 (see section ‘Intergroup relation without second order coupling’ in the electronic supplementary material on parameter choices). The relationship between intragroup and intergroup coordination (electronic supplementary material, figure S8A; β1(0 Hz) = 0.974, t199 = 53.2, p < 0.001; β1(0.3 Hz) = 0.292, t199 = 4.52, p < 0.001; β1(0.6 Hz) = −0.011, t199 = −0.41, p > 0.05) resembles the case of b ≠ 0 (a = b = 0.105, figure 3c). A difference remains that for b = 0, β1(0.6 Hz) is not significantly less than zero (p = 0.68; electronic supplementary material, figure S8A yellow). The average level of intragroup and intergroup coordination also varies with diversity in the same way as the case of b ≠ 0 (electronic supplementary material, figure S8b for b = 0, interaction effect F2,19194 = 3737, p < 0.001, compared to figure 3d for b ≠ 0). In short, group-level statistical features can be mostly preserved without second-order coupling (i.e. b = 0). However, this is no longer the case when it comes to interpersonal relations. The distributions of dyadic relative phases are shown in figure 5g–i. Without second-order coupling, the model does not show a preference for antiphase in any of the three diversity conditions, thereby missing an important feature of human social coordination (for additional comparisons between human and model behaviour, see section ‘Additional analyses on the coexistence of inphase and antiphase preference’ in the electronic supplementary materials). Analytically, we find that the coupling ratio κ = 2b/a determines whether antiphase is preferred (for the simple case of identical oscillators, see section ‘Multistability of the present model’ in the electronic supplementary materials). A critical coupling ratio κc = 1 demarcates the regimes of monostability (only all-inphase is stable for κ < 1) and multistability (any combination of inphase and antiphase is stable for κ > 1). This critical ratio (for equation (2.3)) is identical to the critical coupling of the HKB model [29], where the transition between monostability (inphase) and multistability (inphase and antiphase) occurs (equation (2.2); parameters in the two equations map to each other by a=a~/2 and b=b~). This shows how equation (2.3) is a natural N-dimensional generalization of the extended HKB model, in terms of multistability and order-to-order transitions. So far, our model has captured very well experimental observations with the simple assumption of uniform coupling. However, loosening this assumption is necessary for understanding detailed dynamics. Here is an example from [24] (figure 6a), where coordination among three agents (1, 3 and 4, labels of locations on LED arrays) is visualized as the dynamics of two relative phases (ϕ13 red, ϕ34 yellow). Agents 3 and 4 coordinated inphase persistently (10–40 s yellow trajectory flat at ϕ34 ≈ 0), while agents 3 and 1 coordinated intermittently every time they passed by inphase (red trajectory ϕ13 becames flat, i.e. dwells, near inphase around 10, 20 and 35 s). Curiously, every dwell in ϕ13 (red) was accompanied by a little bump in ϕ34, suggesting ϕ34 was periodically influenced by ϕ13. In the framework of our model, we can approximate the dynamics of ϕ34 from equation (2.1) by assuming ϕ34 = 0 (thus ϕ13 = ϕ14), ϕ˙34=f(ϕ34)+(a31−a41)sinϕ13+(b31−b41)sin2ϕ13⏟=:K(ϕ13),2.4 where f(ϕ34) is the influence of ϕ34 on itself, K(ϕ13) the influence of ϕ13 on ϕ34. From K(ϕ13), we see that ϕ13 has no influence on ϕ34 if the coupling is completely uniform (i.e. K(ϕ13) ≡ 0 if a31 = a41 and b31 = b41), making it impossible to capture the empirical observation (red relation influencing yellow relation, figure 6a). To break the symmetry between agent 3 and 4, we ‘upgrade’ equation (2.3) to the systemφ˙i=ωi−ai∑ j=1Nsinϕij−bi∑ j=1Nsin2ϕij,2.5 where each oscillator can have its own coupling style (oscillator specific coupling strength ai and bi). In the present case, we are interested in what happens when a3 ≠ a4 for i ∈ {1, 3, 4}. Two simulated trials are shown in figure 6b and c with non-uniform versus uniform coupling (same initial conditions and natural frequencies across trials, estimated from the human data). The bumps in ϕ34, accompanying dwells in ϕ13, are reproduced when a3 ≫ a4 (figure 6b) but not when a3 = a4 (figure 6c; see section ‘Additional triadic dynamics’ in the electronic supplementary materials for more analyses). This example shows that to understand interesting dynamic patterns in specific trials, non-uniform coupling strength is important.Figure 6. Model with non-uniform coupling captures detailed relative phase dynamics observed in human social coordination. (a) Experimental observation of the coordination dynamics between three persons (agent 1, 3, 4, spatially situated as in legend) in terms of two relative phases (ϕ13, ϕ34; y-coordinates) as a function of time (x-coordinates). ϕ34 (yellow) persisted at inphase for a long time (10–37 s trajectory flattened near ϕ = 0) before switching to antiphase (40 s; inphase and antiphase are labelled with thick and thin dashed lines, respectively, throughout this figure). ϕ13 (red) dwelt at inphase intermittently (flattening of trajectory around 10, 20, and 35 s). Three bumps appeared in ϕ34 during its long dwell at inphase (near 15, 25, 37 s), which followed the dwells in ϕ13, indicating a possible influence of ϕ13 on ϕ34. (b,c) Two simulated trials with identical initial conditions and natural frequencies, estimated from the human data. In (b), agent 3 is more ‘social’ than agent 4 (a3 > a4). More precisely, agent 3 has a much stronger coupling (a3 = 1) than all others (a1 = a4 = b1 = b3 = b4 = 0.105, as in previous sections). The recurring bumps in ϕ34 are nicely reproduced. In (c), agents 3 and 4 are equally ‘social’ (a3 = a4 = 0.5525, keeping the same average as in (b)). ϕ34 is virtually flat throughout the trial. (Online version in colour.)
The present model successfully captures key features of multiagent coordination in mid-scale ensembles at multiple levels of description [24]. Similar to the HKB model [29], second-order coupling is demanded by the experimental observation of antiphase (and associated multistability) but now in eight-person coordination; and similar to the extended HKB [37], the model captures how increasing frequency difference δf weakens inphase and antiphase patterns, leading to segregation but now between two groups instead of two persons. This cross-scale consistency of experimental observations may be explained by the scale-invariant nature of the critical coupling ratio κc = 1, the transition point between monostability (only an all-inphase state) and multistability (states containing any number of antiphase relations). The scale invariance suggests that experimental methods and conclusions for small-scale coordination dynamics have implications for multistability, phase transitions and metastability at larger scales, and enables a unified approach to biological coordination that meshes statistical mechanics and nonlinear dynamics. Another generalization of the classical Kuramoto model by Hong & Strogatz [45] also allows for antiphase-containing patterns (π-state) by letting the sign of the first-order coupling (a) be positive for some oscillators (the conformists) and negative for others (the contrarians). However, in contrast to our model, antiphase induced this way does not come with multistability, nor the associated order-to-order transitions observed in human rhythmic coordination [13,46]. The second-order coupling in our model allows each individual to be both a conformist and a contrarian but possibly to different degrees [47]. The simple addition of a second stable state may not seem like a big plus at N = 2 (2 stable states), but it rapidly expands the system’s behavioural repertoire as the system becomes larger (2N−1 stable states for N oscillators; with only first-order coupling, the system always has 1N−1 = 1 stable state, and therefore does not benefit from scaling up). This benefit of scale may be how micro-level multistability contributes to the functional complexity of biological systems [43,48]. Outside of the mathematical context of stability analysis, we have to recall that spontaneous social coordination is highly metastable (e.g. figure 2a) [24], captured by the model when frequency diversity is combined with weak coupling (e.g. figure 4a, in contrast to b,c under stronger coupling). Individuals did not become phase-locked in the long term, but coordinated temporarily when passing by a preferred state (inphase and antiphase) [14,43] (e.g. red trajectory in figure 6a). For N > 2, an ensemble can visit different spatial organizations sequentially (see examples in [24]), forming patterns that extend in both space and time (electronic supplementary material, figure S4 for intragroup patterns), which further expands the repertoire of coordinative behaviour (see section ‘A note on metastability’ in the electronic supplementary materials). By allowing complex patterns to be elaborated over time, metastability makes a viable mechanism for encoding complex information as real-world complex living systems do (e.g. the brain) [13,14,22,49–52]. By contrast, highly coherent patterns like collective synchronization can be less functional and even pathological [53,54]. Our results call for more attention to these not-quite coherent but empirically relevant patterns of coordination. Besides the multistability or multi-clustering in micro patterns (a general feature endowed by higher-order coupling, e.g. [55–57]), existing mathematical studies suggest that the presence of second-order coupling should also manifest at the macro level in large-scale coordination. Naturally, second-order coupling induces multistability of the order parameter in the thermodynamic limit [58–61]. It also alters the critical scaling of macroscopic order (see [41] for a summary), i.e. for coupling strength K > Kc near Kc, the order parameter ∥H∥ (norm of the order function [62]) is proportional to (K−Kc)β, with β = 1/2 for the classical Kuramoto model and β = 1 when second-order coupling is added [63,64]. For complex biological systems like the brain which appears to operate near criticality [65], these two types of scaling behaviour may have very different functional implications. When modelling empirical data of biological coordination, one may want to have a closer examination or re-examination of the data for multistability and critical scaling of the order parameter, especially if finer level details are not available. Key experimental observations are captured by our model under the assumptions of uniform coupling (everyone coordinates with others in the same way) and constant natural frequency, but these assumptions may be loosened to reflect detailed dynamics. For example, introducing individual differences in coupling style (equation (2.5)) gives more room to explain how one metastable phase relation may exert strong influence on another (figure 6a). Long timescale dynamics observed in the experiment (see section ‘Additional triadic dynamics’ in the electronic supplementary materials) may also be explained by frequency adaptation, which has been observed in dyadic social coordination [66]. A systematic study of the consequences of asymmetric coupling and frequency adaptation on coordination among multiple agents seems worthy of further experimental and theoretical exploration. To conclude, we proposed a model that captured key features of human social coordination in mid-sized ensembles [24], and at the same time connects empirically validated large-scale and small-scale models of biological coordination. The model provides mechanistic explanations of the statistics and dynamics already observed, as well as a road map for future empirical exploration. As an experimental–theoretical platform for understanding biological coordination, the value of the middle scale should not be underestimated, nor the importance of examining coordination phenomena at multiple levels of description. A complete description of the methods of the ‘Human Firefly’ experiment can be found in [24]. Here we provide as many details as necessary for understanding the present paper. A total of 120 subjects participated in the experiment, making up 15 independent ensembles of eight people. The protocol was approved by Florida Atlantic University Institutional Review Board and is in agreement with the Declaration of Helsinki. Informed consent was obtained from all participants prior to the experiment. For an ensemble of eight people, each subject was equipped with a touchpad that recorded his/her tapping behaviour as a series of zeros and ones at 250 Hz (1 = touch, 0 = detach; green rectangles in figure 1), and an array of eight LEDs arranged in a ring (yellow in figure 1), each of which flashed when a particular subject tapped. For each trial, subjects were first paced with metronomes for 10 s, later interacting with each other for 50 s (instructed to maintain metronome frequency while looking at others’ taps as flashes of the LEDs). Between the pacing and interaction period, there was a 3 s transient, during which subjects tapped by themselves. Participants were instructed to match their own tapping frequency to the metronome frequency during the 10 s pacing period, and remain tapping at that frequency throughout the rest of the trial even after the metronome disappeared. During pacing, four subjects received the same metronome (same frequency, random initial phase), and the other four another metronome. The metronome assignments created two frequency groups (say, group A and B) with intergroup difference δf = |fA − fB| = 0, 0.3 or 0.6 Hz (same average (fA + fB)/2 = 1.5 Hz). This gives rise to three conditions: (1) 1.5 Hz versus 1.5 Hz, (2) 1.65 Hz versus 1.35 Hz, and (3) 1.8 Hz versus 1.2 Hz. Each ensemble completed six trials per condition (a total of 18 trials in random order). From a single subject’s perspective, the LED array looks like the legend of figure 2a (all LEDs emit white light; colour-coding only for labelling locations): a subject always saw his/her own taps as the flashes of LED 1, members of his/her own frequency group LED 2–4, and members of the other group LED 5–8 (members from two groups were interleaved to preserve spatial symmetry). From the tapping data (rectangular waves of zeros and ones), we obtained the onset of each tap, from which we calculated instantaneous frequency and phase. Instantaneous frequency is the reciprocal of the interval between two consecutive taps. Phase (φ) is calculated by assigning the onset of the nth tap phase 2π(n − 1), then interpolating the phase between onsets with a cubic spline. The relative phase between the ith and jth subject at time t is ϕij(t) = φi(t) − φj(t). Human subjects have variable capability to match the metronome frequency and maintain it, which in turn affects how they coordinate. To reflect this kind of variability in the simulations, the oscillators’ natural frequencies were drawn from a probability distribution around the ‘metronome frequency’ (central frequencies fA and fB for groups A and B). To estimate this distribution from human data, we first approximated the ‘natural frequency’ of each subject in each trial with the average tapping frequency during the transient between pacing and interaction periods (see Methods of the human experiment), and subtracted from it the metronome frequency (see blue histogram in electronic supplementary material, figure S3 from the ‘Human Firefly’ experiment [24]). We then estimated the distribution non-parametrically, with a kernel density estimator in the form of P^(x)=1nh∑i=1nK(x−xih),4.1 where the Kernel Smoothing Function is Normal, K(y)=(1/2π)e−y2/2. Here n = 2072 (259 trials × 8 subjects) from the experiment. We choose the bandwidth h = 0.0219, which is optimal for a normal density function according to [67],h=(43n)1/5σ,4.2 where σ is the measure of dispersion, estimated byσ~= median{|yi−median{yi}|}0.6745,4.3 where yi’s are samples [68]. The result of the estimation is shown in electronic supplementary material, figure S3 (red curve).The (short-windowed) phase-locking value (PLV) between two oscillators (say x and y) during a trial is defined as PLVxy=1W∑w=1W1M|∑m=1Mexp(iϕxy[(w−1)M+m])|,4.4 where ϕxy = φx − φy, W is the number of windows which each ϕ trajectory is split into, and M is number of samples in each window (in the present study, W = 16 and M = 750, same as [24]).Intragroup PLV (PLVintra) is defined as PLVintra=((|A|2)+(|B|2))−1(∑x,y∈A PLVxy+∑x,y∈B PLVxy),4.5 where A and B are two frequency groups of four oscillators, corresponding to the design of the ‘Human Firefly’ experiment [24], A = {1, 2, 3, 4}, B = {5, 6, 7, 8} and |A| = |B| = 4.Intergroup PLV (PLVinter) is defined as PLVinter=1|A∥B|∑x∈A,y∈B PLVxy.4.6 In both the human and simulated data, comparisons of PLVintra and PLVinter for different levels of δf were done using two-way ANOVA with Type III sums of squares, and Tukey honest significant difference tests for post-hoc comparisons (shown in figure 3b,d).The level of integration between two frequency groups is defined based on the relationship between intragroup coordination (measured by PLVintra) and intergroup coordination (measured by PLVinter). The groups are said to be integrated if intragroup coordination is positively related to intergroup coordination, and segregated if negatively related. Quantitatively, for each combination of intergroup difference δf and coupling strength a (assuming a = b for our model, assuming b = 0 for the classical Kuramoto model), we use linear regression PLVinter,k(δf,a)=β0(δf,a)+β1(δf,a) PLVintra,k(δf,a)+ errork(δf,a),4.7 where PLV⋅,k(δf,a) is the inter/intra-group PLV for the kth trial simulated with the parameter pair (δf, a), and the slope of the regression line β1(δf,a) is defined as the measure of the level of integration between two frequency groups. If β1 > 0, the groups may be said to be integrated; if β1 < 0, segregated. The set {(δf,a)|β1(δf,a)=0} is the critical boundary between the domains of intergroup integration and segregation. Theoretical analyses (section ‘Choosing the appropriate coupling strength’ in the electronic supplementary materials) show that this measure is meaningful (i.e. reflecting qualitative differences between dynamics; figure 4a–c).All simulations were done using the Runge–Kutta 4th-order integration scheme, with a fixed time step Δt = 0.004 for duration T = 50 (matching the sampling interval and the duration of interaction period of the human experiment [24]; second may be used as unit), i.e. for system X˙=f(X), with initial condition X(0) = X0, the (n + 1)th sample of the numeric solution can be solved recursively X[n+1]=X[n]+16(k1+2k2+2k3+k4),4.8 wherek1=Δt f(X[n]),4.9 k2=Δt f(X[n]+k12),4.10 k3=Δt f(X[n]+k22)4.11 andk4=Δt f(X[n]+k3).4.12 The solver was implemented in CUDA C++, ran on a NVIDIA graphics processing unit, solving every 200 trials in parallel for each parameter pair (δf, a). For each trial, initial phases (of eight oscillators) were drawn randomly from a uniform distribution between 0 and 2π, and natural frequencies from distributions defined by equation (4.2) (reflecting the design of and variability observed in the human experiment [24]). Here 200 trials are used per condition, greater than that of the human experiment (see [24] and section ‘Design of the human experiment’ in the electronic supplementary materials for details) to obtain a more accurate estimate of the mean.The protocol of the human experiment was approved by Florida Atlantic University Institutional Review Board and is in agreement with the Declaration of Helsinki. Informed consent was obtained from all participants prior to the experiment. Data of the human experiment are available online at doi:10.17605/OSF.IO/SC9P6. Simulated data are available at doi:10.17605/OSF.IO/PB7DH. M.Z. designed and conducted the human experiment, developed the theoretical model, performed simulations and mathematical analysis of the model, analysed human behavioural data and simulated data, and drafted the manuscript; C.B. participated in the design of the human experiment, participated in the development and mathematical analysis of the model and critically revised the manuscript; J.A.S.K. participated in the design of the human experiment, participated in the development of the model and critically revised the manuscript; E.T. participated in the design of the human experiment, participated in the development of the model and critically revised the manuscript. The authors of this work were supported by NIMH grant no. MH080838, NIBIB grant no. EB025819, the FAU Foundation, FAU I-SENSE and FAU Brain Institute. The authors thank Dr Roxana Stefanescu for assisting in data collection of the human experiment. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4585724. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 17Human rhinovirus (HRV) is the cause of the common cold that affects billions of people each year. The small viral particle, about 30 nm in diameter, has a protein shell, called the viral capsid, that encapsulates and thus protects its single-stranded RNA genome. An essential step in the infection process is the structural rearrangement of the proteins in the capsid shell, as this rearrangement results in the formation of pores through which the genomic RNA is extruded during the infection. A better understanding of this mechanism may therefore point to novel targets for anti-viral therapy and prevention. Conformational changes occur in a number of viruses during infection [1–4]. In each case, the spatial rearrangement of coat protein or coat protein domains results in the expansion of the capsid and the opening of pores through which the genomic material, and in some cases also other viral components, are released. We focus here on the pseudo T = 3 icosahedral capsid of HRV type 2 (HRV2), which is made of 60 protomers, each composed of four coat proteins VP1 to VP4 (figure 1a). While the three larger proteins (VP1, VP2, VP3) form the exterior surface of the particle, the smaller VP4 is located at the interior capsid surface. A characteristic feature of the protomer is the presence of a hydrophobic pocket located at the core of VP1, that in the native virus is occupied by the so-called pocket factor, presumably a fatty acid, which is believed to stabilize the capsid [5–10]. Figure 1. The capsid. (a) Sketch of the arrangement of the coat proteins VP1–VP3 in the native virion. Small-case Latin letters label coat proteins in the same protomer: 5 protomers are highlighted, labelled by (a, b, c, g, m). (b) The HRV2 native particle (PDB ID 1fpn). VP1s are displayed in blue, VP2s in green and VP3s in red. The black circle at the centre of the figure highlights a twofold position where pores are going to open as a consequence of protein displacement. (c) The expanded particle (PDB ID 3tn9—same colour coding as in (b) applies). (d) Details of the hinge movement within the protomer due to the relative rotation of the subunits of VP1 (adapted from [10]): VP1, VP2 and VP3 are coloured in blue, green and red, respectively. (Online version in colour.)
During infection, HRV attaches to the membrane of the cell at a receptor site, is internalized into the endosome, where it loses its pocket factors and the capsid expands, leading to the formation of the pores. The RNA exits the capsid through pores at the twofold axes and enters the host cell through channels in the endosomal membrane [11]. The mechanism by which RNA is released has not been completely elucidated, but experimental evidence suggests that exit occurs by an organized mechanism and is preceded by a substantial reorganization of the RNA inside the capsid [12]. The details of the conformational changes leading to the expansion of the capsid and the opening of the pores are likely to be related to the mechanism by which RNA is released. Moreover, it is known that for some viruses belonging to the same genus of HRVs, i.e. the Enterovirus genus, the engagement with the receptor triggers the conformational change [13–16]. In addition to the phylogenetic classification HRVs are divided into a major group and a minor group based on which cellular receptor they use for viral entry: major-group HRVs bind to the intercellular adhesion molecule 1 receptor, while the minor-group HRVs bind the low-density lipoprotein receptor. In the major group, the conformational changes of the capsid leading to RNA release are triggered by the interaction with the cellular receptor, whereas in the minor group, the low endosomal pH induces the removal of the pocket factor and the consequent viral structural rearrangements that leads to the expansion of the capsid. We focus here on HRV2, a minor group rhinovirus, and we study the transition pathways between the native particle (figure 1b) and its genome-containing expanded form, the A-particle, which is a metastable intermediate on the pathway to the empty B (80S) particle (figure 1c). During this transition, individual protomers of the capsid undergo a conformational change, during which the two domains of VP1 move apart by an angle θ (figure 1d; called opening event in the following), and VP2 and VP3 move in concert with one of the domains. Opening of all protomers collectively results in expansion of the native particle by about 4% in radius and release of the VP4s. The mathematical model used here is based on the coarse-grained approach developed in [17], in which the capsid proteins (CPs) are viewed as elementary units interacting by weak bonds, and the resulting energy landscape is explored by determining the paths and the energy barriers joining the metastable states. This allows to study the order in which the individual protomer transitions might occur, in order to identify the likely transition kinetics of the capsid expansion event. In particular, we have addressed two issues that are relevant for understanding genome release, an essential part of the infection process (for more details, see the Discussion section). First, we have investigated whether the structural transition of the capsid during its expansion is governed by diffuse nucleation events, with no regularity, or by a more organized domino effect, as suggested by the fact that interactions between the capsomers are relatively weak (the energy cascade hypothesis). Second, and more importantly, given that it has been experimentally established that (i) RNA exit is directional, with the 3′-end exiting first, and (ii) the positions of the 5′- and 3′-ends inside the capsid are fixed in areas of the capsid that are roughly opposite to each other [18,19], the following question arises: What is the relation between the localization of the 3′-end and the opening of the pores in the capsid and in the endosome through which the 3′-end exits? Our model has been designed to address this issue. In particular, we explore whether there are pathways that induce a preferential opening of the pores near the 3′ site, and we show that there is a reasonable parameter range in which this is the case. In fact, there is a clear separation in the parameter space between three possible modes of opening, and this suggests that a fine regulation of the RNA-extrusion process (i.e. the preferential opening of a twofold channel in the capsid near the attachment site of the 3′-end of the RNA) is possible without requiring regulation via the action of receptors. This is consistent with the fact that some strains of human rhinovirus, such as HRV2 that is under consideration here, do not require receptors for genome release [11], and suggests a principle of economy in the release mechanism. Note that, in order to be effective, the above mechanism requires that the cascade is triggered at the protomer at which the 3′-end of the RNA is bound. This could perhaps be triggered by a specific interaction between genomic RNA and CP at that site, that impacts on that CP’s conformation and its interactions with surrounding CPs. Conformational changes in CP in response to contact with an RNA stem-loop have been reported in other viruses before, such as the allosteric conformer switch in MS2 that is a prerequisite for capsid assembly [20,21]. Also, in human parechovirus (HPeV), a different picornavirus that does not cleave its VP0 into VP2 and VP4, we have recently shown that there are multiple dispersed RNA sequence/structure motifs in the viral genome with affinity for CP that we termed packaging signals due to their role in capsid formation [22,23]. Cryo-electron microscopy studies of rhinovirus also show multiple dispersed contacts between genomic RNA and capsid, with a change in the contact pattern upon expansion [12]. In particular, in addition to the contacts close to the twofold axes that are present already in the native HRV2 particle, there are new contacts around the fivefold axes in the expanded A-particle (cf. fig. 2 in [12]). It is, therefore, possible that contacts between genomic RNA and CP play a role in the release mechanism. During infection, HRV2 is internalized by the cell within endosomes, and the decrease in pH triggers a series of structural transformations of the capsid, leading to the expansion of the particle, the formation of pores and to the exit of the viral RNA into the cytosol. The particle has been imaged at different stages of the expansion: the structure of the native particle has been determined at 2.6 Å resolution by X-ray crystallography [6], while X-ray crystal structures at 6.4 Å and cryo-electron microscopy studies [12] showed that A (the genome-containing expanded form—see the Introduction) and B (the expanded and empty form) particles are almost identical, and an X-ray structure has been determined at 3.0 Å resolution for the empty particle [10]. Henceforth, in what follows, we identify the A capsid with the B capsid. The full dynamics of the process is still unclear, although some of the occurring structural transformations have been in part elucidated. It is generally acknowledged that, as a consequence of the decrease in pH, the pocket factor at the core of VP1 is released from its location, and this allows the relative rotation of two domains of VP1 (the α helix and the C-terminus move away from the β-barrel), resulting in a change in the conformation of the monomer and the collapse of the pocket. This hinge movement affects VP2 and VP3 positions as they displace in concert with one of VP1 domains [7,8,11]. The relative motion of the subunits of VP1 together with VP2, VP3 leads to an increase of the diameter of the capsid, and results in the opening of three types of pores, two of which at the quasi-threefold and twofold axes. The other channels are located at symmetry-related positions (around the fivefold axes) and supposedly allow to externalize a portion of VP1 (N-terminal residues) that is thought to be instrumental in the adhesion of the capsid to the interior of the endosomal membrane, through which the RNA must pass in order to enter the cytosol. The pores at the twofold axes are relevant for infection because they are wide enough to allow the transit and exit of VP4 and the RNA. With reference to figure 1a they are located at the interfaces VP2a:VP2m and at positions related by icosahedral symmetry, and their formation is caused by the relative clockwise motion of VP2a and VP2m that move away from each other as a result of the hinge movement within the protomers [10]. There is evidence that the RNA leaves the capsid at one of the pores at the twofold axes [24,25], and the process starts at its 3′ end [18]. Furthermore, in the native virion, the 5′ end, which is the end at which RNA synthesis begins, is bound inside the capsid to a CP at a threefold site roughly opposite the twofold exit side. The expansion of the capsid is the result of a collective rearrangement induced by rigid body motions of protomer domains (figure 1b,c,d). These rigid motions will be parametrized by a single angular variable, one for each protomer, denoted by θi ∈ [0, 1] for i = 1, …, 60, with the convention that θi = 0 and θi = 1 correspond to the closed and open configuration of the protomer, respectively. Also, we assume that the protomers are labelled as in figure 2a on a Schlegel diagram of the capsid. The numbering scheme has been chosen such that protomers in the same pentamer are consecutively numbered, and that the numbers of opposite protomers add up to 61. Each capsid configuration is associated with a vector θ = (θ1, …, θ60). Hence, the two states θi=0 ∀i and θi=1 ∀i correspond to the native (closed) and expanded (open; A particle) capsid. Figure 2. (a) Schlegel diagram of the capsid architecture, showing the numbering scheme used to identify individual protomers in the viral capsid. (b) Intra-pentamer interactions around the particle fivefold axes (adjacency matrix Aij) are shown as pentagons (red in the online version). (c) Inter-pentamer interactions across the particle twofold axes (adjacency matrix Bij) are shown as lines (green in the online version). (d) Inter-pentamer interactions around the particle threefold axes (adjacency matrix Cij) are shown as triangles (blue in the online version). (Online version in colour.)
The changes in the capsid may occur either simultaneously or as a cascade of subsequent events, and we model it as a sequence of elementary transitions between metastable states, i.e. local minima, of a suitable energy function. The energy cascade mechanism has been demonstrated for the maturation of HK97 in [26,27], and mathematical models based on these ideas have been developed in [17,28]. We assume that the free energy is the sum of three components, that take protein–protein interactions into account, and model protein–RNA interactions as parameters modifying these energy terms. The first term, that we call the intra-protomer energy, accounts for the forces that drive the structural transition of a protomer after the removal of the pocket factor, as well as the barrier that has to be overcome to break the bonds necessary for this process. The second term is the intra-pentamer energy, that captures contributions from the bonds between the adjacent protomers within the same pentamer. Finally, the inter-pentamer energy accounts for the bonds between adjacent protomers belonging to different pentamers. The above terms only account for the energy barriers related to the breaking of bonds between different capsomers (or domains of the same protomer, as is the case for VP1). However, the interactions between the capsomers are much more sophisticated, and certainly include mechanical knock-on effects due to their mutual push and pull during the opening of the protomers and the expansion. To include these effects, we introduce a weak constraint on the motion of adjacent capsomers via a penalization term in the total energy. The relative rotation of the subunits of VP1 upon removal of the pocket factor has the effect of widening the angle between the domains by θ, that we will call the opening angle in the following, thereby disrupting and forming a number of bonds between the subunits. The impact of this conformational change on the protein interfacial energies has been quantified in [10] (see also table 1).
We describe the switch between the two configurations of the protomer via a two-well energy function, with minima at the closed and open configuration of the protomer. It is this energetic term that drives the expansion of the capsid in our model. To motivate our choice of two-well energy, we note that while in other viruses (cowpea chlorotic mottle virus, equine rhinitis A virus (ERAV)) there is evidence that the confinement of the RNA inside the capsid induces an internal pressure that drives the expansion, no evidence for such a mechanism is present for HRV2. Rather, the idea here is that, after the removal of the pocket factor, the domains of VP1 within a protomer are free to attain a preferred stable configuration, which suggests that the intra-protomer energy must have a minimum h < 0 in correspondence of this metastable state (θ = 1). However, as mentioned above, in order to reach this state the two domains of VP1 have to break a number of bonds, which suggests that the closed form of the protomer is metastable (hence the first minimum at θ = 0) and that there is a barrier k between these two wells whose size is proportional to the energy necessary to break those bonds. Finally, we know that (for values of the pH as in the endosome and in the absence of the pocket factor) the open capsid is more stable than the closed one, which suggests that the minimum at θ = 1 is deeper than the minimum at θ = 0, so that the reverse transition is more difficult than the forward one. Let henceforth f(θ) be a two-well energy as in figure 3a, parametrized by two coefficients k and h, such that
Figure 3. The individual energy terms for intra-protomer, intra-pentamer, inter-pentamer interactions and the terms for the mechanical and steric interactions. (a) Bistable intra-protomer energy f(θ); (b) plot of the function g(θi, θj) in the intra-pentamer and inter-pentamer interaction energies; (c) plot of the function r(θi, θj) accounting for the steric and mechanical interactions around a threefold axis; (d) plot of the function s(θi, θj) accounting for the steric and mechanical interactions around a fivefold axis. (Online version in colour.)
We assume that the total intra-protomer energy of the capsid has the form Eprotomer(θ)=∑i=160f(θi).2.2 In the simulations described later, we have tested four different explicit forms for the function f, without finding any significant difference. The simplest one is the fourth-order polynomialf(x)=hx2(3x2−4(p+1)x+6p)(2p−1), with p a solution of p3(p − 2)/(2p − 1) = −k/h, where k and h are as in figure 3a. We have also tried piecewise smooth functions obtained by interpolating fixed points with splines and, finally, a Gaussian mixture modelf(x)=−ln(a1 e−(x2/s1)+a2 e−((x−1)2/s2)), with a1 = 1, a2 = 400, s1 = 1/20, s2 = 1/15, inspired by recent techniques developed to reconstruct low-dimensional dynamical transition networks from high-dimensional static samples [29].This term refers to the cohesive interactions among two adjacent protomers in the same pentamer. Specifically, according to table 1, the intra-pentamer interactions are VP1a:VP1b, VP1a:VP3b, VP2a:VP1bVP2a:VP3b, VP3a:VP1b, VP3a:VP3b,2.3 that, referring to the Schlegel diagram in figure 2a, correspond to the red segments in figure 2b.The corresponding intra-pentamer energy is the energy that is required in order to break the intra-pentamer bonds that block the opening of each protomer. We assume it to have the general form Epentamer(θ)=12∑i,j=160Aijg(θi,θ j),2.4 where Aij is the adjacency matrix of the graph in figure 2b, whose nodes are the protomers and whose edges are the intra-pentamer bonds (red segments). We require that the function g has the form g(θi,θ j)=g~(a(θi2+θ j2)) where g~ is smooth and monotonically increasing from 0 to 1 in the interval [0, + ∞), g~(0)=0, and g~(x)≡1 for x ≥ 1; here 1/a > 0 is a measure of a typical interaction radius. In our simulations, we will useg(θi,θ j)=1− e−a(θi2+θ j2),2.5 with a ≫ 1 sufficiently large (cf. also [30] and figure 3b).The interaction energy g(θi, θj) has a sharp minimum at θi = θj = 0, i.e. when both adjacent protomers are closed and undeformed. However, when either θi = 1 or θj = 1, so that at least one of the protomers is open, the energy is maximal, since it is sufficient that one of the protomers is open to break the bonds involved in the interaction with its neighbours. This term accounts for the contributions of the cohesive interactions among protomers belonging to different pentamers, that we identify with the energy of the bonds between these protomers that have to be disrupted in order to expand the capsid. We distinguish between interactions around the threefold and twofold axes. According to table 1, the inter-pentamer interactions around the threefold axes are VP1a:VP2g, VP3a:VP2g, VP3a:VP3g.2.6 Referring to the Schlegel diagram in figure 2a, the inter-pentamer interactions around the threefold axes are indicated by the blue segments in figure 2d. Analogously, the interactions across twofold axes, i.e.VP2a:VP2m,2.7 correspond to the green segments in figure 2d. Consistent with this, we assume that the inter-pentamer energy is the sum of two contributions,E3fold(θ)=12∑i,j=160Bijg(θi,θ j)andE2fold(θ)=12∑i,j=160Cijg(θi,θ j),2.8 where Bij and Cij are the adjacency matrices of the graphs in figures 2c,d respectively, i.e. the graphs whose nodes are the protomers and whose edges are the intra-pentamer bonds, and the function g is as in (2.5).Neighbouring proteins tend to impact on each others motion via steric and mechanical constraints. To simplify, we assume that protomers push and pull each other across interfaces within the pentamer (figure 2b/ fivefold axes), and interfaces between different pentamers (figure 2d/ threefold axes). Because of the relative positions of the VP1 domains involved in the openings of the individual protomers, we only need to consider interactions at the ab (cf. (2.3)) and ag interfaces (cf. (2.6)), neglecting possible steric interactions across the am interface (VP2a:VP2m). This is because the interface lies at the twofold axis where the main pore opens as protomer domains move away from each other; in fact, the expansion of the capsid tends to separate these interfaces, after breaking the cohesive bonds involved in the inter-protomer energy. We account for the steric and mechanical interactions by energetic terms that penalize the separation between adjacent protomers. In our simplified model, we do not study the actual motion of the protomers in detail, but we assume that it can be represented by a function of the variables θi, so that it is reasonable to enforce the constraints by penalizing the difference θi − θj, where i and j are adjacent protomers. To describe the steric and mechanical interactions between different pentamers around the threefold axes, we penalize the relative motion of the protomers through a term of the form ES3(θ)=12∑ij=160Cijr(θi,θ j),2.9 where Cij is the adjacency matrix of the graph in figure 2d and moreover r(θi,θ j)=r~(b(θi−θ j)2) (cf. figure 3c). Here, the function r~(x) satisfies the same requirements as the function g~ in §2.3, and b ≫ 1 is a real constant measuring the rigidity of the proteins, with small values meaning soft and large values meaning rigid. In our simulations, we shall user(θi,θ j)=1− e−b(θi−θ j)2.2.10 As to the intra-pentamer interfaces, since the opening of a protomer induces a clockwise rotation of the pentamers, we can assume that the opening results in a push of the adjacent clockwise protomer. This introduces a chirality in the corresponding penalization of the energy, that has the form ES5(θ)=∑ij=160 A~ijs(θi,θ j),2.11 whereA~ij={Aijif i>j mod 50otherwise is the matrix that takes into account only the clockwise adjacencies within the pentamer, and the push force from i to j is accounted for by the functions(θi,θ j)={r(θi,θ j)if θi>θ j,0otherwise. In summary, the total energy is E(θ)=Eprotomer(θ)+c1Epentamer(θ)+c2E2fold(θ)+c3E3fold(θ)+c4ES3(θ)+c5ES5(θ),2.12 where ci, i = 1 … 5 are parameters indicating the relative strengths of individual energy contributions. In particular, the functions Epentamer, E2fold, E3fold, ES3 and ES5 are all normalized to the same height of the plateau, so that the energy barriers corresponding to the number of bonds lost in the rearrangement of the capsid across the interfaces (2.3), (2.6), and (2.7), respectively, are accounted for by the parameters c1, c2 and c3.Recall now that the intra-protomer energy Eprotomer involves two parameters k and h, with k the energy barrier necessary to break the bonds within the protomer and allow for its opening (cf. figure 3a). The actual value of the barrier is 89.5 kcal mol−1 (cf. table 1), and corresponds to the number of bonds broken during the transition according to table 1. We normalize the energy so that k = 1 by dividing all constants by this value, i.e. c1=92.889.5,c2=24.689.5andc3=48.989.5.2.13 The constants c4 and c5 represent values of the energy required to violate the steric constraints. In the absence of experimental values, we will discuss below their role and the impact of different choices for these values. As to the constant h, which is the depth of the well of the intra-protomer energy corresponding to the open state of the protomer, we have chosen it to be h = −6 in order to ensure that the total energy decreases at each transition event (cf. §3.1). Our model describes the expansion of the capsid via the sequential opening of one or more protomers as a sequence of transitions between metastable states, i.e. local minima, of the total energy. To explore the energy landscape of the capsid we use a technique borrowed from the theory of large deviations [31], based on the assumption that the transitions occur along minimum energy paths (MEPs), which are paths connecting minima that occur with maximum probability when small stochastic fluctuations are included into the model [32]. We briefly review below the above technique, referring to [32–35] for a detailed motivation and description. To describe MEPs between two minima of the energy function, assume that small fluctuations around a minimum are described by a stochastic dynamical system of the form dθ=−∇θE dt+ϵ dW, where ε tends to zero (ε ≪ 1) and W is a multidimensional Brownian motion. Every trajectory of this system starting near a minimum exits its basin of attraction with probability 1 and enters the basin of attraction of another minimum, and the trajectories concentrate with high probability near a special path that connects the two minima, called the MEP. The MEP between two minima can be computed by a technique invoking a curve evolution equation [32–34]. Consider now the collection of all metastable states of the capsid, called the state space S. The cardinality of S is 260, implying that the problem of studying all possible transitions is computationally intractable. Therefore, we have restricted our study only to transitions that involve the opening of a single protomer at a time. We have also numerically explored some reverse transitions, corresponding to the closing of some of the protomers, as well as some of those involving the simultaneous opening of at least two protomers, but we have consistently found larger energy barriers than those corresponding to single protomer openings. Hence, we define a discrete-time Markov chain with state space S such that the transition probability from state θ to state θ ′ is P(θ,θ′)=exp(−2β(θ,θ′)/ϵ2)∑ θ~∈Sexp(−2β(θ, θ~)/ϵ2),3.1 where the states θ and θ′ are such that there exists i for which θi = 0 and θ′i = 1, and θj = θ′j ∈ {0, 1} for j ≠ i, and there is a MEP between θ and θ ′ such that the energy computed along the path has a single maximum at a point θ *. We set P(θ, θ ′) = 0 otherwise. If the transition probability is not zero, we define the barrier between the two states as [35]β(θ,θ ′)=E(θ ∗)−E(θ).3.2 Note that our choice of the transition probabilities implies that we are forcing the transition to be irreversible, which is not usually the case in large deviations theory. Our choice follows from the fact that closed empty HRV2 capsids have not been observed experimentally, which suggests that the actual process of expansion is indeed not reversible, even though there are some viruses, such as ERAV, for which closed empty capsids are observed in vitro under specific experimental conditions. Irreversibility is also implied by the fact that the barriers for the forward transitions (closed to open) in our model are consistently smaller than those for the reverse transition. This is indeed a consequence of our choice of the intra-protomer energy f. In fact, according to our general scheme, the expansion of the capsid is due to the protomer reaching a more stable position, when the obstruction due to the pocket factor is removed. This is modelled by the deep well in the function f(θ). All other energy terms are cohesive and counteract the expansion. The likelihood of the reverse process (i.e. closing versus opening) is related to the relative depths of the energy wells: if the ‘open’ well is deeper than the ‘closed’ well, it is less probable to go back to the closed state than to go forward from the closed to the open state. Hence, in order to account for the difficulty (if not impossibility) of the reverse transition, we have assumed the energy well in the intra-protomer energy f to be deep enough so that the global energy (the sum of all contributions) actually decreases in each protomer opening, and the barrier for closing is higher than the barrier for opening. Note finally that this fact also implies that the energy decreases at each transition step: the energy cascade. We prove in appendix A three basic facts that facilitate the analysis of the transition mechanism. In particular, for large a and b, we have the following results:
We first address the issue of whether the expansion of the capsid is likely to occur either by a sequential process of successive openings of adjacent protomers, or by a disordered generalized opening of many or all protomers at different unrelated sites. To assess the likely transition mechanism, we note that the barrier to opening a single protomer in the native configuration of the capsid according to (3.4) is β(θ0,θ1)=k+2c1+c2+2c3+2c4+c5,4.1 where θ0 = (0, 0, …, 0) and, without loss of generality due to the symmetry of the capsid, θ1 = (1, 0, …, 0). Now consider the second transition event: if the protomer that opens is not adjacent to protomer i = 1, the energy barrier is still (4.1); otherwise, if it is adjacent to 1, then the barrier is smaller than (4.1) (see below). Hence, the most probable transition path is a sequence of openings of adjacent protomers initiated at a single nucleation site, rather than a sequence of random openings at unrelated protomers.Next, we study how the initial stages of the expansion depend on the parameters b, c4, and c5 in the mechanical constraint energy. Here we interpret b as playing the role of an elasticity modulus, with larger b meaning more rigid proteins, and c4, c5 indicating the strength of the mechanical constraints on proximal protomers. Numerical simulations (cf. table 2) and the argument presented in appendix A show that the initial stages of the transition belong to a small number of different classes, depicted in figure 4 and labelled according to their positions relative to five-, three- and twofold axes of the particle:
Figure 4. Classes of transition pathways. Different transition pathways are classified according to the positions of the second and the third protomer undergoing an opening event.
We now show how the above classification can also be derived from (3.4). Assume, without loss of generality, that protomer 1 has opened, and consider now the second opening event. The protomers that are adjacent to i = 1 are 2, 5, 6, 10, 30 (figure 2a), and the corresponding energy barriers are β(θ1,θ1,2)=k+c1+c2+2c3+2c4,β(θ1,θ1,5)=k+c1+c2+2c3+2c4+c5,β(θ1,θ1,6)=β(θ1,θ1,30)=k+2c1+c2+c3+c4+c5andβ(θ1,θ1,10)=k+2c1+2c3+2c4+c5, where we have denoted by θa,b,…,c the state in which θa = θb = … = θc = 1, and θj = 0 for all j ≠ a, b, …, c. Therefore,β(θ1,θ1,2)<β(θ1,θ1,5)⇔0<c5,β(θ1,θ1,2)<β(θ1,θ1,6)=β(θ1,θ1,30)⇔c3+c4<c1+c5andβ(θ1,θ1,2)<β(θ1,θ1,10)⇔c2<c1+c5 (always true since c2<c1). The first observation is that, when c5 > 0, the opening of protomer 2, adjacent to 1 in the counter-clockwise direction, is preferred to the opening of the protomer 5, adjacent to 1 in the clockwise position. Hence, the classes 5 and 2+ can only be realized when c5 = 0. If also c4 is small, the sequence of the openings is then determined just by the competition between the bond energies c1, c2 and c3. Since c1 > c2, c3 the energy decrease due to the disruption of the intra-pentamer interface 1–2 or 1–5 dominates and the barrier for the opening of 2 or 5 is lower than all other barriers.However, when c5 > 0, the intra-pentamer penalization term (2.11) is such that the opening of protomer 1 generates a push on the clockwise adjacent protomer 5 (when this is in the closed state) that costs energy. If protomer 5 is in the open state, there is no push and no corresponding energy penalization. Hence, assume that 1 is open. Then by the above argument the opening of protomer 2 would not generate a push on its clockwise neighbour 1 and would not require energy. On the other hand, the opening of 5 would generate a push on protomer 4 (figure 2a), which is closed, and this would require to overcome an energy barrier corresponding to c5. Hence, even though the opening of the protomers generates a clockwise push, the sequence by which they open propagates counter-clockwise, as in rarefaction waves. Actually, the opening of protomer 5 also should release the energy accumulated in the push from protomer 1 at its counter-clockwise side, so that the net gain of energy should be zero, as for the opening of protomer 2. However, by our assumptions on the form of the chiral constraint energies, the relaxation of the constraint is felt only when the opening of protomer 5 is almost complete, while the expenditure of energy due to the push on 4 operates already at the first stages of the opening, so that it does represent a barrier to overcome to initiate the process (cf. expression (A 4)). A stronger statement is that, if the steric constraint energy thresholds c4 and c5 are such that c4−c5<c1−c3andc5>0,4.2 then, with highest probability, the second protomer to open is necessarily protomer 2 (recall that c3 < c1 by (2.13)).Assume now that the second protomer that opens is protomer 2, and compute the barriers required to open any of the protomers adjacent to the pair 1–2, i.e. 3, 6, 10, 11, 15, 30. We get β(θ1,2,θ1,2,10)=k+2c1+c3+c4+c5,β(θ1,2,θ1,2,3)=k+c1+c2+2c3+2c4,β(θ1,2,θ1,2,15)=k+2c1+2c3+2c4+c5andβ(θ1,2,θ1,2,11)=β(θ1,2,θ1,2,6)=β(θ1,2,θ1,2,30)=k+2c1+c2+c3+c4+c5. Therefore, we have first that alwaysβ(θ1,θ1,2,10)<β(θ1,θ1,2,15),β(θ1,θ1,2,10)<β(θ1,θ1,2,11)=β(θ1,θ1,2,6)=β(θ1,θ1,2,30), andβ(θ1,θ1,2,10)<β(θ1,θ1,2,3)⇔c1+c5<c2+c3+c4,β(θ1,θ1,2,3)<β(θ1,θ1,2,11)⇔c3+c4<c1+c5andβ(θ1,θ1,2,3)<β(θ1,θ1,2,15)⇔c2<c1+c5 (always true since c2<c1). Hence, if0<c1−c2−c3<c4−c5<c1−c3,4.3 the third protomer that opens is protomer 10, with highest probability, and we recover class 2−, which requires the interplay of both mechanical constraints to be operative.Analogously, always in the hypothesis that the second protomer to open is 2, the opening proceeds along the pentamer in a counter-clockwise direction and belongs to class 5− when c4−c5<c1−c2−c3,4.4 since in this case c4 is too small to force the opening along a trimer.Finally, a similar analysis shows that the transition proceeds by opening a full trimer, i.e. it belongs to class 3, if c1−c3<c4−c5andc1+c2−c3<c4.4.5 Hence, as expected, the competition between the steric constraint energies determines the kinetics of the opening sequence of the protomers. As mentioned in §2, during the expansion a number of pores open in the capsid: the pores at the twofold axes seem to be the most important, because they are the widest, and allow both for the VP4 and the RNA to exit from the capsid. The opening of these pores is due to the clockwise rotation of two facing protomers, for instance those labelled a and m in figure 1a. In particular, the pore is formed when VP2a moves away from VP2m. In our schematics in figure 4 showing the first three steps of the transition pathways, this only occurs at the third stage of classes 2+ and 2−. In all other transition paths, this generally occurs at a later stage. Furthermore, in classes 2+ and 2−, the position of the first pore to open is uniquely determined, while in the other expansion modes the location of the first twofold pore to open is random. Viral capsids are metastable structures—stable enough to provide protection for the viral genome, yet able to release their genomic cargo upon environmental cues. An important feature regulating this balance is the cooperative effects of multiple, relatively weak interactions between neighbouring coat proteins. Previous studies demonstrate that the most energetically favourable mechanism for the global conformational change of the capsid during infection is a sequence of elementary conformational changes of single or small groups of protomers, that weaken the cooperative effect of the cohesive energies and destabilize adjacent protomers [28]. At each stage, the total energy decreases, as well as the threshold for each individual transition (figure 5). This mechanism has been experimentally observed in the maturation of HK97 in [26,27], and studied on simple capsid models in [17,28]. Each elementary change is viewed as a path between two energy minima on the energy landscape, and the precise sequence of openings can be determined by exploring the energy landscape via MEPs. Figure 5. A typical energy cascade for the opening of the first 6 protomers in mode 2−. The energy is plotted along a path in R60 that is the union of the first 6 MEPs, and s is the arc parameter along the path.
In this work, we have applied the above approach to HRV2, with the purpose of contributing some working hypotheses to the research on the infection mechanism of HRV2, and improve the understanding of the factors that promote the release of the genome in the host cell. Specifically, for this type of virus, no receptor is involved in the externalization of the RNA and its inclusion into the host cell. Once the virus is internalized into the early endosome, the change of pH within the endosome induces the structural modifications leading to the expansion of the capsid described in §2. As discussed there, as a result of the expansion, pores open at twofold axes of the capsid, the N-terminal arm of VP1 is externalized and VP4 exits the capsid: the N-terminal arm of VP1 is hydrophobic and is thought to be important in the adhesion of the capsid to the interior of the endosomal membrane. Furthermore, the RNA exit from the capsid starts by extruding the 3 ′-end from a pore at one of the twofold axes [18,24,25]. Experimental evidence shows that the endosome is not disrupted during the exit of the RNA into the cell [36]. Hence, it is likely that the RNA exits from the endosome into the cell through pores in the endosomal membrane, and it seems that VP4 is instrumental in the formation of these pores. A first hypothesis is that, in the absence of any regulatory mechanism, the capsid expands and all pores at the particle twofold axes open simultaneously. In this case, the 3’-end of the RNA is free to exit the capsid via any available pore (most likely the one which is nearest to its location in the interior of the capsid), but it faces the problem of exiting the endosome and entering the cell through one of the pores in the endosomal membrane. There are two possible ways to achieve this: either the RNA is extruded into the endosome and must then find a pore (which is an unlikely scenario as the genome would be degraded in the endosome), or the pore in the capsid is directly adjacent to a pore in the endosomal membrane. However, if the pore through which the RNA exits is not defined, any pores at any of the 30 particle twofold axes could contact the endosomal membrane, making this process inefficient and thus unlikely. This suggests the following alternative. If the expansion is initiated at the site at which the 3 ′-end of the RNA is located, and if the expansion pathway is such that it preferentially opens the pore nearest to the 3 ′-end, then VP4 would be externalized at that capsid pore first, enabling it to open a pore in the endosomal membrane just facing this pore. This would then result in formation of a channel through which the RNA could exit both the capsid and the endosome and enter the cell. Then the expansion would proceed, but there would be no need for the opening of further pores in the endosomal membrane, and the problem of the localization and matching of the three players in this game (the 3 ′-end of the RNA, the pore in the capsid and the pore in the endosomal membrane) would be solved. Also, if the first twofold pore that opens during the transition pathway was uniquely determined in this way in proximity to the binding site at the 3 ′-end of the RNA, this would ensure that the RNA could exit even if capsid expansion was stopped or slowed down after a few steps. We have shown here that a simple coarse-grained model can account for initial pore opening immediately near the 3 ′-end of the RNA, and that this automatically occurs for every energy-decreasing transition pathway, if the constants c4 and c5 satisfy reasonable constraints, i.e. c4 ∼ c5. This is highly likely, given that c4 and c5 are related to the mechanical response of two different interfaces of the same protein. However, our coarse-grained model cannot explain how binding of the 3 ′-end of the RNA to a protomer may promote the initiation of the structural capsid transition at this protomer. There are precedent cases for RNA–CP contacts changing the conformation of capsomers and their ability to interact with other capsomers in the capsid shell, such as the allosteric conformers switch of the MS2 dimer triggered by contact with the RNA. Our model suggests that, similarly, the RNA–CP interactions at the protomer in contact with the 3 ′-end of the RNA might account for its role in initiating the transition pathway. However, this suggestion requires experimental validation. In summary, our model is designed to demonstrate that it is possible, for a realistic range of the structural parameters describing the capsid, to envisage a simple mechanism by which the three fundamental players (the 3 ′-end of the RNA, the pore in the capsid and the pore in the endosomal membrane) act in a coordinated fashion without the need for further regulation or any indeterminacy to ensure an efficient and fast release of the genome into the host cell. This article has no additional data. We declare we have no competing interests. R.T. thanks the EPSRC for an Established Career Fellowship (EP-R023204-1), the Royal Society for a Royal Society Wolfson Fellowship (RSWF-R1-180009) and the Wellcome Trust for a Joint Investigator Award with Prof. Peter Stockley (University of Leeds) (110145 and 110146). G.I. and P.C. acknowledge support from the grant ‘Stochastic methods for applications’ (University of Torino, DINE-RILO-17-01) and from PRIN 2017 project ‘Mathematics of active materials: from mechanobiology to smart devices’. We are grateful to Prof. Peter Stockley for his valuable comments and feedback on the manuscript. We provide here a justification for assertions (a), (b) and (c) in §3.2. To prove (a), note that when θi ∈ {0, 1} for all i, each term in the total energy (2.12) either attains a local minimum or is locally constant. Therefore, since the minimum of a sum is greater than or equal to the sum of the minima, it follows that all θ ∈ {0, 1}60 are minimizers. Other minimizers could in principle exist, but they cannot be found analytically. We have therefore performed a set of global numerical searches (trust-region algorithm with Matlab Global Optimization Toolbox), which showed no evidence of other local minima. Assertion (b) has been also validated by numerical simulations, but an analytical proof can be sketched using an asymptotic expansion as b → ∞ of the action functional based on the notion of Γ-convergence [37]. The argument is as follows: it is known that the MEP connecting two minima θ0 and θ1 is a minimizer of the geometric action functional (cf. [34]) J(θ)=∫01{|∇E(θ(s))∥θ ′(s)|+∇E(θ(s))⋅θ ′(s)} ds=∫01|∇E(θ(s))∥θ ′(s)| ds+E(θ(1))−E(θ(0)),A 1 among all smooth curves θ(s) such that θ(0) = θ0 and θ(1) = θ1. Since the last term is independent of the path, the MEP is also a minimizer of the functional I(θ)=∫01|∇E∥θ ′| ds.For simplicity, we write b=n∈N, En in place of E, and split the energy as the sum of two terms, the first of which is independent of n: En=E0+c4ES3+c5ES5,E0=Eprotomer+c1Epentamer+c2E2fold+c3E3fold. Introducing the functionalsIn(θ)=∫01|∇En(θ(s))∥θ ′(s)| ds,I~0(θ)=∫01|∇E0(θ(s))∥θ ′(s)| dsandI0(θ)=I~0(θ)+Φ(θ(0),θ ′(0))+Ψ(θ(1),θ ′(1)), with Φ and Ψ depending only on the boundary values of θ and θ ′, we first note that the MEP relative to I~0 is a straight line segment. In fact, let θ0 and θ1 be two metastable states such that θ j0=θ j1∈{0,1} if j ≠ i and θi0=0, θi1=1 for some protomer i. We claim that the minimizer θ(s), s ∈ [0, 1], of I~0 connecting θ0 to θ1 isθi(s)=s,θ j(s)≡θ j0=θ j1∈{0,1}for j≠i, s∈[0,1].A 2 To see this, we resort to the characterization of the MEPs in [34] as parametrized curves θ = θ(s) with the property that ∇E0(θ(s)) is parallel to θ ′(s) for s ∈ [0, 1]. For j ≠ i, we have that∂E0∂θj=f ′(θ j)+2a∑k(c1A jk+c2B jk+c3C jk)θ jexp(−a(θ j2+θk2)),A 3 and since f ′(0) = f ′(1) = 0 and θ jexp(−a(θ j2+θk2))∼0 whenever θj ∈ {0, 1}, this term vanishes on (A 2), which is therefore the MEP for I~0. This implies that also the minimizers of I0 are the straight-line segments (A 2): consider in fact a putative minimizer of I0 that is not a straight line segment, and modify it on the interval (δ/2, 1 − δ/2) by a smooth curve that coincides with (A 2) on (δ, 1 − δ), so that the boundary values of θ and θ ′ do not change. Since (A 2) is a minimizer of I~0, as δ → 0 this would decrease I0, which is a contradiction.We now turn to the asymptotics of the action functional. We say that InΓ-converges to I0 if (cf. [37])
This notion of convergence for the action functional has the advantage that, if a sequence of minimizers of In converges to some θ, then this is a minimizer of I0 (cf. [37]). In terms of the MEPs, this means that we can approximate the MEP for large b in terms of the minimizer of the action functional I0, which is a rectilinear path. The choice of the topology that defines the convergence of the sequences θn must guarantee that sequences of minimizers converge, and, at the same time, that g1 and g2 hold. To simplify matters, we choose here the C1([0,1],R60) norm, which is admittedly too strong to ensure the convergence of minimizers, but makes g1 and g2 simpler to prove. With some additional technical effort the result below can be extended to weak convergence in W1,∞([0,1],R60), that guarantees the convergence of minimizers. We sketch here a proof of the fact that InΓ-converges to I0 for a simplified problem in two dimensions, with θ = (θ1, θ2), assuming that the energy reduces to En(θ)=f(θ1)+f(θ2)+K(1−exp(−n(θ1−θ1)2), with E0(θ) = f(θ1) + f(θ2) and K a positive constant.Consider now any sequence θn such that θn → θ and (θn) ′ → θ ′ uniformly on [0, 1], such that, to fix ideas, θ0 = (0, 0) and θ1 = (1, 0): expanding the square of the first factor in the action functional In and letting ρn(s)=θ1n(s)−θ2n(s) and Rn(s)=2nKρn(s)exp(−nρn2(s)), we find |∇En(θn(s))|2=(f ′(θ1n(s))+Rn(s))2+(f ′(θ2n(s))−Rn(s))2. The idea is now to decompose the integration domain [0, 1] into the union of Iδn={s∈[0,1]:|ρn(s)|<δ} and its complement. On [0,1]∖Iδn it straightforward to see that Rn converges uniformly to zero, while on Iδn the sequence Rn is not bounded in the sup-norm and dominates the terms involving f ′. Hence, In can be approximated byIn(θn)∼∫Iδn2|Rn(s)||(θn) ′(s)| ds+∫[0,1]∖Iδn|∇E0(θn(s))∥(θn) ′(s)| ds for n large. Now, ∇E0(θn(s)) and (θn) ′(s) converge uniformly to ∇E0(θ(s)) and θ ′(s) on [0, 1], and therefore, by the arbitrariness of δ, the second term above converges to I~0(θ). On the other hand, if we assume for simplicity that the only point at which ρn(s) vanish is s = 0, on Iδn we have that ρn(s)=θ1n(s)−θ2n(s)∼(c1n−c2n)s, with cin=(θin) ′(0)→ci=θi ′(0), so that the first term above becomes, if c1 ≠ c2,∫Iδn22nK|c1n−c2n|s e−n(c1n−c2n)2s2(c1n)2+(c2n)2 ds∼K2((c1n)2+(c2n)2)|c1n−c2n|→K2(c12+c22)|c1−c2|, which is the function Φ(θ(0), θ ′(0)) in the expression for I0.Third, we prove (c). Given that, as b = n → ∞ the MEP is approximately the straight line segment (A 2), we can compute the barriers by evaluating the energy on it: E(t)=f(t)+∑ j(c1Aij+c2Bij+c3Cij)(1−exp(−a(t2+θ j2)))+∑ j(c4Cij+c5 A~ijH(t−θ j)+c5 A~ jiH(θ j−t))×(1−exp(−b(t−θ j)2))+ const., where H is the Heaviside step function. Therefore,E(t)−E(0)=f(t)+∑ j/θ j=0(c1Aij+c2Bij+c3Cij)(1−exp(−at2))+∑ j/θ j=0(c4Cij+c5 A~ij)(1−exp(−bt2))−∑ j/θ j=1(c4Cij+c5 A~ ji)(1−exp(−b(1−t)2)).A 4 The above function is the sum of f and three monotone functions, two increasing and one decreasing. For b and a large enough, E has a single maximum in [0, 1], at the point at which f has its maximum, and the value of the maximum of E is exactly the sum of the maxima of its constituents, which yields the relation (3.4).Finally, we list in table 2 the results of numerical simulations showing how the initial opening mode depends on the constants c4 and c5. Footnotes© 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 18One of the questions that has interested evolutionary theorists for almost 100 years is the generation and spread of mutants. Starting from early works of Haldane [1], Fisher [2] and Wright [3], researchers focused on the probability and timing of mutant fixation under various assumptions, see e.g. seminal work of Kimura on neutral evolution [4,5] and Patwa & Wahl [6] for a review on fixation probabilities of advantageous mutants. Around the 1950s, mutant evolution in random environments attracted attention of evolutionary biologists and mathematicians. Different aspects of this problem have been investigated, such as the evolution of mutation rates in random environments [7–10]. In [11], the role of a randomly fluctuating environment was studied by assuming that while the wild-types had constant numbers of offspring, mutants’ numbers of offspring were random (with the same mean), and changing every time step. It was shown that in this case the mutants were negatively selected. A more general setting of randomly fluctuating environments was studied by Frank & Slatkin [12] and Frank [13]. It was shown that even if the environmental changes affected the division rates of mutants and wild-types in a similar way, minority mutants had an intrinsic advantage due to frequency-dependence effects. A similar conclusion was reached in the study of Melbinger & Vergassola [14], who also obtained results on the timing of mutant fixation in randomly changing environments affecting the agents’ division rates. In [15], it was shown that a fluctuating environment changed the forces of selection and allowed mutants that were on average weaker to become fixated with a probability much larger than expected (based on their mean fitness). In ecology, a set of profound results have been obtained in the framework of the Modern Coexistence Theory, which is concerned with the instantaneous rate of increase of a rare species [16–18]. In particular, Chesson & Warner [19] have shown analytically, that temporal variability in division rates would lead to a positive rate of increase of the rare species (even if it is an inferior competitor), and that both species would (in the symmetric fitness scenarios) be attracted to a fixed point of equal abundance. This is known as the ‘storage effect’. Chesson & Warner [19] have further shown that variability in death rates would lead to the opposite effect, leading to a negative rate of increase of the rare species. Analytical results for extinction times have been derived by Kessler et al. [20], Hidalgo et al. [21] and Danino & Shnerb [22]. A different type of randomness is associated with the existence of ‘spots’ that may be characterized by different (random) conditions for the agents. For example, agents can be assumed to interact on a heterogeneous or random network, where not all vertices have the same number of connections and hence their fitness values are based on different numbers of interactions, making some vertices more advantageous than others. Several groups studied related problems, especially in the context of the game theory and cooperation (e.g. [23–28]). In a different style of modelling, fixation of mutants was studied in a spatially heterogeneous environment, represented by a multi-patch (finite island) model, where fitness values of mutants and wild-type agents were different depending on the patch. Both the high migration rate limit [29] and the low migration limit [30] have been studied. Gavrilets & Gibson [31] derived formulae for the probability of fixation for both beneficial and deleterious alleles in a symmetric two-deme system. These results were extended and generalized in the study of Whitlock & Gomulkiewicz [32], who reported that under symmetric migration, spatially non-homogeneous fitness in a metapopulation never decreased—and sometimes substantially increased—the probability of mutant fixation, compared to metapopulations experiencing homogeneous selection with the same mean selection intensity. The role of spatially variable environments in species coexistence has been also addressed by the Modern Coexistence Theory. In particular, the environmental influence on the growth rate of the rare invader has been studied analytically [33], finding a generally positive effect. In [34], we studied the effect of spatial randomness on mutant fixation probability. Our setting was different from the finite island or the random graph models. We assumed that for wild-type and mutant cells, the division rates at different locations were drawn from identical probability distributions, and for a given realization they did not change in time. We showed that under the Moran or haploid Fisher–Wright processes on a complete graph or on a circle, as long as initially mutant cells were in a minority, their fixation probability was significantly higher than their initial frequency (which is the fixation probability under the neutrality assumption). We further demonstrated that the fixation time (conditioned on mutant fixation) was significantly affected by the randomness, and that randomness increased the mean conditional fixation time on a circle but decreased it on a complete graph [35]. In this paper, we explore aspects of mutant dynamics that have not been previously addressed in the literature. This includes systematic comparisons of (1) spatial versus temporal variation; (2) division rates versus death rates being affected by environmental randomness, (3) mass action versus spatially restricted systems and (4) birth–death (BD) versus death–birth (BD) Moran processes. To put our work in the context of the existing results, below we list some of the key novel findings reported in this paper. (a) Classical results in non-spatial systems with temporal fluctuations can be destroyed by spatial interactions. It is well known that under temporal randomness in a completely mixed system, minority mutants experience (i) higher than neutral fixation probability and a higher mean conditional fixation time, if the division rates are affected by randomness and (ii) lower fixation probability and lower mean conditional fixation time if the death rates are affected. In this paper, we show that when spatial interactions among individuals are added, the above effect is weakened, and can be completely destroyed. Mutants in a circular graph experience neutral dynamics for the DB update rule in case (i) and for the BD rule in case (ii) above. (b) The effect of spatial randomness is not the same as that of temporal randomness. We show that in the case of spatially variable environment, both for BD/DB processes, both for complete/circular graph, and both for division/death rates affected, minority mutants experience a higher than neutral probability of fixation. This is in contrast to the temporal fluctuation case, where, for example, when the death rates are affected by randomness, minority mutants experience a disadvantage. Fixation time results are also different under spatial compared to temporal randomness. (c) Spatial and temporal correlations explain the observed differences. Finally, in this work, we discuss the nature of the two types of variability (temporal and spatial) and note a fundamental difference between the two, namely, spatial correlations of fitness values in the temporal case and temporal correlations of the fitness values in the spatial case. This paper adds to the previous work on spatial and temporal environmental variability [14,16,17,19,21,22,36–51], and attempts to generalize and explain a large amount of results for the behaviour of mutants under spatial and temporal randomness. We implement modifications of the classical, constant-population Moran process [52]. Let us denote the population size by N, the division and death rates of the wild-type cells as rA and dA, respectively, and division and death rates of the mutant cells as rB and dB, respectively. In the system with random environment, division and/or death rates of wild-type cells and mutants are subject to random change, see the electronic supplementary material, section S1 for complete details of the implementation. In particular, to model temporal fluctuations, we assume that for each time-step, the division values rA and rB are chosen from a given probability distribution, see figure 1a for a schematic illustration. In this paper, we focus on the case where the probability distributions of rA and rB are the same. Similarly, the death rate values dA and dB are chosen from a single probability distribution. We are interested in mutant fixation probabilities and times, averaged over all realizations of the rate values. Figure 1. A schematic illustrating temporal (a) and spatial (b) randomness. Cells (assumed to reside on a ring) are represented by circles. Mutants are coloured by reddish tones (marked by m) and wild-types by bluish tones (marked by w). Four consecutive time-steps are shown, and for simplicity it is assumed that the locations of the four mutant cells do not change (which is not the case in general). Saturation of the colour represents cells’ rate values, and the division rates and the death rates can each take two possible values. In (a), at each time step, all mutants and all wild-type cells have the same rate values, but they randomly change from step to step. In (b), both mutants and the wild-type cells have random fitness values determined by the cells’ locations, but they remain constant in time. (Online version in colour.)
To model spatial fluctuations, we assume that fitness values of each cell are defined by (a) its type and (b) its location. Each realization of the evolutionary process is characterized by a fixed set of wild-type fitness values, rA1,…,rAN, and a fixed set of mutant fitness values, rB1,…,rBN, where the superscript is referring to a specific location, and the values rAi and rBi are i.i.d. random variables for 1 ≤ i ≤ N. These values, once assigned, remain constant throughout the realization, see figure 1b for a schematic illustration of this model and a comparison with the setting with temporal randomness (panel (a)). Similarly, death rate values, dAi and dBi are assigned randomly for each realization. We start by studying the statistics of mutant fixation for the process on a complete graph, under temporal randomness. Assume that the division rates of wild-type and mutant cells are i.i.d. with standard deviation σr, and the death rates of wild-type and mutant cells are i.i.d. with standard deviation σd. In this paper, we are using a specific type of probability distribution, a two-valued, zero skewness distribution, where value 1 + σ occurs with probability 0.5 and value 1 − σ with probability 0.5 (here σ = σr for division rates and σ = σd for death rates). Other distributions were investigated in [34,53]; the results were found to be qualitatively similar. Results of numerical simulations are presented in figure 2. All simulations start with 1 mutant and N − 1 wild-type cells. In figure 2(a,b), the mean fixation probability is shown for both BD and DB processes, as a function of the standard deviations of the division rates (σd, horizontal axis) and the death rates (σr, vertical axes). Panels (c,d) show the corresponding mean conditional fixation times, also as functions of the two standard deviations. Figure 2. Temporal randomness, the case of a complete graph. Evolutionary properties of mutants with random birth/death are studied. Panels (a) and (b) give the results for the fixation probability for the BD and DB processes, respectively. Panels (c) and (d) give the result for the mean conditional fixation time for the BD and DB processes, respectively. We have used N = 5; 106 realizations were run, and the quantities were divided by the corresponding values in the absence of any external randomness. (Online version in colour.)
We observe that, consistent with the findings of [12–14], the probability of fixation is an increasing function of σr. In other words, a minority mutant experiences a selective advantage in the presence of temporal randomness in the division rates. The mean conditional fixation time is also an increasing function of the randomness in the division rates, as reported in [14]. A very different result is observed as we increase the amount of randomness in death rates. Minority mutants experience a disadvantage in the presence of temporal randomness in death rates (we can see a decay in the probability of fixation in the horizontal direction). This result has been discussed in the context of the so-called storage effect (e.g. [19]). This effect has been used to explain the finding that the total number of species supported by the ecosystem increases due to the variability of the environment [51]; in order to obtain this mechanism, it is important that environmental stochasticity affects recruitment instead of mortality rates [19,54]. The negative selection acting on minority mutants under random death rates which we report here, is consistent with a decreased diversity of species under this type of randomness. Results for the statistics of mutant fixation for temporal randomness on a circle are presented in figure 3, and they are strikingly different from the patterns observed in the case of the complete graph. Spatial restrictions imposed by the circular geometry eliminate any selective effects of temporal randomness in the case where the death rates are random for the BD process, and in the case where the division rates are random for the DB process. The dependence in the other direction retains the same tendency as was found for the complete graph. Figure 3. Temporal randomness, the case of a circular graph. Evolutionary properties of mutants with random birth/death are studied. Panels (a) and (b) give the results for the fixation probability for the BD and DB processes, respectively. Panels (c) and (d) give the result for the mean conditional fixation time for the BD and DB processes, respectively. We have used N = 6; 106 realizations were run, and the quantities were divided by the corresponding values in the absence of any external randomness. (Online version in colour.)
A summary of results for the fixation probability for the model with temporal randomness is given in table 1. Results for the mean conditional fixation time for the model with temporal randomness are given in table 2.
In order to understand intuitively the behaviour of mutants in random conditions, we examine the expected increment of the number of mutants. Suppose that initially, there are m mutants in a population of N cells. Let us denote by P(1)(+i) and P(1)(−i) the probability that after one update, the number of mutants will increase (decrease) by i. The superscript in these notations refers to the number of steps considered. Then the expected increment of the number of mutants after one update is given by Q(1)=⟨P(1)(+1)−P(1)(−1)⟩,3.1 where averaging is performed over all realizations of the process. In general, the expected increment of the number of mutants after n updates is defined asQ(n)=⟨∑i=−nniP(n)(i)⟩.3.2 How do we explain that in the case of temporal randomness, for random divisions, minority mutants are ‘selected for’ (that is, fixate with a probability greater than their initial proportion), and for random deaths, majority mutants are ‘selected for’ (this holds both for DB and BD on the complete graph)? Consider the process on the complete graph, and observe that the statistics of the probability of mutant increase in BD, equation (3.3), in the absence of randomness in death, are identical (up to the sign) to the statistics of the probability of mutant decrease in DB, equation (3.4), in the absence of randomness in divisions: QBD,σr,compl. graph(1)=⟨rB−rAmrB+(N−m)rA×m(N−m)N−1⟩3.3 and−QDB,σd,compl. graph(1)=⟨dB−dAmdB+(N−m)dA×m(N−m)N−1⟩.3.4 For the BD process, expression (3.3) contains the fraction of total fitness contributed by a mutant minus the fraction of the total fitness contributed by a wild-type cell. Assume that m < N/2. If in a particular situation, mutants have a smaller division rate than wild-types, the difference is measured against the total population fitness, which is relatively large, due to a majority of advantaged types. If mutants have a larger division rate than wild-types, this is measured against the total population fitness that is smaller due to a decreased contribution of the majority. This results in a larger contribution of terms corresponding to mutants dividing faster, and an overall positive mutant increment.To show this more precisely, consider the case of the BD rule with random divisions. Suppose first that rB = b, rA = a and b > a, such that the contribution to QBD,σr(1) is positive. The opposite situation where rB = a, rA = b, happens with the same probability, and its contribution is negative. The absolute values of the two contributions are however different, and the sum of the two is given by (b−a)2(N−2m)ab(N+m(b/a−1))(N−m(1−a/b)).3.5 Since b > a and m < N, both terms in the denominator are positive, and the total quantity is positive as long as m < N/2. Summing up over all possible values of a and b with b > a and the corresponding probabilities, we can see that the expected increment is positive for m < N/2. This argument shows that for any probability distribution of the division rates, mutants are effectively positively selected as long as they are in a minority. Exactly, the same argument shows that under the DB rule with random deaths, mutants are negatively selected as long as they are in a minority.The other two cases on a complete graph (BD with random deaths and DB with random divisions) can be handled in a similar way. This argument breaks down however once we consider a circular graph, in two out of four cases. Neutrality on the circle is expected when the random case corresponds to the second event: random deaths in BD and random divisions in DB. For the second event, there are only two cells that are competing, and there is no minority or majority in this case. This makes the mutant dynamics neutral for the BD process with random deaths and for the DB process with random divisions. To illustrate this, let us consider the case of random division rates on a circle. For the BD and DB processes, the expected increments are given by QBD,σr,circle(1)=⟨rB−rAmrB+(N−m)rA⟩3.6 andQDB,σr,circle(1)={⟨2NrB−rArB+rA⟩=0,1<m<N−1,⟨1NrB−rArB+rA⟩=0, m=1 or m=N−1.3.7 We can see that in the former case, the expression is similar to the expression in (3.3), and results in a positive increment for minority mutants. The latter case averages out to zero, thus leading to neutral dynamics of mutants regardless of whether they are in a minority or a majority. The cases corresponding to random death rates are analysed similarly.For completeness, we have also considered the case of a different update rule, the pair competition. In this model, two individuals are chosen randomly, and then one of them dies and the other one reproduces according to their relative fitnesses (which is subject to the influence of the variable environment). We have observed that the fixation probability remains constant (and equal 1/N) for the temporal randomness. Results for mean conditional fixation time are summarized in table 2. We can see that the time to fixation exhibits exactly the same trends as the probability of fixation, see table 1: whenever the mutant behaves as if it is selected for (i.e. its probability of fixation increases), its mean conditional time also increases, and whenever the mutant’s fixation probability decreases, so does its mean conditional fixation time. For further details, see electronic supplementary material, section S2. Next we turn to the case of spatial randomness. Results for evolutionary properties of mutant dynamics are presented in figure 4 for the complete graph, and in figure 5 for the circle. As before, for the numerical studies, we focus on the case where initially, there is one mutant in the system, and consider both the mean mutant fixation probabilities and the mean conditional fixation time. Again, both BD and DB processes are studied (BD in the left panels and DB in the right panels). Figure 4. Spatial randomness, the case of a complete graph. Evolutionary properties of mutants with random birth/death are studied. Panels (a) and (b) give the results for the fixation probability for the BD and DB processes, respectively. Panels (c) and (d) give the result for the mean conditional fixation time for the BD and DB processes, respectively. We have used N = 6; 106 realizations were run, and the quantities were divided by the corresponding values in the absence of any external randomness. (Online version in colour.)
Figure 5. Spatial randomness, the case of a circular graph. Evolutionary properties of mutants with random birth/death are studied. Panels (a) and (b) give the results for the fixation probability for the BD and DB processes, respectively. Panels (c) and (d) give the result for the mean conditional fixation time for the BD and DB processes, respectively. We have used N = 6; 106 realizations were run, and the quantities were divided by the corresponding values in the absence of any external randomness. (Online version in colour.)
The dependence of the probability of mutant fixation on the standard deviation of the distribution of the division rates (σr, the vertical axes in all panels) are consistent with the results reported in [34,35]. We observe that for minority mutants, both for BD and DB processes, and for both complete and circular graphs
It can be seen that the dependence on σr is stronger for the BD process (figures 4a and 5a) compared to the DB process (panels (b) of the two figures). Compared to σr, σd appears to have a stronger effect on fixation probability. We also notice that while the probability of mutant fixation is a monotonically increasing function of σd, the dependence on σr is somewhat more complex: as a function σr, the probability increases for low σd, but this trend slows down and even reverses for higher σd. For completeness, we have also performed simulations with the pair competition rule (when in a randomly chosen pair of connected individuals, the progeny of the one with the higher fitness replaces the one with the lower fitness). In this model, mutant fixation probability grows with the standard deviation, and the magnitude of the effect is stronger than that for the DB and BD processes (not shown). We next turn our attention to the mean conditional fixation time, figures 4c,d and 5c,d. As the standard deviation, σd, increases, the fixation time grows for both processes (BD and DB), both for the circle and for the complete graph. This means that randomness in the death rates slows down mutant fixation in both mass action and the one-dimensional nearest neighbour scenarios.1 Randomness in the division rates, however, affects the mean conditional fixation times differently, depending on the underlying graph [35]. For the complete graph, randomness in the division rates accelerates fixation (the fixation time decreases in the vertical direction in figure 4c,d). For the circle, the result is the opposite: randomness in the division rates decelerates mutant fixation (the fixation time increases in the vertical direction in figure 5c,d). An important question is whether any of these effects become negligible with growing system size, N. Interestingly, the amount of advantage enjoyed by a minority species increases in large populations, allowing this ‘selection’ force to overcome random drift [34]. One way to measure the size of the effect as N increases is to compare the probability of mutant fixation (PN) multiplied by N with unity. In a constant environment, we have PN × N = 1 or neutral mutants. In a system under variable environment with N = 4 or N = 5, this quantity is less than 1% larger than 1. As the population size reaches N = 50, the value of PN × N becomes 2.85 (i.e. 280% of that of a neutral mutant in a constant environment). For N = 600, the quantity PN × N is about 15 (about a 1500% increase). A summary of results for the fixation probability for the model with spatial randomness is presented in table 3. Results for the mean conditional fixation time for the model with spatial randomness are given in table 4.
In order to explain the results of table 3 on an intuitive level, let us study the statistics of mutant cell dynamics. The mean increment of the number of mutants after n updates, Q(n), is defined in equation (3.2). In the case of temporal randomness (§3.2), quantity Q(1) served as an indicator of the system’s behaviour. In particular, if minority mutants behaved as if they were selected for, we had Q(1) > 0 for m < N/2. Calculations change in the case of spatial randomness, as explained below. For the BD process under spatial randomness, we can show that Q(1) = 0. This means that the expected increment in the number of mutants after 1 update is zero. Therefore, we check the expected increment after two steps. We obtain that Q(2) > 0 if m < N/2 and Q(2) < 0 if m > N/2, that is, minority mutants are selected for, but this manifests itself after two steps. For the DB process under spatial randomness, the calculations are different depending on whether division rate or death rates are random. If only divisions are random and death rates are constant, we obtain that both Q(1) = 0 and Q(2) = 0. Therefore, in this case we need to check the expected increment after three steps. We obtain that for N = 3, Q(3) = 0 for all m, and for N > 3, Q(3) > 0 if m < N/2 and Q(3) < 0 if m > N/2, that is, a minority mutant is advantaged for N > 3, which is consistent with our findings reported in [34]. If deaths are also random, then for N ≥ 3 and m < N/2, we can show that Q(2) is an increasing function of randomness. To justify this approach, we focus on the correlations among division/death rates of cells. We will refer to these values simply as fitness values because they define reproductive success of cells. In the temporal randomness case, there is spatial correlation of mutant fitness values with each other (they are simply the same), and similarly, the wild-type fitness values at different spatial points are correlated (i.e. are the same), see figure 1a. Competition happens between two groups of individuals that experience this type of correlation, leading to the non-zero selection for (or against) one of the groups, see the arguments built around equations (3.3)–(3.5). In the case of spatial randomness, there are no correlations among mutant or among wild-type fitness values at different locations, see figure 1b. So if we consider an individual snapshot, it is not clear what the ‘minority’ even means. All cells have random fitness values, and it appears that we have a number of different types (four types in the case of figure 1b, and possibly more types in the case of other distributions). Now, if we consider several temporal updates in a row, correlation patterns start to emerge. Unlike the temporal case, fitness values of individual spots remain the same, but of course this is only observed if we consider more than one update. That was the motivation to look at expected increments after more than one update. As time goes by, mutant and wild-type cells redistribute in space, and this process (probabilistically) has a directionality. In order to explain this concept, let us consider a complete graph and assume for simplicity that fitness values of mutant and wild-type cells are anti-correlated, that is, each spot that is characterized by a high (low) mutant fitness will have a low (high) wild-type fitness. It was shown in [34] that such anti-correlated systems are characterized by stronger effects of randomness compared with uncorrelated systems. We claim that with each update, any change that happens in the configuration of wild-types and mutants is more likely to increase the fitness of cells than to decrease it. In general, cells of higher fitness are more likely to divide, and cells of lower fitness are less likely to divide and therefore are more likely to die (as division protects a cell from death). A death of a low fitness cell can either lead to no change (if the cell is replaced by a new cell of the same type), or it can lead to an increase of fitness at that spot in the cell is replaced by a new cell of the opposite type. Therefore, with each update, the expected fitness at each spot increases. What this means in practice is that mutant cells redistribute such that they tend to occupy spots with higher mutant fitness values, and similarly, wild-type cells redistribute trying to occupy spots with higher wild-type fitness values. Now, it is clear that in the context of gaining higher fitness, a minority has an advantage compared to a majority. Indeed, the numbers of ‘good’ and ‘bad’ spots are equal on average for different configurations, and therefore it is much easier to find a configuration where a minority enjoys a large advantage than a configuration where a majority has a higher average fitness. In the case of the complete graph, this is illustrated schematically in figure 6, where we fix a fitness configuration with dark orange circles denoting mutant favourable spots and light blue circles denoting wild-type favourable spots. Starting from an initial state of two mutants, panel (a) shows a possible path where mutants grow and redistribute and occupy only favourable spots. This is not possible for the majority wild-type (panel (b)), because for any configuration some of them would be forced to occupy unfavourable spots. The same argument (with modifications) applies to systems with uncorrelated fitness values and to processes on a circle. Figure 6. A schematic showing possible evolutionary paths where cells occupy spots that are favourable for them (the case of the complete graph). Here dark orange circles denote spots that are favourable for mutants and light blue circles—spots favourable for wild-type cells. (a) The locations of mutants are marked by ‘m’, the rest of the spots are assumed to be occupied by wild-type cells. Minority mutant cells expand and occupy favourable spots. (b) The majority wild-type cells are denoted by ‘w’. As they expand, it is impossible for them to redistribute in such a way that they only occupy favourable spots (because there are more wild-type cells than such spots). In general, it is easier for minority cells to explore fitness configurations in their favour compared to majority cells. (Online version in colour.)
Figure 7 shows the mean division rates of mutant and wild-type populations (averaged over many runs) at several consecutive time steps. In panel (a), we start with one mutant in the system of nine cells, and we see that the mutants clerkly gain fitness advantage. The same trends holds for panels (b) and (c), where we start with two and three mutants out of N = 9, respectively; the fitness advantages gain by the mutants is smaller in panels (b) and (c) compared to panel (a), and also, in panel (c) we observe that the wild-type fitness is also experiencing an increase (but a smaller increase compared with the mutant minority). Finally, in panel (d) we start with six mutants out of nine cells (a majority). We can see that the mean division rates of the mutant majority grows slower than that of the wild-type minority. Figure 7. Spatial randomness: mean fitness of mutants and wild-type cells at several consecutive time steps, starting from (a) m = 1 mutant, (b) m = 2 mutants, (c) m = 3 mutants and (d) m = 6 mutants. The total number of cells is N = 9. Simulations with random division rates are performed under the BD update rule on a circle, with σ = 0.9 and mutant and wild-type division rates anti-correlated; 10 000 realizations were used. (Online version in colour.)
Results for fixation timing are summarized in table 4. We observe that the relationship between the probability of fixation and mean conditional fixation time is different under spatial randomness, compared to the case of temporal randomness (tables 1 and 2). We refer the reader to electronic supplementary material, §3 for further details. In this study, we present analysis of mutant evolutionary dynamics in random environments, and compare the cases of temporal and spatial randomness. We find that the two are quite different in the way they affect the mutant fixation probability and the mutant mean conditional fixation time. The concept of temporal randomness is well established. There is large and growing literature on evolutionary dynamics in fluctuating environments [22,54–56], in addition to the papers already mentioned in the Introduction. The importance of temporal fluctuations of the environment has been emphasized in ecology for several decades, see e.g. [57] who introduced environmental fluctuations in the context of population extinctions. Others (e.g. [58]) used environmental fluctuations to explain observed population sizes and diversity. Changing environments play a role in the evolution of bacterial colonies residing in a host [59,60], marine plankton dynamics [61], tropical island ecosystems [54] and many other ecological contexts. In microbial communities, organisms experience changes in chemical composition, local temperature or illumination of their surroundings [62]. For larger size species, fluctuations of temperature, light, precipitations, humidity, available nutrients, etc., can strongly influence the dynamics of the evolving community [21]. Here we focused on the dynamics of non-favoured mutants, that is, mutants whose (possibly random) division/death rates come from the same probability distributions as those of the wild-type cells. Consistent with previous findings, temporal randomness (random environmental fluctuations) can significantly affect the mutant fixation probability. In particular, if the environment directly affects division rates (making them random), then a non-favoured minority mutant experiences positive selection [12,14]; if the death rates are affected, then a non-favoured mutant experiences negative selection [19,51,54]. It turns out that once spatial considerations are taken into account, these results can be weakened or even completely suppressed. This happens if the process whose rates are affected by the environmental randomness does not involve competition of individuals in a group, where one species is a minority and the other a majority. In the case of circular geometry, for example, in a DB process, divisions that follow each death event are restricted to the two nearest neighbours of the removed individual, and thus if it is only division rates that are affected by temporal environmental fluctuations, no effects described for the non-spatial systems will manifest themselves, and a minority mutant will not enjoy a selective advantage. Similarly, in a BD process on a circle where divisions are random, a minority mutant will not experience negative selection, in contrast to the same case in mass action. While the phenomenon of selection suppressors has been described in [63,64] in the context of evolution on directed graphs, selection suppression reported here is of a different nature. In terms of mean conditional fixation times (where averaging is performed both over all the realizations of the process and also over all the realizations of the rate values), we observe that under random division rates, the fixation times are longer and under random death rates, they are shorter, compared with the system in a constant environment. Longer fixation times indicate longer coexistence of the two species and also point at a possibly increased diversity under randomly fluctuating division rates. Similarly, short fixation times point towards a reduced diversity under randomly fluctuating death rates [19,21,51,54]. Again, these effects can be suppressed by spatial interactions. We next turn to the phenomenon of spatial randomness. Spatial variability and its effect of local reproduction and death parameters have been studied in the past (e.g. [65,66]). An idea of ‘source’ and ‘sink’ habitats has been developed [65] where local regulation occurred through an outflow of organisms from sources (i.e. areas of enhanced reproduction) to sinks (locally insufficient reproduction). In some cases, heterogeneity of the environment manifests itself as a number of separate, discrete patches, such as separate islands containing populations of a plant species, or different hosts containing a type of parasite; then, metapopulation, or ‘deme’, models, have been used by ecologists successfully to describe the dynamics of such structured environment (e.g. [67]). In other contexts, however, one cannot assume that each uniform patch contains a large population of interacting individuals. One example of such an environment is a biofilm. Biofilms are characterized by microscale heterogeneities in physiologically important parameters, such as chemical gradients of nutrients, oxygen, waste products and signalling compounds, as well as heterogeneities in the flow of the interstitial fluid [68,69]. Localized zones that vary widely in their physiological conditions over microscope distances create a complex evolutionary environment for the bacteria [70,71]. Another example of a spatially heterogeneous environment is a tumour. Solid cancers are characterized by a highly complex microenvironment; there are e.g. regions of acidosis and hypoxia resulting from variable blood flow through leaky immature vessels [72,73]. Apart from active tumour cells, stroma, necrotic cells and blood vessels contribute to the complex ecology of tumours. The nutrients are distributed in a complex, non-uniform fashion. Inflammatory factors (such as cytokines, chemokines and growth factors) are constantly produced, which in turn attract tumour infiltrating cells, including macrophages, myeloid-derived suppressor cells, mesenchymal stromal cells and TIE2-expressing monocytes. All these non-malignant cell populations create the environment where the evolutionary tumour dynamics unfold [74]. Another relevant context where the microscale heterogeneity of organs’ environments is recognized is building the so-called organs-on-a-chip. One of the important aspects of building organs-on-a-chip is providing various cell types and extracellular matrix environments that approximate spatial heterogeneity of the real tissues [75,76]. It is situations like these, where the spatial scale of change of environmental factors is comparable with the scale of the patches occupied by a single individual, that are the focus of the present study. Unlike temporal randomness, spatial randomness always promotes (i.e. provides positive selection for) non-favoured mutants. This holds in the cases where division and/or death rates of individuals are affected by the random environment. Intuitively, this can be understood by envisaging the dynamics of birth and death in a spatially heterogeneous (but temporally constant) environment. Statistically speaking, rounds of deaths and divisions tend to increase the mean fitness of cells. Cells of each type are more likely to be removed from spots that are not favourable and to be gained (through reproduction) at spots that are favourable for them. The fundamental asymmetry between a minority and a majority type is the opportunity to colonize more and more such favourable spots. A small minority (on average) will have access to yet unexplored favourable spots. A large majority, as it expands even further, is actually likely to lose in average fitness, because typically there are not enough ‘good’ spots to accommodate all. This type of exploration gives rise to the evolutionary dynamics whereby minority mutants behave as if they were advantageous, no matter what processes (divisions or deaths or both) are affected by the randomness. A different situation is observed when we study fixation times. It turns out that the geometry of the underlying network is the key. For the circular graph, spatial randomness in divisions and/or deaths delays fixation. For the stringent constraints imposed by the circular (one-dimensional) geometry, any disruption of the path to fixation presents an obstacle, with the overall effect of increasing the mean conditional fixation time. In contrast to this, randomness in division rates on the complete graph has the opposite effect of speeding up fixation. Since the complete graphs presents multiple paths to fixation, variation in division rates can actually open up opportunities for a faster colonization of the space, while the ‘dead zones’ that presented a serious problem in one-dimensional can now be overcome by going around them. Finally, we note that it is partially the aim of this paper to emphasize the difference between the BD and DB processes. The most striking qualitative differences between BD and DB processes are observed for temporal randomness: (1) under temporal randomness (random divisions) on a circle, a minority mutant is advantageous for BD and neutral for DB. (2) Similarly, under temporal randomness (random deaths) on a circle, a minority mutant is neutral for BD and disadvantageous for DB. We have shown that the expected mutant increment after one time-step is zero, and steps are uncorrelated. In the case of spatial randomness, consecutive steps are correlated, and even if after one update the expected mutant increment is zero, it may not be after two or more updates. As a consequence, the two processes (BD and DB) are characterized by subtler, mostly quantitative differences under spatial randomness. These differences manifest themselves computationally in how many steps it takes to observe a non-zero expected mutant increment. For example, one has to go up to three steps to see a non-trivial effect for the DB process with random divisions, as opposed to only two steps for BD, accounting for an overall weaker effect of spatially random divisions under DB updates, compared to BD updates. For other interesting differences between DB and BD processes, see [77], where we showed that the isothermal theorem [63] broke down for the DB process with differing division rates of wild-types and mutants, and for the BD process with differing death rates. Given how widely the Moran process is used, it is important to understand that in light of the above findings, the modelling choice (BD versus DB) may significantly affect the results. How does one interpret this difference? One may conclude that one of the formulations is wrong. More likely, different modelling choices may be suitable under different biological circumstances. For example, the DB process represents a death-driven system, where divisions only occur when death creates an empty space; on the other hand, the BD process assumes that cell divisions can happen regardless of space availability, but they lead to death of other cells (e.g. by crowding). It appears, however, that reality is more complicated, and simplified models such as the (constant population) Moran process, or the Wright–Fisher process, or the contact process are but idealizations of the real biological process of cellular turnover. Therefore, when using such idealizations, one has to be aware of the consequences of the details of the modelling processes. In our previous papers, we have demonstrated that, for example, the Wright–Fisher model is characterized by a qualitatively similar behaviour when exposed to a random environment [35]; further, we argued (in a different context) that the contact process could be viewed as a hybrid between the DB and BD Markov models [78], but in other contexts, it exhibited qualitatively different trends compared to the Moran model [79]. There are a number of extensions of this study that are subject of current and future work. In this paper, we concentrated on the temporal randomness whose timescale is similar to that of divisions and deaths. It will be important to explore the influence of timescales on the results presented in this paper, see also [15] and references therein. Further, in the present study, we focus on two extreme cases: the nearest neighbours (circular) graph and the complete graph. The former model represents the case where cells are restricted to certain locations and the only movement occurs through cell renewal. The latter case (complete graph or mass action) represents the opposite end of the spectrum, where cells can divide and fill a space very far from their origin. This is an implicit way of incorporating migration in the system. By exploring both cases, we hope to get the range of phenomena to be expected in models that explicitly include migration. While we expect that such explicit migration models will not lead to qualitatively new phenomena, this is subject of our current work, which will build on our previous modelling of cell migration [80]. Finally, we note that in reality, the fitness of an individual may depend not only on the present state of the world, but also on its past states (for example, unfavourable conditions today may reduce an organism’s fitness tomorrow, even if the conditions change to favourable). We did not take these effects into account in this paper, but this will be an interesting future extension. To conclude, this paper compares and contrasts temporal and spatial types of randomness and their role in the evolution of non-favoured mutants. Further work is required to introduce more realism in the system, by combining the two types of randomness and also introducing migration as a natural extension of the current models. This article has no additional data. We declare we have no competing interests. No funding has been received for this article. Footnotes1 For completeness, we have also performed simulations for the one-dimensional geometry with reflective boundary conditions. All the results are qualitatively the same (not shown). Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4593836. References
Page 19It is commonplace to assume that the principal effect of fluid flow on an oceanic ecosystem is to mix biological populations and the nutrients that they rely on. Indeed, such mechanisms lie at the heart of our understanding of annual cycles in primary productivity, whereby seasonal interactions between an upper photic mixed layer and deeper nutrient-rich waters can cause rapid increases in algal biomass over a few weeks [1]. It is natural, then, to question whether oceanic flows have significant effects upon the population dynamics, either quantitatively or qualitatively, particularly in the absence of gradients in nutrient or light or detailed behavioural responses. Is it reasonable to assume that fluid circulation ensures an essentially well-mixed environment over a range of ecologically meaningful length and time scales? While there is much work on individual zooplankton–phytoplankton interactions in shear flows [2,3] and many observations of plankton heterogeneity associated with large-scale currents [4], there is little consensus about the impact of general flows on population dynamics. A traditional view is that flows and associated effects should either wholly mix the biology or separate the biology into distinct well-mixed patches (e.g. in circulating flow structures), each with a full complement of interacting species [5]. However, simulation evidence [6] suggests that turbulence can actively drive small-scale patchiness for motile phytoplankton and experimental evidence in Palma Bay in the Balearic Islands finds a causative relation between plankton size structure and slowly varying annual flow features [7]. Here, we show that physical effects can disaggregate foodweb components and that this effective segregation can in principle dictate large-scale ecological dynamics. Planktonic organisms have different physical characteristics from the fluid in which they are found. For example, they have different densities, sizes and shapes [8]. As a result, different species will experience different drift relative to the surrounding fluid (figure 1); particle trajectories will not match streamlines of fluid flow; and inertia and sedimentation drive potentially complex trajectories [9,10]. The magnitude of these effects has to be carefully assessed, but it is clear that even a small amount of drift perpendicular to streamlines in regions of high shear can lead to very large dispersion in the direction of the flow (the well-known Taylor dispersion [11]). Non-swimming organisms do not simply follow fluid flow streamlines; depending on their relative density and shape they can accumulate or spend more time in regions of high shear or vorticity [12] or in flow regions collinear with gravity [13] that naturally arise in marine flows. Figure 1. Sketch illustrating inertial drift (red arrows) of zooplankton (black) out of eddies, allowing phytoplankton (green) within the eddies to escape grazing control by zooplankton. The responsive radius of each zooplankton is illustrated by a dashed circle. Flow streamlines are given by black lines and arrows.
If predator and prey species have different densities, or differences in size and shape, then these variable inertial characteristics will lead to them being transported differently by the flow. This can, in principle, have ecological consequences. For example, Reigada et al. [12] demonstrated that spatial segregation of predators and prey driven by inertial effects in synthetic turbulence can allow a local prey population to grow in a relatively unconstrained manner. The predator population responds at the fringes of the burgeoning prey population; the local predator population increases, the original inertial flux is countered by a diffusive flux and, eventually, the predators consume a large part of the prey population. In this way, the requirement present in many existing mathematical descriptions of bloom phenomena (for example, [14]) for a large external perturbation to kick-start the system away from dynamic equilibrium is superfluous. Here, we build on Reigada et al. [12] and investigate the effects of inertia on advected excitable phytoplankton–zooplankton dynamics for a simple two-dimensional cellular flow. This flow is chosen because it incorporates streamline curvature, and can represent an array of eddies, while the relevant flow magnitude and length and time scales can be transparently controlled. We, systematically, investigate a range of physical flow scales and ecological parameters. Further to this, we explore the spatially averaged bifurcation structure from regions of parameter space that impart non-bloom dynamics through to regions with ecologically realistic stationary or oscillatory bloom behaviour. We discover that small amounts of inertia can not only kick-start algal bloom formation in a circulating flow, but that as a consequence they can drastically alter phytoplankton–zooplankton interactions and thus mean population dynamics. Our results suggest that studies of phytoplankton–zooplankton dynamics that assume that turbulence simply mixes species in small regions may not tell the whole story. In contrast to seasonal ocean-scale cycles in productivity, harmful algal blooms (HABs) occur on a smaller spatial scale and are difficult to predict. Nevertheless, these major biological events lead directly to extensive ecological and socio-economic damage on a global scale. They occur when local algal populations undergo a period of rapid growth, causing toxic or damaging effects to surrounding ecosystems. HABs have been shown to have widespread health impacts on fish and shellfish [15], marine mammals [16], birds [17] and humans [18]. HAB events are happening more often and in more places than ever before [19], and they particularly endanger small communities in the developing world that depend on a healthy catch of seafood to sustain the local population. With HABs having such increasing negative impacts on public health and the worldwide economy, it is of growing importance that we discover and fully understand the mechanisms by which they may be triggered. Here, we argue that the temporal and spatial scales associated with simple oceanic flow features, combined with realistic physiological differences between phytoplankton and zooplankton, are likely to be important drivers of HAB dynamics. We follow Reigada et al. [12] in constructing a model for the trajectories of plankton species subject to inertial effects. Font-Muñoz et al. [7] provide experimental evidence that size structure is directly affected by kilometre-scale flow structures over a yearly cycle in a real-world coastal system. They conclude that inertial effects alone can account for the observed heterogeneity. For a spherical particle at position X with velocity V in a non-stationary fluid velocity field U(X, t), the equation of motion for the particle determined by Maxey & Riley [9] is given by m pdVdt=m fDU(X)Dt+6πaη(U(X)−V)−m f2(dVdt−dU(X)dt)−6πa2η∫−∞td(V−U(X))/dτ(πν(t−τ))1/2 dτ,2.1 where mp is the particle mass and mf is the mass of fluid displaced by the particle, η and ν are the dynamic and kinematic viscosity of the surrounding fluid, respectively, and a is the radius of the particle. Terms on the right-hand side account for the Bernoulli force from the undisturbed flow, the Stokes viscous drag, an added mass effect and the Basset history force (see [9] for details). It is clear that consideration of Kolmogorov scales may become important for larger plankton in the open ocean, and the strict validity of equation (2.1) is open to question in this range (see [20]). However, we avoid these complexities and exploit the leading-order drift of particles across streamlines in larger rotating flows. The main aim in our study is to provide a simple model of species segregation due to an inertial effect in a well-defined flow. Reducing the size of the plankton will reduce the rate of drift, but will not change the qualitative dynamics.We simplify significantly by making approximations proposed by Taylor [21], Auton et al. [22] and Druzhinin & Ostrovsky [23] (see [10]) to equation (2.1), which yields an equation for velocity that depends only upon the position of the particles. This enables us to define an effective (non-dimensional) particle velocity field V(r)=U(r)+R−1A[U(r)⋅∇]U(r)+O(A−2),2.2 where U(r) is the ambient fluid velocity field, R = 3mf/(2mp + mf) is the (non-dimensional) Bernoulli number describing a ratio of masses, and A = 12πaη/(2mp + mf) is the reciprocal of the characteristic viscous drag time of the particle. The Stokes number St is given by a ratio of 1/A and the characteristic flow time scale, such that St=u0/l0A, where u0 and l0 are flow velocity and length scales, respectively. (To derive (2.2), St is considered small, the approximate form of (2.1) is integrated and exponential transients are neglected.) Note that Font-Muñoz et al. [7] contains a typographical error on the left-hand side of their governing equation (2.1), but the simulation results remain accurate (I Tuval 2019, private communication).Following Reigada et al. [10], the divergence of (2.2) can be written in terms of the magnitude of the local strain-rate S and vorticity Ω of the original flow U. Hence, ∇⋅V=R−1A(2S2−|Ω|22).2.3 Therefore, particles move across streamlines and tend to aggregate in regions of negative divergence. If the organism is more dense than the fluid, (R < 1), then accumulation is expected in regions where S2 > |Ω|2/4, meaning there is high strain and low vorticity, while less dense organisms, (R > 1), accumulate in regions of low strain and high vorticity (inside eddies). Neutrally buoyant particles (R = 1) are passively advected by the ambient flow and do not accumulate in any particular region (unless other terms in (2.1) are retained). Equation (2.2) provides the leading-order effect of inertia in a relatively simple Eulerian flow field.The above approach is of real practical use as it allows us to consider population dynamics with a spatially continuous description across large length scales of interest (approx. 100 m). Note that in moving from a description describing individual organisms to a continuum, we must consider length scales much larger than the distance between organisms. To achieve this, we construct a system of reaction–advection–diffusion equations of the form ∂P∂t=−∇⋅(VPP−DP∇P)+fP(P,Z)2.4 and∂Z∂t=−∇⋅(VZZ−DZ∇Z)+fZ(P,Z),2.5 where VP and VZ are the effective velocity fields for the phytoplankton and zooplankton, respectively, and DP and DZ are the diffusivity coefficients. Typically, we set DP and DZ equal as effective eddy diffusivity is likely to be significantly larger than that due to swimming.The choice of excitable plankton dynamics is inspired by the general Truscott & Brindley [14] model of plankton blooms in a well-mixed system. The model considers two interacting trophic levels, phytoplankton (P) and zooplankton (Z). It consists of two nonlinear ordinary differential equations for P and Z. The model exhibits excitable dynamics: small perturbations return to the non-trivial steady state whereas larger perturbations can instigate a large excursion around phase space over an extended period, corresponding to a bloom. We therefore use the reaction terms fP(P, Z) and fZ(P, Z) from the Truscott & Brindley [14] model, such that fP(P,Z)=rP (1−PK)−RmZP2κ2+P22.6 andfZ(P,Z)=ϵRmZP2κ2+P2−μZ,2.7 where r is the maximum growth rate of phytoplankton, K is the phytoplankton carrying capacity, Rm is the maximum specific predation rate, κ is the half-saturation constant, ε is the efficiency of the zooplankton and μ is the linear death rate of the zooplankton.There are three steady states of the system: a trivial equilibrium at the origin, zooplankton extinction at (P, Z) = (K, 0) and a coexistence state at (P,Z)=(κζ,rκϵμζ(1−κKζ)),2.8 where ζ=(ϵRm/μ−1), meaning that coexistence is not possible unless εRm > μ. At ζ = κ/K, the coexistence state collides with the zooplankton extinction state and the system undergoes a transcritical bifurcation. The trivial equilibrium is a saddle point of the dynamical system, while the zooplankton extinction point is a stable node when ζ < κ/K and a saddle point otherwise.Following Truscott & Brindley [14], we find that the points at which Hopf bifurcations occur for the coexistence equilibrium are determined by the solutions to the cubic equation Kζ3−Kζ+2κ=0.2.9 Descartes’ rule of signs tells us that there must be one negative and two positive real roots. However, by definition, ζ cannot be negative, and so there are two Hopf bifurcations. As ζ increases from 0, the first bifurcation causes a stable limit cycle to come into existence, and the second results in the coexistence equilibrium regaining linear stability as a stable spiral.The effective velocity fields Vξ, ξ = P, Z, are given by Vξ=U+βξ[U⋅∇]U,2.10 where βξ, ξ = P, Z, are Maxey–Riley coefficients, with βξ < 0 for negatively buoyant particles and βξ > 0 for positively buoyant particles.Reigada et al. [12] used a turbulent stationary flow as a background flow. We shall instead employ a (stationary) cellular flow [24]. This gives us some advantages as it allows us to have full control over the length and flow speed scales of the eddies in our model, meaning we can vary the maximum flow speed as a bifurcation parameter. This enables us to perform a full bifurcational study of the spatial averages of the two species with respect to both physical and ecological processes, and leads to a simplified one-dimensional system. Here, we shall consider the simplest case where inertial effects become relevant for two-dimensional flow in a horizontal plane. However, for a vertical plane one must also include sedimentation and the Lambert–Beer law for light attenuation and thus growth dependent on light absorption by other organisms above a given position in space. Hence, the ambient fluid velocity U = (U1, U2) is given by U1=U0sin(2πxL) cos (2πyL)2.11 andU2=−U0cos(2πxL)sin(2πyL) 2.12 for (x, y) ∈ [0, L]2, where U0 is the maximum speed of the flow and L is the diameter of a single circulatory cell, which can be written in terms of the streamfunction ψ = (U0L/2π)sin(2πx/L)sin(2πy/L) and so satisfies incompressibility.Typically, submesoscale eddies of horizontal diameter 0.1–10 km (smaller than large mesoscale eddies, 10–200 km) and vertical extent 0.01–1 km can persist in the ocean for days, with some submesoscale coherent vortices even persisting for years [25]. Constrained regions can also contain circulating flows; Font-Muñoz et al. [7] indicate in their study that they observed flow features with length scales of the order of a kilometre that were relatively stable and switched around annually. Statistical measures of spatial features of the plankton are necessary to be able to compare the different spatial distributions resulting from various parameter values. We use a measure of aggregation Πp defined as Π p=1−⟨ p⟩2⟨ p2⟩,2.13 where 〈 · 〉 represents a spatial average. Πp ranges from 0 to 1 − 1/N2, where a value of 0 means there has been no aggregation (the distribution is homogeneous), and a value of 1 − 1/N2 means that all the plankton have aggregated to a single point in the grid, which has N2 mesh points.We choose a realistic scenario where the phytoplankton are assumed to be neutrally buoyant so that βP = 0, while the zooplankton are taken to be negatively buoyant with βZ = −2.22 [12]. The size L of the (sub-mesoscale) eddies was taken to be 100 m across, and the maximum flow speed U0 was varied as a bifurcation parameter to investigate the response of the system to increasing spatial segregation caused by the Maxey and Riley term in equation (2.10). DP and DZ are set at a value of DP = DZ = 1.6 m2 s−1, in line with estimates of marine turbulent eddy diffusivity [26] for flow features of the order of a few kilometres. We chose this value for the diffusivity rather than the empirically derived value of 0.04 m2 s−1 suggested by Okubo [26], so that we could make a direct comparison with the results of Reigada et al. [12] as well as allowing rapid convergence of the numerical scheme. We have repeated the simulations with diffusivity DP = DZ = 0.04 m2 s−1, a reduction by a factor of 40. The results are qualitatively unchanged, and the bifurcation value for the flow parameter decreases by less than 10%. Importantly, this points to our mechanism of bloom formation being even more biophysically relevant than the results presented below. Equations (2.4) and (2.5) were solved subject to periodic boundary conditions using a staggered mesh solver for the advection and diffusion components of the advection–diffusion–reaction equation, with an explicit Euler method used for the reaction terms. The numerical scheme was tested for convergence by repeating the simulations using a variety of grid spacing and time steps. While the two-dimensional system provides archetypal solutions for an array of eddies, it is possible that the coupling between physical and ecological dynamics may be represented well in just one dimension, with a concomitant reduction in numerical complexity. Such a simplification would allow us to examine whether the observed two-dimensional dynamics are in any way attributed to the geometry associated with stagnation points and heteroclinic connections (corners) or stream-wise instabilities around the eddy. Axisymmetric eddies are an option but also require consideration of stagnation points at their outer boundaries if placed in a periodic array. Therefore, we avoid these topological issues and develop a simple approach to investigate the effect that drift into or out of an eddy has on the population dynamics. We model the concentration of plankton species across the diameter of a single eddy for a fixed value of y = L/4, so that the x-component U1 of the background flow given by (2.11) and (2.12) vanishes and the only remaining contribution to the zooplankton effective particle flow field in the x-direction comes from the Maxey and Riley drift term βZ[U⋅∇]U from (2.10). We can then calculate its x-component V1Z for y = L/4 from (2.2). Recall that P is neutrally buoyant and so experiences no drift. Hence, (2.5) becomes ∂Z∂t=−∂∂x(V1ZZ−DZ∂Z∂x)+fZ(P,Z),2.14 with no-flux boundary conditions at x = 0 and L/2, and similarly for ∂P/∂t in (2.4).We use realistic values for the ecological parameters [14] and they can be found in table 1. The zooplankton death rate μ was chosen as a bifurcation parameter, taking values from 0.001 d−1 to 0.035 d−1. The first Hopf bifurcation occurs at μ = 0.0185 d−1 for the chosen parameter values, the transcritical bifurcation occurs at 0.0349 d−1, and so our range of values for μ ensures that we capture all qualitative behaviours of the excitable dynamical system.
In order to test the excitability of the system for different flow speeds, U0, we use the ideas of Truscott [27] and slowly change the value of r from an initial value r0. This acts as a perturbation to the system and allows us to find the critical value of dr/dt that results in the triggering of a bloom and leads to an excursion around phase space indicative of the system undergoing an excitation. All simulations were carried out using an initially homogeneous phytoplankton and zooplankton distribution corresponding to the ODE steady state (2.8), and iterated forward in time until all transients had decayed (1000 days) and the system exhibited stable limit cycle behaviour. The choice of t = 0 is arbitrary, and corresponds to the time of minimum spatially averaged phytoplankton population in the limit cycle. Figure 2 shows snapshots of the model’s typical spatial output at four instants during the bloom cycle. At t = 0 days, the phytoplankton population P remains close to its homogeneous steady state, but spatial structure is apparent in the zooplankton population Z, with individuals advected away from the centre of the eddies. At t = 66 days, Z is sufficiently depleted within the eddies that P can increase in these regions due to the local removal of grazing control; a local bloom is initiated. By t = 117 days, the local P bloom has reached its peak and spreads diffusively towards the edges of the eddies. This gives Z an increased opportunity to consume its prey in regions of lower vorticity, leading to increased predator growth on the fringes of the circulations and a decrease in P back towards its minimum. Figure 2. Large-scale blooms can be triggered via local flow effects. Spatio-temporal evolution of a plankton bloom triggered by spatial separation of predator and prey populations due to flow. An initially homogeneous distribution of zooplankton (bottom row) becomes concentrated in regions of low vorticity. This allows the phytoplankton population to escape grazing control in regions of high vorticity (top row), initiating a local bloom on an ecologically realistic time scale. (Online version in colour.)
Figure 3 is an alternative depiction of these dynamics, detailing the temporal evolution of the mean population size (normalized by maximum P population) and aggregation measures of the P and Z populations, ΠP and ΠZ, respectively, over a bloom cycle. At around t = 50 days zooplankton aggregation ΠZ reaches its maximum. As a consequence, the P population undergoes a rapid increase on a time scale of the order of days, indicating that the accumulation of zooplankton towards the edges of eddies provides enough space for the phytoplankton in the centre of the eddies to escape local grazing pressure. This leads to a decrease in P aggregation; phytoplankton spread diffusively across the eddy, and a local minimum in ΠP is reached soon after the maximum point of P at location (c). The zooplankton are then able to eat prey on the edges of the eddies and their population begins to rise before reaching a maximum at location (d). The bloom persists for a time of the order of months for the chosen parameter values. Figure 3. Evolution of spatial aggregation and population dynamics driven by flow. Dynamics of mean population size (blue for P, red for Z) and aggregation (yellow for P, purple for Z) of phytoplankton and zooplankton showing that the phytoplankton bloom occurs for a time of the order of weeks and is initiated shortly after zooplankton aggregation reaches a maximum. (Online version in colour.)
Figure 4 provides a bifurcation diagram with the inertia parameter βZ. Here, the flow speed U0 is set to 3 m s− 1, the zooplankton death rate μ is fixed at 0.012 d−1 and the value of the inertia parameter βZ is varied from −3.5 to 0. We plot the absolute value |βZ| to allow a direct comparison of the shape of the graph with those found in figure 5. The diagram demonstrates that there are no stable limit cycles for |βZ| below a value of approximately 1, at which a Hopf bifurcation occurs. A region of oscillatory solutions exists for|βZ| greater than this value but less than approximately 3. Beyond this value the spatially averaged inhomogeneous zooplankton population settles instead to a larger steady-state solution, which increases with the inertia parameter. These results allow us to establish a causal relationship between inertia and the initiation of oscillatory blooms in our model. It should, however, be noted that similar instabilities (in different physical regimes) can be induced in the absence of inertial effects, for example by a stretching flow with positive divergence [28] or through the interaction of Hopf and Turing mechanisms [29]. Figure 4. The Hopf bifurcation does not occur when inertial effects are too small. A bifurcation diagram showing the minimum and maximum spatially averaged phytoplankton density 〈P〉 for the two-dimensional system with inertia parameter |βZ| varying from 0 to 3.5, fixed maximum flow speed U0 = 3 m s−1 and zooplankton death rate μ = 0.012 d−1. (Online version in colour.)
Figure 5. Flow-induced blooms exist across a large region of parameter space. Bifurcation diagrams showing the maximum and minimum spatially averaged phytoplankton density 〈P〉 for the one-dimensional (a) and two-dimensional (b) systems with maximum flow speed U0 varying from 0 to 4.2 m s−1 and zooplankton death rate μ = 0.012, 0.015 and 0.018 d−1 in blue, red and yellow, respectively. (Online version in colour.)
The bifurcation diagrams in figure 5 show that flow-induced blooms are a phenomenon which persists across a wide range of parameter values, and also that the essential features of the two-dimensional system (2.4), (2.5) are captured by the simpler model in one spatial dimension (2.14). The figures show the steady state of the system, or the maximum and minimum values of the oscillations in spatially averaged population density for regimes exhibiting oscillatory behaviour. As the flow magnitude U0 increases, we see that there exists a critical background flow speed above which an oscillatory domain of solutions (corresponding to cyclic blooms) is initiated via a Hopf bifurcation. The bifurcation point depends on both ecological and physical parameters, and the figures show that larger zooplankton death rates μ increase the likelihood that relatively small amounts of flow may induce blooms. It is interesting to note that all three of these μ values are beneath the Hopf bifurcation point of the underlying excitable dynamical system (μ=0.0185 d−1), meaning that the physical flow effects are influencing the large-scale dynamics in all cases. Note that the critical flow speeds needed to induce oscillations (for each value of μ) are very similar in the one- and two-dimensional numerical models, indicating that the one-dimensional approximation is able to capture the behaviour of the full two-dimensional system for these parameter values. For larger flow speeds (U0 > 3.5 m s−1), the behaviour of the two systems starts to differ, with the one-dimensional system indicating a persistent bloom while the transition is more gradual in the spatially averaged output from the two-dimensional system. This is due to two-dimensional spatial effects, which can be understood by looking at the dynamics in more detail as explained below. Figure 6 depicts a two-parameter bifurcation diagram of U0 against μ for both the one-dimensional and two-dimensional systems. The lower (red) lines indicate the critical parameter value pairs corresponding to the initiation of oscillatory solutions, while the upper (yellow) lines indicate parameter value pairs beyond which oscillations no longer can be found in the solutions. In both systems, region (A) is of a very similar shape and size, providing more evidence that the one-dimensional approximation is reasonable for small background flow speeds. However, differences appear for larger values of μ and U0 with an extra region of oscillatory solutions occurring in the two-dimensional system for μ > 0.03. Figure 7a helps to explain this extended oscillatory region, showing that the strict dichotomy between stable equilibrium and stable limit cycle regimes is not perfectly inherited from the non-spatial and one-dimensional systems. Instead, there are parameter choices containing additional small oscillatory modes at the spatially averaged scale. Figure 7b shows that these oscillations are caused by differences in local spatial dynamics. We plot the mean phytoplankton concentration within a small box inside an eddy (blue), and contrast this with the mean concentration outside the eddy (red) for a region of parameter space where secondary P − Z oscillations exist. Interestingly, while the P population outside the eddy has a regular oscillation and crashes approximately every 280 days, the population within the eddy crashes only every other cycle; the eddy provides some protection from grazing. These descriptions are valuable both in demonstrating the general predictive utility of the one-dimensional model and in illustrating the secondary local flow-induced structures which may arise. Figure 6. Regions of stable and oscillatory solutions, for a range of Z mortality μ and flow speed U0. (a) One-dimensional system. (b) Two-dimensional system. Oscillatory solutions, corresponding to flow-induced bloom dynamics, exist over a large area of parameter space (b). Region (A) corresponds to stable non-bloom dynamics for low flow speed (U0) and zooplankton mortality (μ) and region (C) to persistent blooms for large values of the same parameters. Critical combinations of flow and mortality are required to start a bloom. The bifurcation structure is complex close to the upper (yellow) boundary and while there is some numerical sensitivity there is convergence to a curve with geometrically interesting features. (Online version in colour.)
Figure 7. Local secondary oscillations in the two-dimensional model. (a) Bifurcation diagram showing spatially averaged phytoplankton concentration for μ=0.03 d−1 and varying U0. (b) The mean phytoplankton concentration within a small box inside an eddy (blue), compared with that outside the eddy (red) for U0 = 1.3 m s−1; note the contrasting local period-2 oscillations, which account for the observed period doubling at the spatially averaged scale in (a). (Online version in colour.)
As a measure of excitability, in figure 8 we plot the critical value of (1/r0)dr/dt required for a trajectory to take a large excursion around mean P − Z phase space rather than returning directly to the coexistence equilibrium point (2.8). This value is plotted against U0 with μ=0.012 d−1. The curve is seen to meet the x-axis at around U0 = 1.98 m s−1, which corresponds to the lower Hopf bifurcation point in figure 5b. Therefore, even if flow speeds are not sufficient to cause the system to oscillate, an increase in flow speed can result in enhanced bloom excitability in the presence of an auxiliary environmental perturbation. Figure 8. The perturbation threshold for excitable behaviour decreases with maximum flow speed. Plot showing the value of (1/r0)(dr/dt) against U0, with μ=0.012 d−1, for values between U0 = 0 and 1.98 m s−1, the Hopf bifurcation point. (Online version in colour.)
We have shown that the inclusion of physical effects, such as small differences in inertia or buoyancy between predators and prey, can dramatically affect encounter rates between planktonic species, and that these changes can have consequences at ecological scales. For illustration, the one-dimensional model indicates that the cross-stream velocity of a copepod of radius a = 5 mm of density ρp that is 10% more dense than water, ρf, in an eddy of radius L/2 = 100 m with maximum flow velocity U0 = 1 m s−1 is given by V=−(2/9)((ρ f−ρ p)a2U02π/ηL)sin(4πx/L) m s−1, giving a maximum drift speed of 9 mm s−1. Even with a mean drift speed of 1 mm s−1 the organisms will migrate to the fringes of the eddy in a time of order 1 day. Over a time scale of several days, segregation between species and thus a significant reduction of grazing can occur. This has the effect of forcing trajectories in population dynamic phase space. If the underlying system is excitable then large excursions away from steady states are expected. Moreover, population dynamics where phytoplankton–zooplankton cycles are present (limit cycles or more complex attractors) in fully mixed systems can be quenched by inertia-induced drift (e.g. figure 5). The numerical results show that blooms can be triggered by increased circulation flow speeds leading to greater spatial segregation between predator and prey. Hence, the flow itself can not only induce plankton bloom formation but can also qualitatively impact the population dynamics, shifting oscillatory dynamics to steady states and vice versa. One criticism of the current approach is that the population dynamics depend only on local concentrations and not fluxes. Clearly, higher contact rates may increase grazing of phytoplankton by zooplankton, an effect that could be considered in future investigations, in line with Lewis et al. [30]. For mid-range flow speeds (typically, 1 − 3 m s−1 with our set of parameters), the inertial terms drive solutions away from steady states into oscillatory bloom solutions. Essentially, slightly dense zooplankton are gradually drawn out of eddies where there is a relatively low mean phytoplankton number density. The resultant reduction in grazing in the centre of the circulation reduces the constraint on phytoplankton growth and they are observed to bloom. However, large local gradients of phytoplankton concentration inevitably drive diffusion down the gradients. The zooplankton graze the phytoplankton at the edge of the eddies and grow in number themselves, generating diffusive fluxes of zooplankton that swamp inertial fluxes, and leading to consumption and repression of the eddy-focused bloom down to levels below the steady state. For the parameter ranges explored, any non-negative predator death rate (below the rate at which the coexistence equilibrium disappears) permits oscillatory solutions for some range of flow speeds. Sufficiently large flow speeds (typically U0 > 3 m s−1) lead quickly to disaggregation of species, with a zone of overlap between P and Z. Oscillatory dynamics are lost and phytoplankton are seen to reach high concentration in the centre of eddies, bounded above by the carrying capacity K. The observed mean phytoplankton concentration reflects the increasing size of the zooplankton-absent zone with flow speed. The ecological model presented herein is a simple and mathematically tractable way to capture the excitable plankton dynamics between two trophic levels, predator and prey. It is notable, however, that many HABs involve mixotrophic species [31]. For example, the bloom-forming dinoflagellate Noctiluca scintillans is a mixotrophic species which both feeds on phytoplankton and exploits the photosynthetic ability of ingested Pedinomonas noctilucae living in their vacuoles. Indeed, because the ingested microalgae may themselves be toxigenic, this mixotrophic relationship has been postulated as a mechanism which may increase HAB toxicity [32]. For species whose flow-related biophysical parameters are known, the methods of Hammer & Pitchford [33] can be adapted to quantify the joint role of mixotrophy and fluid motion in HAB formation, and will form a useful subject of future work. The results in this paper are for a horizontal two-dimensional flow, and demonstrate that the interaction between physical and ecological systems gives rise to consequences unaccounted for by either system on its own. The model takes a simplified view of mixing by only including effective eddy diffusivity as a means for cells to spread out across the spatial domain, while the ability for cells to accumulate due to the Maxey and Riley effects is the cause of spatial segregation between predator and prey. However, much mixing occurs in the vertical direction. In order to consider the impact of vertical mixing one must also give careful consideration to sedimentation, light dependence and physical effects at the upper and lower boundaries. Behrenfeld & Boss [34] give a comprehensive overview of the effect of nutrient and light availability on phytoplankton biomass and how these change with mixed layer depth, building on the seminal work of Sverdrup [1]. At leading order one might assume that sedimenting organisms are spherical and that gravitational torques and biased swimming motion can be neglected. However, this is generally not the case. For instance, many plankton, such as diatoms, are markedly elongated and this can have a dramatic effect on sedimentation velocity [13]. Also, many species are bottom heavy or subject to sedimentary torques due to body asymmetry and swim in biased directions relative to gravity [6,35]. The growth of phytoplankton is very much dependent on the light availability, and the phytoplankton may themselves be phototactic [36]; models could include the well-known Lambert–Beer law for light attenuation and thus growth, and may also incorporate upward motion. Finally, there are different scenarios regarding the lower boundary condition: no-flux and no-slip conditions suggest a shallow sea whereas to model a mixed layer overlying deeper seas requires careful consideration of biomass loss and nutrient upwelling events [37]. All of these aspects merit further detailed study. Model code files and data for plots are included in the electronic supplementary material. J.R.W. conducted the analysis and numerics. All authors contributed to the design of the study and to the writing of the manuscript. We declare we have no competing interests. This work was supported by the EPSRC (grant no. EP/N509802/1). We thank our collaborator Prasad Perlekar (TIFR Hyderabad) for his assistance with numerical methods. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4576175. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 20Many animals rely on olfaction for detecting food, natural predators and mating partners. The odorant is initially recognized by olfactory receptor neurons (ORNs). The information is then transferred to a secondary region, either the antennal lobe in insects or olfactory bulb in vertebrates. Projections from the secondary region extend to higher order brain regions, the mushroom body and lateral horn in insects and the orbitofrontal cortex, amygdala, entorhinal cortex and ventral striatum in vertebrates. The architecture of the olfactory circuit differs from that of other sensory modalities (for a review, see [1,2]); for example, the olfactory circuit consists of fewer layers. Therefore concepts derived from experimental and theoretical studies on other systems may not be applicable to olfaction. Computational models that can replicate the behaviour of real ORNs during odorant stimulation may generate testable hypotheses on mechanisms underlying olfactory transduction and encoding. Indeed, computational models have enhanced our understanding of the mechanisms underlying odorant detection in both invertebrates and vertebrates [3–8] and facilitated investigations of olfactory pathway functions [9–12]. Such models have also been used to clarify the coding properties of ORNs such as the stimulus–response relationship of the ORNs [13,14] and the implications of the efficient coding hypothesis [15]. Pheromone detection in moth ORNs occurs in two stages: receptor activation by the odorant and action potential (spike) generation. Odorant molecules are first absorbed by the sensillum lymph, where they initiate a cascade of complex biochemical interactions. Receptor activation and related downstream signalling cascades leading to membrane depolarization have been described by various mathematical models [3,14,16], including detailed biophysical models [4–7,17,18]. To understand the mechanisms of pheromone detection, it is essential to develop a computational model that replicates odorant-evoked ORN responses. Reduced neuronal models, such as the leaky integrate-and-fire (LIF) neuron [19–21], can be good approximations of real neurons [22,23] and therefore useful tools for simulating and investigating prominent features of network dynamics [24,25]. A few models incorporating receptor activation into a simple spike generation mechanism based on the LIF model have been developed [13,26] in order to study steady-state ORN behaviour. However, the LIF model cannot accurately replicate the response dynamics. Here, we develop a computational model for individual ORNs that generates spikes in response to dynamic odorant stimulation. We demonstrate that an adaptation mechanism in spike threshold is necessary to reproduce the response dynamics of ORNs. The mathematical tractability and simplicity of the proposed model allows for efficient simulations and analysis of ORN spiking activity. Experimental data were obtained from ORNs by applying different pheromone doses to antennae of the moth Agrotis ipsilon (see Methods for details). To simulate the fluctuating odorant concentration in a natural environment [27], the pheromone was applied in short intermittent pulses (puffs) separated by stimulus-free periods (blanks) of random duration (figure 1a). Figure 1. Experimental data for the responses of olfactory receptor neurons (ORNs) to pheromone stimulation. (a) ORNs were stimulated by intermittent delivery of the sex pheromone (four pheromone doses ranging from 1 to 1000 pg) to mimic fluctuating odorant concentration in a pheromone plume. (b) Examples of spike trains generated by two ORNs (cells A and B) in response to 0.5 s of constant pheromone stimulation at 100 pg. Top: The average firing rate of each cell. Bottom: Raster plots of 10 trials (rows) from each cell. Note the heterogeneity in firing rates between the two ORNs despite stimulation by the same pheromone pulse. (c–f) The average firing rate across cells in response to the same 0.5 s pulse stimulus of pheromone at different doses (1–1000 pg). The shaded area represents the range between the lower and upper quartile trajectory. (Online version in colour.)
Responses of different ORNs to the same pheromone pulse exhibited marked cell-to-cell variability (figure 1b) as reported in previous studies [28,29]. This response heterogeneity of ORNs might be caused, for example, by differences in the density of olfactory receptors (ORs), odorant-binding proteins and odorant-degrading enzymes among ORNs. Nonetheless, averaged responses across cells demonstrated a typical phasic–tonic time course regardless of pheromone dose (figure 1c–f). From a baseline rate near 0 Hz, the firing rates increased rapidly (phasic period), reaching a peak around 100 ms after stimulus onset, and then slowly decaying toward a steady-state firing rate that was higher than the spontaneous firing rate (tonic period). The peak firing rate increased with pheromone dose, but the delay of the peak firing rate (latency) and the phasic–tonic response time course did not change. The proposed ORN model (figure 2) consists of two main parts: (i) receptor activation due to pheromone stimulation and (ii) spike generation according to an integrate-and-fire mechanism. Figure 2. Proposed model of an olfactory receptor neuron (ORN). Stimulus. The odorant concentration fluctuating in time is the input to the model neuron. (1) Receptor activation. The odorant molecules in the air Lair are adsorbed in the lymph at the receptor site. The adsorbed molecules L either bind to receptors R resulting in activated receptors R* or they are degraded by an enzyme N, which converts them into an inactive product P. (2) Spike generation. Activated receptors R* induce a receptor current in a single-compartment model. The model neuron generates action potentials when the membrane potential reaches a threshold θ(t). Note that a time-dependent spike threshold model (dotted) can reproduce experimentally observed ORN responses. Response. The model provides spike times from which the firing rate can be calculated. (Online version in colour.)
Receptor activation. We describe the process of receptor activation by the following chemical reactions, derived by Kaissling and coworkers [15,30,31] Lair→kiL2.1 nL+R⇌k−1k1RL⇌k−2k2R∗2.2 andL+N⇌k−3k3NL→k4P+N.2.3 Equation (2.1) describes an absorption of odorant molecules in the air Lair by the sensillum lymph at a rate ki, which yields odorant molecules at the receptor site L. Equation (2.2) describes the binding of n molecules of odorant L to a receptor. Odorant molecules L reversibly bind to free receptors R at rates k1 and k−1, which yields the receptor–ligand complex RL. Then, the complexes RL are reversibly activated (R*) at a rate k2 and k−2. Finally, equation (2.3) describes the kinetics of odorant degradation at the receptor site by an odorant degrading enzyme N. The odorant and enzyme reversibly form a complex NL according to rate constants k3 and k−3, and the complex is degraded into an inactive product P at a rate k4. The chemical kinetics (2.1)–(2.3) can be described by a system of differential equations (see Methods, equations (4.1)–(4.6)).Spike generation. We describe the ORN by a single-compartment model. The membrane potential V(t) evolves according to [32] CmdVdt=−gL(V−EL)+IR(t),2.4 where Cm is the cell capacitance, gL is the leak conductance and EL is the reversal potential of the leak current. The current from the odorant receptors IR(t) is determined by the quantity of activated receptors according to [13]IR(t)=−γR∗(t)(V−ER),2.5 where R*(t) is the concentration of activated receptors R* at time t, ER is the reversal potential of the receptor current and γ represents the conductance induced by a single activated receptor R*. A spike is generated when the membrane potential V(t) reaches a threshold θ(t). After each spike, the membrane potential is reset to a value Vreset. In the following sections, we consider two types of spike thresholds, a constant threshold and an adaptive threshold.First, we considered the model with a constant spike threshold, θ(t) = θ0, known as the leaky integrate-and-fire (LIF) model [32]. We investigated whether the LIF model with receptor dynamics (2.1)–(2.5) can reproduce the average response of ORNs to a pheromone pulse stimulus (figure 1c–f). We observed that the firing rates of the model increase monotonically, whereas the firing rates of ORNs always exhibited a peak followed by a slower decline to steady state (phasic–tonic response) (figure 3a). The model firing rates increase monotonically because the number of activated receptors R*(t) increases during the stimulation period. Thus, the model based on (2.1)–(2.5) with a constant spike threshold cannot reproduce the time course of the average ORN response. Figure 3. Model with an adaptive spike threshold can reproduce the phasic–tonic response of ORNs to a pulse odorant stimulation. Average responses of ORNs (dashed lines) were compared with the responses of the model neurons (solid lines), i.e. the model with a constant threshold (a) and the model with an adaptive threshold (b). The unit receptor conductance was γ = 41 nS · μM−1 in (a) and γ = 99 nS · μM−1 in (b). Each spike generated by the model with a constant threshold (a) was followed by a 3 ms refractory period. The pheromone concentration in the air, Lair, was set to 0.1 pM, 1 pM, 10 pM, 100 pM for the pheromone doses 1 pg, 10 pg, 100 pg, 1000 pg, respectively. See tables 1 and 2 for the other parameters. (Online version in colour.)
Except for non-decreasing firing rate profiles, the model has another issue of being able to reproduce correctly only either the peak firing rate or the first-spike latency, but not both of them simultaneously. This problem could only be numerically resolved by allowing an unphysiologically long refractory period after each spike. Figure 3a shows a compromise fit that could be achieved with a realistic 3 ms refractory period, where both the peak firing rate and the first-spike latency are much larger than in real ORNs. Since the LIF model with constant spike threshold could not replicate the qualitative characteristics of ORN responses, it was modified by including an adaptive spike threshold [33–36], which depends on previous spike times. The threshold θ(t) increases by Δ/τ after each spike and decreases exponentially to an asymptotic level θ0 with the time constant τ. The parameter Δ represents the strength of adaptation (see Methods for a formal mathematical description). Unlike the LIF model, the model with the adaptive spike threshold is able to accurately reproduce the time course of the average ORN responses under each odorant concentration (figure 3b). In addition, the model captures the dependence of the response characteristics of ORNs, i.e. the peak firing rate (figure 4b) and the first-spike latency (figure 4c), on the odorant concentration over a wide range of odorant doses (1000-fold). Figure 4. Model with an adaptive spike threshold can reproduce the odorant response characteristics of ORNs. (a) A scheme illustrating two salient characteristics of the response time course: the peak firing rate and the first-spike latency. (b,c) The effect of odorant concentration on the response characteristics. The peak firing rate (b) and the first-spike latency (c) obtained from experimental data (dashed blue, mean with inter-quartile range) were compared with those obtained from the model (solid black). (Online version in colour.)
The model parameters are summarized in tables 1 and 2. Most of them were adopted from previous studies [7,15,16,30–32,37], while the two rate constants for receptor activation (equation (2.3)), k3 and k4, were chosen to achieve rapid deactivation of L. The remaining four parameters (n, τ, Δ and γ) were determined by minimizing the integrated squared error between the average response of ORNs and the model response (see Methods). In the natural environment, odorant concentrations fluctuate rapidly; therefore, it is crucial to replicate the response dynamics of an ORN to such stimulation. To mimic the natural pheromone plume under experimental conditions, we stimulated the antennae by intermittent delivery of the pheromone [41,42]. The firing rates of individual ORNs were then compared with those generated by the model with the adaptive spike threshold. Since we wanted to reproduce the activity of individual ORNs, we had to take into account a cell-to-cell variability in ORN responses (figure 1b). The heterogeneity among ORNs can be captured by fitting some of the model parameters to the experimental recording of each individual ORN (see Methods), while keeping all the other parameters fixed as in tables 1 and 2. As for the choice of which parameters should be allowed to vary across the cells, we tested three options. First, we let γ vary (heterogeneity in γ); second, we let the pair of threshold parameters Δ and τ be cell specific (heterogeneity in (Δ, τ)); and third, we fitted all three parameters γ, Δ and τ to each neuron (heterogeneity in (γ, Δ, τ)). Finally, we examined the prediction performance of each heterogeneous model by the coefficient of determination (see Methods). The prediction performances of the three heterogeneous models with cell-specific parameters were compared with the model where all parameters were fixed for all cells as in table 2 (homogeneous model); see figure 5a. The median prediction performance of the homogeneous model was 0.13 (inter-quartile range: −0.02 to 0.30). Fitting only γ led to a mild improvement in the prediction performance (median 0.26, inter-quartile range: 0.18 to 0.35). The prediction performance improved substantially with heterogeneous τ and Δ (median 0.6, inter-quartile range: 0.46 to 0.67). Having all three parameters γ, τ, Δ heterogeneous did not bring any improvement compared with heterogeneity only in (Δ, τ) and the median prediction error was even slightly lower (median 0.59, inter-quartile range: 0.47 to 0.66), most likely because too many free parameters led to overfitting. Figure 5. Heterogeneity in ORN model parameters. (a) Prediction performance of the model with all parameters fixed (homogeneous model) and three models with heterogeneous parameters (heterogeneity in γ, heterogeneity in (Δ, τ) and heterogeneity in (γ, Δ, τ)). (b) Scatter plot of the threshold parameters (Δ and τ) adjusted to individual neurons. The red dot represents the parameters fitted to the average ORN response (table 2). (Online version in colour.)
Therefore, we concluded that the cell-to-cell heterogeneity among ORNs is best captured by fitting the threshold parameters (Δ and τ) to the experimental recording of each individual ORN, since this yields a significant improvement in the prediction performance over the homogeneous model (Wilcoxon’s rank sum test, p < 0.001, n = 84). Figure 6a illustrates an example of the model fit to recordings of two neurons. While the temporal pattern of the observed responses is similar, the amplitudes are different. The model with the adaptive spike threshold reproduces the response time course of the two neurons accurately. The distribution of the response time course of the fitted model neurons (n = 84) to the same stimulus is shown in figure 6b. Owing to the heterogeneity in threshold parameters, the amplitudes of the responses are highly variable among the model neurons, but the temporal patterns of the responses remain similar. Figure 5a shows the threshold parameters obtained from all ORNs. The mean values (+/− the standard deviation) of the parameters are 1.2 ± 0.38 s for the threshold time constant τ and 0.5 ± 0.23 mV s for the adaptation level Δ. Values of τ and Δ are negatively correlated (correlation coefficient −0.48). This finding can be intuitively explained as that these two parameters can compensate for each other to some extent. A similar firing rate may be achieved by combining either a small step increase and a slow relaxation time or a big increase and fast relaxation. Although the threshold parameters exhibit high variability among the ORNs, they are comparable to the parameters fitted to the average response (table 2). Figure 6. Fit of the model with an adaptive spike threshold to individual ORN responses. (a) Top: Time course of the pheromone stimulus. The stimulus was switching between ON and OFF states. In the ON state, the pheromone dose was 100 pg. Bottom: Firing rate time courses of two neurons (cells 1 and 2) obtained from experiments (black) and those of the model with individually tuned threshold parameters (red). (b) The distribution of firing rates of the model neurons whose threshold parameters were derived from 84 ORNs. The dark blue line represents the mean trajectory and the light blue area represents the range between the first and the third quartile. The individual trajectories vary only in the amplitude of the fluctuations, not in the temporal pattern. (Online version in colour.)
We present a computational model of a moth ORN that reproduces the firing rate dynamics of an ORN under intermittent pheromone stimulation over a 1000-fold range of concentrations. Further, our model captures cell-to-cell response variability of ORNs by tuning only two model parameters controlling the spike threshold. The model is less accurate for longer stimulations, where the model firing rate increases more slowly than the true firing rate. The model also mildly underestimates maximal spike rates. The response heterogeneity of moth ORNs, manifested by different dose–response properties among cells, and its impact on neuronal coding were thoroughly studied by Rospars et al. [29]. In addition, cell-to-cell response variability among ORNs has been investigated in other animal species such as mice [43]. This variability is captured in our model by setting different threshold parameters, i.e. the strength and the time constant of the adaptation. Previous works [36,44] suggested that the biophysical origins of the adaptive threshold are the slow K+ currents in the neuron, such as the Ca2+-activated K+ current [39] and M-type K+ current. Thus, our results imply that differences in the slow K+ current density might contribute to the response heterogeneity among ORNs. The model presented here serves as an efficient tool for simulating moth ORN responses. First, the model captures the typical response properties observed experimentally, particularly the phasic–tonic response pattern characterized by a rapid increase and a slow decay to a steady-state firing rate, as well as the effect of odorant concentration on the peak firing rate and first-spike latency. Second, our model can simulate cell-to-cell response variability among ORNs by individually setting only two parameters controlling the adaptive spike threshold. Third, our model provides the spike times, unlike linear–nonlinear models, which can capture only the firing rates [41,42,45]. Hence, our model could be useful for investigating the possibility of latency coding in olfactory information processing [46,47] and the role of spike-timing-dependent plasticity in olfaction [12,48,49]. Consequently, the proposed model can be applied to simulate a network of heterogeneous ORNs in order to investigate how ORN populations process olfactory information in the moth. Experimental evidence suggests that adaptation occurs at the level of both the receptor potential and action potential generators [50,51]. This is effectively achieved in our model by including the chemical kinetics of activated receptors, which is dependent on the stimulation history, and by the adaptive threshold dependent on the spiking history. However, the proposed model does not consider detailed biochemical pathways downstream of odorant-receptor binding that also play a role in adaptive processes, since a comprehensive picture of the olfactory transduction does not emerge yet and since it is notoriously difficult to fit parameters of detailed biophysical models from limited experimental data. In such cases, even slight differences in initial parameter settings can lead to highly disparate results [52,53]. Sliding adjustment of odour response threshold and kinetics has several molecular actors, such as ion channels, second messengers and ORs. ORs make non-selective cation channels, which are permeable also for Ca2+. First, adaptation in Drosophila OR-expressing ORNs is mediated by the Ca2+ influx during odour responses [54] and Ca2+-dependent channels may also serve for odour adaptation as in vertebrate ORNs [55]. Second, G-protein signalling cascades can both increase or decrease the ORN sensitivity [56,57]. Finally, ORs also adjust their sensitivity according to previous odour detections [58,59]. Insect ORs are formed by an odour-specific OrX protein and an odorant co-receptor, Orco, which plays a central role in both downregulating and upregulating the ORN sensitivity. In moth pheromone-sensitive ORNs, Orco was proposed to function as a pacemaker channel, controlling the kinetics of the pheromone responses [60]. One or a combination of mechanisms of modulation of ORN sensitivity may contribute to expand the dynamic range of olfactory detection and thus allow the temporal structure of odour plumes to be encoded independent of their concentration [14]. In spite of its simplicity, our model effectively captures the adaptation process, since it can predict the response dynamics of ORNs recorded in experiments. However, the feedback mechanism of our model might be fundamentally different from that induced by the second messenger signalling pathways. For instance, the adaptation process due to the adaptive spike threshold model depends solely on previous spike history and is different from the adaptation process in real ORNs caused by Ca2+ influx and the following transduction cascade [7]. An investigation of more physiological feedback mechanisms could allow for further improvements of the model. One possibility may be to include explicit formulae describing the interaction of OR–Orco complexes and the adaptation of the rates of switching between the inactive and the active state, such as in the model by Gorur-Shandilya et al. [14]. Here, we provide the details of the proposed neuron model. Receptor activation by the pheromone (2.1)–(2.3) is described by the following reaction-rate equations: dLdt= kiLair−nk1LnR+nk−1RL−k3LN+k−3NL,4.1 dRdt= −k1LnR+k−1RL,4.2 dRLdt= k1LnR−(k−1+k2)RL+k−2R∗,4.3 dR∗dt= k2RL−k−2R∗,4.4 dNdt= −k3LN+(k−3+k4)NL4.5 anddNLdt= k3LN−(k−3+k4)NL,4.6 where ki, k1, k−1, k2, k−2, k3, k−3 and k4 are the rate constants, Lair and L are the odorant concentrations in the air and in the sensillum lymph, respectively, R, RL and R* are the concentrations of the receptors in the free, receptor–ligand complexed and activated states, respectively, N and NL are the deactivating enzyme concentrations in the free and complexed states, respectively. The total amounts of receptors Rtot and the deactivating enzyme Ntot do not change over time. UsingRL= Rtot−R−R∗4.7 andNL= Ntot−N,4.8 the system of equations (4.1)–(4.6) can be reduced todLdt= kiLair−n(k1Ln+k−1)R−nk−1R∗−(k3L+k−3)N+nk−1Rtot+k−3Ntot,4.9 dRdt= −(k1Ln+k−1)R−k−1R∗+k−1Rtot,4.10 dR∗dt= −k2R−(k2+k−2)R∗+k2Rtot4.11 anddNdt= −(k3L+k−3+k4)N+(k−3+k4)Ntot.4.12 The model parameters are listed in table 1.The membrane voltage V(t) of an ORN is described by the following equation: CmdVdt=−gL(V−EL)−γR∗(t)(V−ER),4.13 where Cm is the cell capacitance, gL is the leak conductance, γ is the unit receptor conductance, R*(t) is the concentration of activated receptor, and EL and ER are the reversal potentials of the leak and the receptor currents, respectively (parameter values shown in table 2).The model neuron generates a spike when the voltage V(t) reaches the spike threshold θ(t), and, then, the voltage is instantaneously reset to a value Vreset. We consider two descriptions for the threshold. In the first description, the threshold is constant, θ(t) = θ0. This description is equivalent to the standard LIF model [13,32]. In the second description, the spike threshold is modulated by previous spikes and is formally described as follows [33,35,36].
Equations (4.9)–(4.12), (4.13), (4.14) and (4.16) were solved numerically using the forward Euler integration method with a time step of 0.01 ms. The initial conditions were R(0) = Rtot, N(0) = Ntot, V(0) = EL and θ(0) = θ0, that is, all of the receptors and the degrading enzymes were in the free state, the voltage was at the resting value and the threshold was at the asymptotic level. The simulation code was written in R [61]. Insects. Experiments were performed with laboratory-reared 4–5-day-old (sexually mature) adult male Agrotis ipsilon fed 20% sucrose solution ad libitum [62]. Pupae were sexed, and males and females were kept separately at 22°C under an inversed light–dark cycle (16–18 h light–dark photoperiod). Electrophysiology. Insects were immobilized with the head protruding. One antenna was fixed with adhesive tape on a small support and a tungsten electrode (TW5-6; Science Products, Hofheim, Germany) was inserted at the base of a long pheromone-responding sensillum trichodeum located on an antennal branch. The reference electrode was inserted in the antennal stem. The electrical signal was amplified (×1000) and band-pass filtered (10 Hz to 5 kHz) with an ELC-03X (NPI electronic, Tamm, Germany), and sampled at 10 kHz by a 16-bit acquisition board (NI-9215; National Inst., Nanterre, France) under Labview (National Inst.). One sensillum was recorded per insect. Stimulation. ORNs were stimulated with the major A. ipsilon sex pheromone, (Z)-7-dodecenyl acetate (Z7-12:Ac). Pheromone was diluted in decadic steps in hexane and applied to a filter paper introduced in a Pasteur pipette. The antenna was constantly superfused by a humidified and charcoal-filtered air stream (70 l · h−1). Air puffs (10 l · h−1) were delivered through a calibrated capillary (ref. 11762313; Fisher Scientific, France) positioned 1 mm from the antenna and containing the odorant-loaded filter paper (10 × 2 mm). An electrovalve (LHDA-1233215-H; Lee Company, France) was controlled by custom-made Labview programs reading sequences generated by Matlab scripts. The time resolution of the sequence was 1 ms. The characteristic response time of the valves, i.e. the time to switch from open to closed or closed to open, was less than 5 ms. The durations of the pheromone puffs and pauses were randomized. Time was divided into bins of a fixed duration (50 or 100 ms). In each bin, the probability of the valve being open was 0.5. Unique sequences of puffs and pauses were generated for each ORN. The dose of pheromone was constant throughout one recording session. In total, recordings of 84 moth ORNs were obtained: 41 recordings with a 50 ms minimum puff/pause duration, 43 recordings with a 100 ms minimum puff/pause duration. Each combination of pheromone dose and minimum puff duration was tested on six or more ORNs. The first 100 s of each recording was discarded because the ORN activity was not stationary. We first fitted the four parameters n, γ, τ and Δ to the average response time courses of ORNs under a pulse stimulation. For each odorant concentration, we extracted all recording segments where a neuron was stimulated with a puff longer than 0.5 s after a no-stimulation period longer than 0.1 s. Then we estimated the firing rate f(t) by convolving the spike train at the extracted segment with a Gaussian kernel function (standard deviation 0.03 s) [63,64]. The mean firing rate was calculated by aligning the individual firing rates with the stimulus onset and averaging across the cells stimulated by the same pheromone dose. The firing rate of the model neuron was obtained similarly by assuming a 0.5 s stimulation with the odorant concentration Lair equal to 0.1, 1, 10 and 100 pM that corresponds to the pheromone doses 1 pg, 10 pg, 100 pg, 1000 pg, respectively. The firing rate of the model was also calculated by convolving the spike train with a Gaussian kernel function (standard deviation 0.03 s). The parameters n, γ, τ, Δ were tuned by minimizing the integrated square error ϵave2=∑Lair∫(fd(t|Lair)−fm(t|Lair))2 dt,4.17 where fd(t|Lair) is the average firing rate for the experimental data, fm(t|Lair) is the firing rate of the model and the summation was conducted across all concentrations of Lair. The minimization was performed using the Nelder–Mead algorithm [65].Subsequently, we fitted threshold parameters (Δ and τ) to the recording from each neuron. These parameters were tuned by minimizing the integrated square error in the 10 s training period ϵind2=∫(fd(t)−fm(t))2 dt,4.18 where fd(t) is the firing rate of the recorded neuron and fm(t) is the firing rate of the model neuron. The model simulation was initiated 1 s before the start of the training period to reduce the influence of the initial conditions. Finally, the model performance was evaluated by the coefficient of determination in the subsequent 10 s prediction period. The coefficient of determination was defined asR2=1−∫(fd(t)−fm(t))2 dt∫(fd(t)−⟨fd⟩)2 dt,4.19 where 〈fd〉 is the average firing rate of the experimental data.Data and R code are available from GitHub at: https://github.com/MarieLevakova/Adaptive-integrate-and-fire-model.git. We declare we have no competing interests. This work was supported by the Institute of Physiology RVO:67985823, by the Czech Science Foundation project no. 17-06943S, by Agence Nationale de la Recherche grant ANR15-CE02-010-01 ‘Odorscape’, by Mobility Project between France and the Czech Republic through grant no. 7AMB17FR059, by JSPS KAKENHI grant nos. JP17H03279, JP18K11560 and JP19H01133, by JST ACT-I grant no. JPMJPR16UC, and by the Okawa Foundation for Information and Telecommunications. We are also grateful for the open collaborative research and MOU grant from the National Institute of Informatics. We thank P. Lansky for helpful discussions and critical reading of the manuscript. Footnotes© 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 21Life has undergone a number of major organizational transitions, from simple self-replicating molecules into complex societies of organisms [1]. Social insects such as ants, with a reproductive division of labour between the egg-laying queen and non-reproductive workers whose genetic survival rests on her success, exemplify the highest degree of social behaviour in the animal kingdom: ‘true’ sociality or eusociality. The workers' cooperative genius is observed in diverse ways [2] from nest engineering [3] and nest finding [4], to coordinated foraging swarms [5] and dynamically adjusting living bridges [6]. This has inspired a number of technological applications from logistics to numerical optimization [7,8]. All of these behaviours may be understood as solving particular problems of information acquisition, storage and collective processing in an unpredictable and potentially dangerous world [9]. Movement (the change of the spatial location of whole organisms in time) is intrinsic to the process. Here, we consider how optimal information processing is mapped to movement, at the emergent biological levels of the organism and the colony, the ‘superorganism’. We develop a Bayesian framework to describe and explain the movement behaviour of ants in probabilistic, informational terms, in relation to the problem they are having to solve: the optimal acquisition of resources in an uncertain environment to maximize the colony's geometric mean fitness [10–12]. The movement models are compared with real movement trajectories from Temnothorax albipennis ants. Scientists have studied animal movement for many years from various perspectives, and in recent years, attempts have been made to unify insights into overarching frameworks. One such framework has been proposed by Nathan et al. [13]. We describe it briefly to set the research context for the reader. Their framework identifies four components in a full description: the organism's internal state, motion capacity, navigational ability and influential external environmental factors. This framework also characterizes existing research as belonging to different paradigms, namely, ‘random’ (classes of mathematical model related to the random walk or Brownian motion), ‘optimality’ (relative efficiency of strategies for maximizing some fitness currency), ‘biomechanical’ (the ‘machinery’ of motion) and the ‘cognitive' paradigm (how individuals' brains sense and respond to navigational information). However, scientists have yet to create a theoretical framework that convincingly unifies these components. Frameworks such as proposed by Nathan et al. are also focussed on the individual, and so for group-living organisms, especially for eusocial ones, they are incomplete. The concepts of search and uncertainty also need to be better integrated within the foraging theory, so that the efficiency of different movement strategies can be evaluated [14]. Here, we contend that animal foraging (movement) models should be developed with reference to the particular information processing challenges faced by the animal in its ecological niche, with information in this context referring to the realized distribution of fitness-relevant resources: in particular, the location and quality of foraging patches, which are unknown a priori to the organism(s). Furthermore, an important ‘module’ in any comprehensive paradigm for animal movement is the role of the group and its goals in determining individual movement trajectories; there has been much research on collective behaviour in recent years, with information flow between individuals identified as an important focus of research [15]. Eusocial insects like ants exhibit a highly advanced form of sociality, even being described as a ‘superorganism’, that is, many separate organisms working together as one [16]. Their tremendous information processing capabilities are seen clearly in their ability to explore and exploit collectively their environment's resources. Ants thrive in numerous ecological niches and alone account for 15–20% of the terrestrial animal biomass on average and up to 25% in tropical regions [17]. The collective behaviour of tight-knit groups of animals like ants has been described as collective cognition [18]. Because a Bayesian framework seems natural for a single animal's decision-making [19], an obvious challenge would seem to be applying its methods to describe the functioning of a superorganism's behaviour. First, we identify a simple model that describes the foraging problem that ants, and presumably other collectives of highly related organisms, have evolved to solve. Evolution by natural selection should produce organisms that can be expected to have an efficient foraging strategy in their typical ecological context. In the case of an ant colony, although it consists of many separate individuals, each worker does not consume (all) the food it collects and is not independent, but there is rather a colony-level foraging strategy enacted without central control that ultimately seeks to maximize colony fitness [20]. Following the colony founding stage comes the ‘ergonomic' stage of a colony's life cycle [21]. This is when the queen is devoted exclusively to egg-laying, while workers take over all other work, including collecting food. Thus, the colony becomes a ‘growth machine' [21], whereby workers collect food to increase the reproductive rate of the queen, who transforms collected food into increased biomass or more numerous gene copies. Ultimately, the success or failure of this stage determines the outcome of the reproductive stage, where accumulated ‘wealth' (biomass) correlates with more offspring colonies. This natural phenomenon has parallels with betting, where the winnings on a game may be reinvested to make a bigger bet on the next game. In the context of information theory, John Kelly made a connection between the rate of transmission of information over a communications channel, which might be said to noisily transmit the outcome of a game to a gambler while bets can still be made, and the theoretical maximum exponential growth rate of the gambler's capital making use of that information [22]. To maximize the gambler's wealth over multiple (infinite) repeated games, it is optimal to bet only a fraction of the available capital each turn across each outcome, because although betting the whole capital on the particular outcome with the maximum expected return is tempting, any losses would quickly compound over multiple games and erode the gambler's wealth to zero. Instead, maximizing logarithmic wealth is optimal, since this is additive in multiplicative games and prevents overbetting. Solving for this maximization results in a probability matching or ‘Kelly’ strategy, where bets are made in proportion to the probability of the outcome [23]. For instance, in a game with two outcomes, one of 20% probability and one of 80% probability, a gambler ought to bet 20% of his wealth on the former and 80% on the latter. This does not depend on the pay-offs being fair with respect to the probabilities of the outcome, or σi=1/pi, which in the aforementioned case would be 5 and 1.25. Instead it simply requires fair odds with respect to some distribution, or ∑(1/σi)=1, where σi is the pay-off for a bet of 1, so they could, for instance, be 2 to 1 or uniform odds in the case of a game with two outcomes (electronic supplementary material, methods). For the purposes of our foraging model, we can simply impose the constraint of fair odds, and any distribution of real-life resource pay-offs can be mapped to this when renormalized. In the case of ants choosing where to forage, the probability matching strategy can be directly mapped onto their collective behaviour. With two available foraging patches having a 20 and 80% probability of food being present at any one time, the superorganism should match this probability by deploying 20 and 80% of foragers to the two sites (although it is also possible to follow a Kelly strategy while holding back a proportion of wealth; electronic supplementary material, methods). Regardless of the particular pay-off σi available at each site, provided ∑(1/σi)=1, this strategy is optimal over the long term, with the evolutionary timescale of millions of years favouring its selection. Figure 1 shows a simulated comparison of the Kelly strategy where probabilities of receiving a resource pay-off are matched, regardless of the pay-off, with a strategy that allocates foragers proportional to the one-step expected return piσi, which does take the pay-off into account. Figure 1. A comparison of the Kelly strategy with an expected return matching strategy, over the long term (identical one-step pay-offs for a ‘win' in both cases). (a) The proportion of ants ‘bet' (yellow bars) matches the probability of success (grey). (b) The proportion of ants is allocated by the expected return (probability × pay-off). The Kelly strategy increasingly outperforms any other strategy as time goes by ((c), example simulation). (Online version in colour.)
Previous analysis of the behaviour of Bayesian foragers versus those modelled using the marginal value theorem indicated that, rather than abandoning a patch when instantaneous food intake rate equals foraging costs, a forager should consider the potential future value of a patch before moving on, even when the current return is poor [24]. The priority of resource reliability over immediate pay-off in our model, when long-term biomass maximization is the goal, is itself an interesting finding about the superorganismal behaviour; but here we go further and specify models of movement to operationalize this strategy. Certain methodologies designed to sample from probability distributions—Markov chain Monte Carlo (MCMC) methods—may be used as models of movement that also achieve a probability matching (Kelly) strategy. Exploring the environment and sampling from complex probability distributions can be understood as equivalent problems. MCMC methods aim to build a Markov chain of samples that draw from each region of probability space in correct proportion to its density. Such a well-mixed Markov chain is analogous to a probability matching strategy. Once the Markov chain has converged on its equilibrium distribution (the target probability distribution, or resource quality distribution in our ant model), it spends time in each location proportional to the quality or reliability (probability) of each point. There is a central ‘social’ (colony-level) element in attempting to enact a Kelly strategy of allocating ‘bets’ in proportion to the probability of their pay-off. This is because it requires a ‘bank’ (collection of individuals) that can be allocated. This logic does not seem to apply when one is thinking of a single individual, which might instead prefer (or need) to pursue high expected returns to survive in the short term. Therefore, our model is relevant to groups of individuals who have aligned interests in terms of their fitness function—this is notably true in the social insects such as the ants, because workers are (unusually) highly related, or in clonal bacteria, for instance. However, using MCMC as a model of movement does not, in itself, imply that social interactions are necessary. Multiple MCMC ‘walkers’ can sample in the parallel from a space and still achieve sampling (foraging patch visitation) in proportion to probability. Nevertheless, social interactions could be highly advantageous in expediting an efficient sampling of the space, through, for example, ‘tandem running’ [25] to sample important areas [26], or pheromone trails to mark unprofitable territory [27]. We use our data [28] from the previous work examining the movement of lone T. albipennis ants in an empty arena outside their colony's nest [29]. T. albipennis ants have been used as a model social system for study in the laboratory, because information flow between the environment and colony members, and among colony members, can be rigorously studied. The ants typically have one queen and up to 200–400 workers [30]. The colony inhabits fragile rock crevices and finds and moves into a new nest when its nest is damaged. With workers being only about 2 mm long, relatively unconstrained trajectories of individuals can be tracked on the laboratory workbench (for example [29]). Behavioural state-based models have been developed that account for the flow of individuals between states with differential equations [31,32], but these lack an account of the ants' movement processes. We run simulations of our MCMC movement models in MATLAB 2015b (pseudocode is available in the electronic supplementary material). Each new model is introduced to explain an important additional aspect of the ants’ empirical movement behaviour. In our movement data [28], there are two experimental regimes, one in which the foraging arena was entirely novel to exploring ants, and one in which previous traces of the ants' activities remained. We use the data from the former treatment, where each ant encounters a cleaned arena absent of any pheromones or cues from previous exploring ants. We restrict our analysis to the first minute of exploration, well before any of the ants have an opportunity to reach the boundary of the arena. Log-binned root mean square displacement is calculated, and a linear regression is made against log time. A gradient equal to a half indicates a standard diffusion process (Brownian motion), whereas greater than a half indicates superdiffusive movements. This approach to characterizing ant search behaviour has been adopted from the study by Franks et al. [33]. We present simulation results from three different models of ant movement. Each model is directly based on a known MCMC. This follows the recognition that we can consider the problem of sampling from probability distributions of two continuous dimensions as analogous between animal movement and statistics (for example). The trajectories produced by each model are compared with the real ant movement data. The development of MCMC methods from the 1950s onwards, to become more efficient, might be considered to parallel the evolutionary history of animal foraging strategies. Some more details on the methods are found in the electronic supplementary material, methods. The first MCMC method to be developed was the Metropolis–Hastings (M–H) algorithm ([34,35], which is straightforward to implement and still commonly used today. We are trying to sample from the target probability distribution (resource quality distribution) P(x), which can be evaluated (observed) for any x, at least to within a multiplicative constant. This means we can evaluate a function P*(x), such that P(x) = P*(x)/Z. There are two challenges that make it difficult to generate representative samples from P(x). The first challenge is that we do not know the normalizing constant Z=∫dNxP∗(x), and the second challenge is that there is no straightforward way to draw samples from P without enumerating most or all of the possible states. Correct samples will tend to come from locations in x-space, where P(x) is large, but unless we evaluate P(x) at all locations, we cannot know these in advance [36]. The M–H method makes use of a proposal density Q (which depends on the current state x) to create a new proposal state to potentially sample from. Q can be simply a uniform distribution: in a discretized environment, these can be x(t) + [ − 1, 0, 1] with equal probability. After a given proposed movement is generated, the animal compares the resource quality at this new location with the resource quality at the previous location. If the new location is superior, it stays in its new location. By contrast, if the resource quality is worse, it randomly ‘accepts' this new location or ‘rejects' this location based on a very simple formula based on the ratio of resource quality (if it is far worse, the animal very rarely fails to return, whereas if it is not much worse, it often accepts this mildly inferior location—see also electronic supplementary material, methods). What is important about this extremely simple algorithm is that, as long as the environment is ergodic (all locations can be potentially reached), given time, the exploring animal will visit each location eventually. Visits will be made with a probability proportional to its resource quality: it will execute an optimal Kelly exploration strategy. The problem here, however, is the time taken. While the M–H method is widely used for sampling from high-dimensional problems, it has a major disadvantage in that it explores the probability distribution by a random walk, and this can take many steps to move through the space, according to Tϵ, where T is the number of steps and ϵ is the step length. T. albipennis ants were found to be engaged in a superdiffusive search in an empty arena (electronic supplementary material, methods), and similarly MCMC methods also have been developed to explore the probability space more efficiently. Random walk behaviour is not ideal when trying to sample from probability distributions because it is more time consuming than necessary. One popular method for avoiding the random walk-like exploration of state space is hybrid Monte Carlo [37], also known as Hamiltonian Monte Carlo or HMC. This simulates physical dynamics to preferentially explore regions of the state space that have higher probability. Unlike the M–H model of movement, HMC makes use of local gradient information such that the walker (ant) tends to move in a direction of increasing probability. How T. albipennis may measure this is explored in the Discussion section. For a system whose probability can be written in the form: P(x)=1Zexp[−E(x)], the gradient of E(x) can be evaluated and used to explore the probability space more efficiently. This is defined as follows:E(x)=−lnP(x). By using this definition, the local gradient ∇E(x) can be calculated numerically. The Hamiltonian is defined as H(x, p) = E(x) + K(p), where K(p) is a ‘kinetic energy', which can be defined as follows: K(p)=pTp2. In HMC, this momentum variable p augments the state space x, and there is an alternation between two types of proposals. The first proposal randomizes the momentum variable, with x unchanged, and the second proposal changes both x and p using simulated Hamiltonian dynamics. The two proposals are used to create samples from the joint densityPH(x, p)=1ZHexp[−H(x, p)]=1ZHexp[−E(x)]exp[−K(p)]. As shown, this is separable, so the marginal distribution of x is the desired distribution exp[ − E(x)]/Z, and the momentum variables can be discarded and a sequence of samples {x(t)} is obtained that asymptotically comes from P(x) [36]. We set the variable number of leapfrog steps (see electronic supplementary material, methods [38])) to L = 10; after following the Hamiltonian dynamics for this number of steps, a new momentum is randomly drawn and a new period of movement begins. This behaviour of moving intermittently in between updating the walker (ant) behaviour captures the behaviour observed in real ants [29] (see Discussion section on gradient sensing). We set the leapfrog step length ε = 0.3 (see electronic supplementary material, methods for further introduction to L and ε). For N = 18 simulated HMC ‘ants' sampling from a sparse probability distribution (a gamma-distributed noise; electronic supplementary material, methods), for 600 iterations, the r.m.s. displacement was again found and its log was regressed on log time. The gradient was found to be 0.567, 95% confidence interval (CI) (0.528–0.606), which is significantly greater than 0.5, so in this respect, it is more similar to the superdiffusive search found in real ants [33]. We can also examine the correlation of velocities between successive movement periods. Since momentum p = mv is a vector in two-dimensional space, we can set m = 1 and find a magnitude for the momentum to determine the ‘speed' of each movement (over the course of L = 10 leapfrog steps). In the previous research on ant movements [29], the correlation between successive average event speeds in the cleaning treatment was found to be 0.407 ± 0.039 (95% CI). As expected for the HMC model, because the momentum is discarded and replaced with a new random momentum after each movement, the correlation of successive event speeds is equal to zero in this model. We can make the HMC model more ‘ant-like'—and potentially more efficient—by only partially refreshing this momentum variable after the end of a movement period. HMC with one leapfrog step is referred to as Langevin Monte Carlo after the Langevin equation in physics (e.g. [39]) and was first proposed by Rossky et al. [40]. However, these methods do not require L = 1, so we use L = 10 to enhance comparability with the previous HMC model. The momentum at the end of each movement can be updated according to the equation p′ = αp + (1 − α2)1/2n, where p is the existing momentum, p′ the new momentum, α is a constant in the interval [− 1, 1] and n is a standard normal random vector. With α less than 1, p′ is similar to p, but with repeated iterations, it becomes almost independent of the initial value. This technique of partial momentum refreshment (PMR) was introduced by Horowitz [41]. Such models are well described in Brooks et al. [38]. Setting α equal to 0.65 (for example) and simulating with N = 18 results in speed correlations equal to 0.387 ± 0.012 (95% CI), which overlaps with the CI for the real ant data. The PMR method can be compared to an ant moving with a certain momentum (direction and speed) and then intermittently updating this momentum in response to its changing position in the physical and social environment, with a degree of randomness also included. The momentum changes as per the HMC method along a single trajectory, according to its subjective perception of foraging quality and potentially influenced by the pheromonal environment. If at the end of the trajectory it does not find itself in a more attractive region than before, it returns to its previous position, and with the correct model parameters (step size and number of leapfrog steps), this should be a relatively infrequent occurrence (see methodological discussion in Brooks et al. [38]). Real ants have been predicted, and found, to leave ‘no entry' markers when they turn back from an unprofitable location [42,43]. Its starting momentum in a particular direction is maintained to some degree but with some randomness mixed in, and so, its previous tendency to move towards regions of high probability (quality) is not discarded as in HMC but used to make more informed choices about which direction to move in next. This is because foraging patches are likely to show some spatial correlation in their quality, with high-quality regions more likely to neighbour other high-quality regions [44,45]. Previous empirical research [29] found evidence that ant movements may be predetermined to some degree in respect of their duration. This implies that periods of movement are followed by a more considered sensory update and decision about where to move to next. A series of smaller movements (like 10 leapfrog steps) followed by a larger momentum update, as in the PMR model, would seem to correspond well with this intermittent movement behaviour. The performance of the three MCMC models developed here can be measured in the following way. As discussed, the foraging ants should pursue a probability matching strategy, whereby they allocate their numbers across the environment in proportion to the probability that it will return (any) pay-off. This will maximize the long-term rate of growth of the colony or its biological fitness. Matching the probability distribution of resources in the environment can be understood as minimizing the distance between it and the distribution of resource gatherers. In the domain of information theory, the difference between two probability distributions is measured using the cross-entropy H( p, q)=H(p)+DKL(p||q). where H(p)=−∑ip(i)logp(i) is the entropy of p and DKL(p∥q) is the Kullback–Leibler (K–L) divergence of q from p (also known as the relative entropy of p with respect to q). This is defined as follows:DKL(p||q)=∑ip(i)log p(i)q(i). If we take p to be a fixed reference distribution (the probability of collecting resources in the environment), cross-entropy and K–L divergence are identical up to an additive constant, H(p), and is minimized when q = p, where the K–L divergence is equal to zero. Cross-entropy minimization is a common approach in optimization problems in engineering and in the present case can be used to represent the task that the ant foragers are trying to perform: match their distribution q with the distribution p of resources in the environment. Therefore, the magnitude and the rate of reduction of the cross-entropy are used to compare the effectiveness of the MCMC models (M–H, HMC and PMR) presented here. However, as noted later, for dynamic environments (where the distribution of resource probabilities p is not fixed), K–L divergence is the suitable cost function to minimize. We perform example simulations of the three MCMC models. Each samples from a target distribution with three ‘resource patches’, of high, medium and low reliability (probability). This example distribution is generated by combining a gamma-distributed background noise (shape parameter = 0.2, scale parameter = 1) on a 100 × 100 grid given a Gaussian blur (σ = 3, filter size 100 × 100), what we refer to later as the ‘sparse distribution', in equal 50% proportion with three patches of resources, which are single points of increasing relative magnitude of 1, 2 and 3 that have been given a Gaussian blur (σ = 10, filter size 100 × 100). The distribution p is thus also on a 100 × 100 grid. The simulations are run for 50 000 time steps, a reasonable period of time to explore this space of 10 000 points. Figure 2 shows the M–H model, which converges rather slowly on the target environment p. Figure 3 shows the performance of the HMC and PMR models, which show an improvement in the convergence rate because they avoid the random walk behaviour of the M–H model. Figure 2. Performance of the M–H model as it generates a sample distribution q that approximates the target distribution p, the location of resources in the environment. The minimum cross-entropy, where q = p, is shown as a dotted line. (Online version in colour.)
Figure 3. Performance of the HMC and PMR models, compared with that of M–H. In general, HMC and PMR outperform M–H because random walk type exploration of probability space is avoided, by following local gradient information and making larger steps. Their performance depends on the nature of the target distribution and choosing suitable values for step length ε and number of steps L. (Online version in colour.)
Figure 4 shows example trajectories from real ants [28] for a period of 100 s and for 100 time steps of the three models. The ants are in an empty arena, and the models are sampling from a sparse distribution (electronic supplementary material, methods). The random walk behaviour of the M–H model is evident, while the greater tendency to make longer steps in one direction is evident in the PMR model in comparison with the HMC model. Figure 4. Comparison of example trajectories from real ants (100 s) and for the three MCMC models (100 simulated time steps). The model trajectories become increasingly superdiffusive.
Figure 5 shows the distribution of directional changes (change in angle heading) between steps. The distribution of direction changes is known as the phase function in statistical physics and has been applied to ant trajectory analysis by, for instance, Khuong et al. [46]. Real ants can make large changes of direction, of course, but this is rarely done with an abrupt heading shift. The M–H model moves grid-wise in single steps; the HMC model has no correlation between step directions, while the PMR model tends to make each new step in a similar (correlated) direction to the prior one. In this respect, too, PMR is a better model of ant movement. Figure 5. The distribution of direction changes between steps in real ants (N = 18) and the three MCMC models (simulated for N = 18 ‘ants' for 1000 time steps). (Online version in colour.)
We have presented a new class of foraging model based on MCMC methods, which operationalize movement for a Kelly strategy (probability matching) in a two-dimensional space. There is an extensive theoretical and empirical literature examining the distribution of step lengths for foraging animals that considers the hypothesis that a Lévy distribution is optimal [47–50]. Lévy flights are a particular form of superdiffusive random walk where the distribution of move step lengths fits an inverse power law, such that the probability of a move of length l is distributed like (l) ≈ l−μ, where 1 < μ ≤ 3. We use the method proposed by Humphries et al. [51] to identify individual movement steps in two-dimensional data, treating monotonic movements in a certain direction in one dimension (i.e. x or y) as a step. We estimate the exponent using maximum-likelihood estimation [52]. The distribution of the ranked step length sizes in both real and simulated data is shown in figure 6. Figure 6. The (apparently) power law distributed step lengths for both real ants and simulated PMR walkers. (Online version in colour.)
There are similar exponents estimated (table 1) for both the real ant data in an empty arena (N = 18 ants from three colonies) and PMR trajectories (100 ‘ants' for 5000 iterations) sampling from a sparse probability distribution (electronic supplementary material, methods). The exponent μ in both cases is in the right region for a Lévy flight 1 < μ ≤ 3. This would seem to be evidence for a Lévy strategy in the ants (although variation in individual walking behaviour can also contribute to the impression of a Lévy flight [53]), but we suggest an alternative in the next section of this paper.
The framework we develop here is an important step in integrating key perspectives in movement research, as described, for example, by Nathan et al. [13]. It incorporates elements of randomness, producing correlated random walks in certain environments; it quantifies optimality in respect of foraging strategies via cross-entropy (Kullback–Leibler divergence); it includes an important aspect of common animal behaviour, namely intermittent movement [54], and specifically for the ants' neural and/or physiological behaviour, apparent motor planning [28]; and it makes explicit the information used by the animal step by step. Finally, and crucially, it explicates cognition at the emergent group level, because individual movement is at the service of a group-level Kelly strategy. One component of Nathan et al.'s framework is the internal state of the organism. This is not included in the models here, although state-dependent behaviours such as tandem running [25] could be included by analogy with particle filtering (e.g. [55]), for instance [26]. Our use of the Markov assumption (movement being memoryless, depending only on the current position) is justifiable with respect both to the worker ant's individual cognitive capacity and its single-minded focus on serving the colony through discovering and exploiting resources. Its motion capacity is linked to the specification of a PMR model; while we specify the navigation capacity in its ability to measure the quality gradient, which is also an externally determined factor. We may consider further the ability of ants to use local gradient information, as in the HMC and PMR models, with respect to the ants' sensory system. T. albipennis is well known for relying heavily on visual information in movement [56,57] and in common with most (or perhaps all) ants on olfactory information. It may be that the intermittent movement examined in Hunt et al. [29] is associated with limitations in the quality of sensory information when moving [54]. We suggest that T. albipennis workers have relatively good eyesight for a pedestrian insect and their small size, having around 80 ommatidia in each compound eye [58] and may be assumed conservatively to have an angle of acuity of 7° [57]. Therefore, movement would seem unlikely to make much difference to how well they can see. Since our model highlights the importance of gradient following, this may be more difficult to measure for the olfactory system during movement. Indeed, in Hunt et al. [29], we suggest that social information from pheromones or other cues is only fully attended during periods of stopping because of motor planning, with the duration of movements being predetermined by some endogenous neural and/or physiological mechanism. Therefore, this may be a mechanistic reason for the stepwise movement in the PMR model, in addition to its informational efficiency, which is its evolutionary origin. Even more sophisticated MCMC models that rely on the second derivative of the probability distribution, such as the Riemann Manifold Langevin method [59], may be relevant, because this property (the rate of change of the gradient) may be only measured with adequate accuracy when the ant is at rest. In a ‘flat' quality landscape, or sparse world, our model generates Lévy-like behaviour as seen for instance in [60]. This remains an adaptive response, but it is not a true Lévy distribution, because there is a finite variance. Much interest has been generated by Lévy flight-based foraging models, which theoretically optimize mean resource collection for certain random worlds; and this would seem to be evidence for just such a strategy in T. albipennis ants. Yet here we make a simple point that rather than being a deliberate strategy, Lévy-like behaviour may result from an organism lacking cues about which way to move. Scale-free reorientation mechanisms have indeed been suggested as a response to uncertainty in invertebrates [61]. Yet the generation of a Lévy-like distribution from our gradient-following model suggests that such observations may not really be scale free. The empirical distribution of momentums provides insight into the length scales on which the world remains smooth. The rate of resource collection can be straightforwardly calculated by finding the cross-entropy (Kullback–Leibler divergence) between the spatial distribution of resources, and the realized foraging distribution resulting from the foraging strategy. The distribution of resources is seen from a ‘genes-eye' view of the animal or superorganism, with respect to maximizing the long-term biomass or number of copies of genes in the environment: this focuses on a location's probability of yielding resources, or reliability, as opposed to the one-off pay-off. The foraging strategy is that chosen by natural selection. Minimizing the cross-entropy (Kullback–Leibler divergence) is achieved by obeying the matching law: foraging proportional to the probability finding the best resources at each available location. This strategy is especially suited to a superorganism like an ant colony, because it can forage in multiple locations simultaneously by allocating worker ants in numbers proportional to the location's reliability, through self-organization [2]. There has been some intimation before that MCMC could be a model for biological processes [62], with some query about whether the requisite randomness is possible in organisms. We think that not only is spontaneous (i.e. non-deterministic, ‘random’) behaviour present, it is necessary for survival in terms of being unpredictable around predators, prey or competitors [63], or for finding food using a ‘strategy of errors' [64]. Indeed, the Bayesian framework developed here allows predictions to be developed regarding the optimal amount of ‘randomness' in behaviour at both the level of the individual and the colony (in the PMR model, this is adjusted with the α parameter) that can be tested in the future empirical research. Further predictions arise from the momentum reversal step in MCMC PMR [38], which may be compared with observations of U-turning in ants [65,66]. Recent literature [67] has developed methods to adjust the path length dynamically, while removing the need to have a parameter L for the number of leapfrog steps. Observing how ants (and other organisms) adjust their step lengths according to different resource distributions will be instructive of their underlying movement model. The major evolutionary transitions [1] can be seen as successive leaps forward in information processing efficiency. The Bayesian framework developed here permits the evaluation and prediction of alternative movement strategies, for groups of high-related organisms, in quantitative, informational terms, in relation to environmental resource distributions. Our framework permits us to make the simple statement that for a movement strategy to be favoured under natural selection: DKL(p||qnew)< DKL(p||qold) ,4.1 i.e. the Kullback–Leibler divergence (measuring the similarity of two distributions) between a potential (genetically accessible) collective movement strategy that results in the equilibrium distribution of foragers qnew, and the organism's resource environment p, has to be lower than under the current strategy found in the population that results in distribution qold. This reduction may indeed be achieved by more sophisticated, coordinated, collective behaviour, notwithstanding higher individual energetic cost. Future research could relate such an expression to concepts in evolutionary genetics such as fitness landscapes [10]. The theoretical relationship between the level of relatedness within a social group, and the relevance of the Kelly strategy, could also be explored in future research.We described the foraging problem as a repeated multiplicative game, where an ant colony has to place ‘bets' on which foraging patches to visit, with an ultimate pay-off of more colonies or copies of their genes being created. Ants are very successful in terms of their terrestrial biomass [17], and so it would seem likely that they are following a highly evolved strategy. We suggest that the theoretical optimum is a ‘Kelly' or probability matching strategy, which maximizes the long-term ‘wealth' or biomass of the colony rather than the resource collection of single ants. By mapping the foraging problem to a set of methods designed to effectively sample from probability distributions, we present models of ant movement that achieve this matching behaviour. These MCMC-based models thus provide spatially explicit predictions for movement that describe and explain how colonies optimally explore and exploit their environment for food resources. We also show how Lévy-like step length distributions can be generated by following a local gradient that is uninformative, suggesting that contrary to being an evolved strategy, Lévy flight behaviour may be a spontaneous phenomenon. While we do not include interactions between ants in the model, past theoretical analysis of information use in collective foraging suggests that totally independent foraging is actually optimal for a broad range of model parameters when the environment is dynamic. This is because information about short-lived food patches may not be worth waiting for [68]. Understanding the logic of information flow at the level of the gene and the cell has been identified as a priority [69]. However, given that no level of organization is causally privileged in biology [70], explicating this at the organismal and superorganismal level should also advance our understanding. Our Bayesian framework operationalizes earlier proposed frameworks (such as that of [13]) for movement in a coherent and logical way, accounting for the uncertainty in both the individual ant and the colony's cognition in relation to the foraging problem. It also allows quantification of the system's emergent information processing capabilities and hypothesis generation for different organisms moving in different environments. Our MCMC models can be used as a foundation upon which further organismal and ecological complexity can be explained in future research and suggest that the movement strategies of animal collectives may be instructive for biomimetic improvements to MCMC methods. The ant Temnothorax albipennis is not subject to any licencing regime for use in experiments. The ants were humanely cared for throughout the experiment. Data are available from the Dryad Digital Repository: https://doi.org/10.5061/dryad.jk53j [28]. E.R.H. drafted the paper, produced the analysis and simulations and suggested equation (4.1); N.R.F. advised on social insect biology; R.J.B. proposed the Bayesian framework and its specific theoretical concepts; all authors contributed to the present draft. We have no competing interests. E.R.H. thanks the UK Engineering and Physical Sciences Research Council (grant no. EP/I013717/1 to the Bristol Centre for Complexity Sciences, EP/N509619/1 DTP Doctoral Prize). FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4569452. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 22Epidemiological modelling plays a vital role in public-health planning, both in terms of generic understanding of infectious disease transmission and control, as well as in terms of detailed predictions for particular situations. The foundation of these models is the compartmental epidemic model of Kermack & McKendrick [1], but modern predictive models generally seek to capture additional heterogeneities within the population [2,3]. These heterogeneities often reflect differential risk structure in terms of the transmission dynamics; high impact examples include age-stratified risk structure [4,5] and spatial structure [6,7]. Producing robust and accurate predictions often relies on an informed choice of which forms of structure to include and the reliable inference of associated parameters. Age-structured models have a rich history, particularly for endemic childhood infections (such as measles) where they have been used to capture the greater rates of transmission between school-age children and the impact of opening and closing of schools [4,5]. In this context, and before vaccination, age structure is a major risk factor as only relatively young children are likely to be susceptible to the disease, with older age cohorts having already been infected and hence having developed immunity. Age-structured models have also been vital in epidemic settings (such as influenza pandemics) where they have been an integral part of disease forecasting [8,9]. For both endemic and epidemic infections, age-structured models benefit from the ease with which the age of cases is recorded and the number of recent studies which quantify social contact patterns with reference to age stratification [10,11]. These studies provide a growing body of empirical data with which to parametrize age-structured models—overcoming the limitations of inference that is based sole on age-distribution data [5]. In addition, it has been suggested that Bayesian hierarchical models can be used to estimate contact patterns from country-level socioeconomic indicators in the absence of detailed survey data [12], vastly increasing the number of settings amenable to this form of age-structured predictive disease modelling. Models that recognize household transmission are dominated by the more theoretical literature, linking household structure to the final distribution of infection following a single outbreak [6,13]. In such models, households are treated as discrete units characterized by their internal composition with respect to the infection (i.e. two susceptibles, one infected and one recovered) and it is generally assumed that transmission is strong within the household but weak and homogeneous between households. This model formulation has provided an important understanding of optimal control in such structured populations [6,13–15], while related approaches have commonly been used to infer the strength of within-household mixing [16,17]. More complex approaches incorporate further levels of structure to model school- and workplace-based contacts [18–20], and have been used to study the efficacy of interventions such as school closure [21]. We motivate the use of household models for endemic infections by the following question: is a child in a large household more at risk of infection than one in a small household? Clearly having an older sibling (compared to being an only child) increases the risk of infection, as older children are a conduit of infection into the home. However, older siblings could already have been infected, and so will be immune and cocoon the younger child. Understanding this complex interaction between infection and household demography requires the use of mathematical models. To date, only a limited number of simulation-based approaches have been able to successfully combine age and household structure [8,9] or to model the impact of household structure for endemic infections [22,23]. In this paper, we formulate a household infectious disease model with demography: a continuous-time deterministic model of infectious disease which incorporates both age- and household structure into its transmission dynamics and captures the evolution of households over time as events in a Markov process. Here, we outline the fundamental processes under-pinning our model, we focus on describing the mechanics of the model while the detailed mathematical description is in the electronic supplementary material. The household infectious disease model with demography combines a Markov chain model for the slow evolution of a household with a Markovian SIR disease model that captures internal transmission within the household, homogeneous between-household transmission and age-structured transmission. The state of a single household is defined by the quadruple (S, I, R, k), where S, I and R are the number of susceptible, infectious and recovered individuals in the household, and k is an integer-valued counter which determines the demographic evolution of the household. At specific values of k, demographic events occur which cause individuals to be added to or removed from the household; k is generally incremented (at exponentially distributed time intervals) but can also be reset to allow repeated demographic events. The demographic status of the household is entirely determined by N( = S + I + R) and k, and these two values can be encoded as a single integer T. To avoid confusion with the states of our Markov chain, and by analogy with the concept of an age class, we refer to the demographic configuration encoded by T(N, k) as the household’s demographic class. Fixing a maximum household size Nmax defines a finite range of values of T and thus a finite state space for the combined demographic–infectious process. We assume an asymptotically large population of households, such that the proportion of households (H_) in each state obeys a set of deterministic ordinary differential equations (ODEs); this allows the calculation of population-level epidemiological quantities from the household-level state distribution. Specifically, the population-level disease prevalence I¯ of the disease is equal to the expected number of infectious individuals per household divided by the expected household size: I¯=∑S,I,R,kIHS,I,R,k∑S,I,R,k(S+I+R)HS,I,R,k, and the infectious prevalence stratified by demographic class T(N, k), denoted by I¯T, is given by the expected number of infectious cases in a household conditioned on that household being in demographic class T, divided by N:I¯T=∑S+I+R=N,kIHS,I,R,k∑S+I+R=N,kNHS,I,R,k, where H refers to the proportion of households in a given state. The proportion of households in demographic class T, denoted by HT, is calculated by summing over all states (S, I, R, T):HT=∑S+I+R=NTHS,I,R,T, where NT is the number of individuals in a household of class T. To fully define the model, we also need to calculate the probability that a child has infectious status R at the instant they leave home, which we denote by PR.Following Ross et al. [24], we denote the household state distribution of the system at time t by H_(t) and the transition matrix by Q(H_). The dependence of Q on H_ arises whenever external conditions impinge upon the household dynamics, as such the dependence is through the three key population-level variables I¯, I¯T and PR. I¯ and I¯T determine the transmission into the household from homogeneous and age-dependent mixing respectively, while new households (counter value k = 1) are formed of susceptible and recovered adults drawn from the pool of children leaving home—hence the immune status of these new households is governed by PR. The transition matrix can be decomposed into three components, Q(H_)=QDemo(H_)+QInt+QExt(H_), each of which is described in more detail below. QDemo contains all the rates for demographic events and depends nonlinearly on PR, QInt contains all the rates for recovery and internal transmission events and is a constant matrix, and QExt contains all the rates for external transmission events and hence depends linearly on both I¯ and I¯T. The evolution of the state distribution is then determined by the nonlinear set of ODEs dH_dt=H_ Q(H_),2.1 with the nonlinearity arising from the dependence of QDemo and QExt on H_. In disease-free situations, the dynamics simplify to a linear problemdH_dt=H_ QDemo(PR=0),2.2 whose long-term equilibrium is given by the eigenvector associated with the largest (zero) eigenvalue of QDemo.Our demographic model describes the evolution of a simple nuclear household, beginning at counter value k = 1. At k = kB, a child is born, adding an extra susceptible individual to the household. The sequence k = 1, …, kB can be repeated a random number of times, with the distribution informed by the family size distribution of the population we wish to model. In this way, multiple children can be born into the household, with k only incrementing to kB + 1 once the last child is born. The household loses an individual at k = kB + kL, when the eldest child leaves home, and subsequently at k = 2kB + kL, when any younger siblings leave home. The eldest child matures during the interval of kL steps between when their youngest sibling being born and them leaving home—hence the rate at which the counter k moves through these kL states is dependent on the number of children in the household. The kB steps between successive children leaving home leads to the same age distribution at leaving for all of the children in a household. The sequence k = kB + kL + 1, …, 2kB + kL is repeated until all of the children have left home; counter values above 2kB + kL are associated with elderly couples. When the counter attains its maximum value of k = 2kB + kL + kR, the household reaches the end of its lifetime and is renewed, with the two remaining individuals in the household being replaced by two new ones whose immunological status is governed by PR (the proportion of recovered individuals leaving home). The demographic process is explained in more detail in electronic supplementary material, S4 and is demonstrated schematically in figure 1. The integers kB, kL and kR are fixed model parameters and can be interpreted as shape parameters for a set of Erlang distributions which define the waiting periods between demographic events. The rate at which k increments depends on the household’s position in the demographic process (i.e. which row of the schematic in figure 1 it is currently on); this allows us to control the mean and the shape parameter of each Erlang waiting time independently. Figure 1. The household demographics is defined by the household size N and phase. Each row corresponds to one phase of the household’s lifespan, moving sequentially from the birth phase, to waiting for the oldest child to leave, to waiting for other children to leave, to the reset-and-replacement phase. The intervals between transitions all follow an Erlang distribution. In this example, the maximum household size is 5.
To fully define the demographic model, we need to specify the infectious status of the individuals who are added to or removed from the household. We will assume for this model that all newborns are susceptible, so there is no maternal immunity or vaccination. The two adults who arrive in the household at k = 0 are chosen at random from the pool of children currently leaving home, so that they are independently recovered with probability PR and susceptible with probability (1 − PR), ignoring the (small) probability that the individuals will be infectious. The calculation of PR relies on knowing the probability that each parent was recovered at the start of a household’s lifetime, which we call P′R. The ‘current generation’ households will therefore be initiated with an average of 2P′R recovered individuals, and so if we choose the infectious status of children leaving home by discounting these recovered individuals from consideration: PR=1∑S,I,R HS,I,R,Tleave∑S,I,R HS,I,R,Tleave R−2PR′N−2PR′, where HS,I,R,Tleave refers to household states where a leave event occurs (k = kB + kL or k = 2kB + kL). In a population with no previous exposure to infection, P′R = 0, which admits a simple calculation for PR; when the dynamics are at equilibrium P′R = PR allowing us to find the probability recursively using a self-consistency procedure. The results presented in this paper are entirely concerned with either the early growth or equilibrium behaviour of the model, making these two values of P′R sufficient for our purposes.Determining the infectious status of fully grown children leaving home follows the same logic as above, but is more complicated as it must be conditioned on the infectious status of the household—for example, it is only possible to removed recovered children if recovered individuals are present in the household. (Full details of the mechanism is given in electronic supplementary material, S4.2). The epidemiological dynamics incorporate three routes of infection, for a household of demographic class T and size N these are
The internal transmission rate βint is calculated by multiplying the per unit-time transmission rate τ by the average time per day spent exposed to within-household contacts, as calculated from contact data. The external transmission rate βext is given by the analogous formula using the average time per day spent exposed to contacts from other households. Age-structured transmission rates for a set of age classes C1, …, CK are calculated by multiplying τ by the average total duration of contacts between an individual in age class Ci and all individuals in age class Cj who are in different households—we label the resulting transmission rate βijext. By defining our total external infection rate to be a convex combination of the homogeneous and age-structured infection rates (controlled by the parameter σ), we can control the level of structure in our model while keeping the total population-level transmission rate constant. The calculation of τ and the required contact durations is covered in electronic supplementary material, S5. Our approach to age-structured mixing takes the typical who-acquires-infection-from-whom approach first introduced by Schenzle [4] but also requires a mapping between age classes and demographic classes in order to model mixing structured by demographic class. Because the counter k increments at exponentially distributed intervals, the time spent in any amalgamation of demographic classes is hypoexponentially distributed. Using this information, we can define a matrix E=ET,i such that ET,i the expected number of individuals in age class Ci in a household of demographic class T. As such, the population-level proportion of individuals in age class Ci is then Pi=∑S,I,R,THS,I,R,TET,i∑S,I,R,THS,I,R,T(S+I+R). The force of infection on individuals in a household of type T is then determined by summing across all associated age classes multiplied by their interaction with other age classes and the chance that those age classes are infectious:λT=∑i,jET,iNTβi,jext∑UHUEU,jI¯UP j, where the second sum approximates the proportion of infected individuals in age class Cj. The full derivation of this equation is given in more detail in electronic supplementary material, S6.Under suitable parameter choices, our model contains the classic homogeneous mixing model (βint = σ = 0), an age-structured model (βint = 0, σ = 1) and a household-structured model as special cases (σ = 0). As a set of (high-dimensional) differential equations, the system is numerically tractable, allowing us to study both its early behaviour following invasion and its endemic equilibrium; both of these can be approached with considerable computational efficiency by relying on the matrix structure of the underlying ODEs (equation 1). In particular, we focus on the early growth rate r and the household reproduction number R* [6] as measures of early dynamics, as well as the equilibrium of the combined demographic and epidemiological system. We compare the full model to other sub-models (homogeneous, age structured and household structured only) to assess the impact of individual forms of structure and contact heterogeneity; this also informs about the likely problems with predictions from simpler models that ignore particular forms of contact structure. To clarify the effects of demography and epidemiology, we study two childhood diseases (a measles-like disease and a mumps-like disease which differ in their transmission rate across a contact [25] and infectious period [2] (table 1)) in two different populations (a UK-like population and a Kenya-like population). Our choice of these two populations is motivated by the availability of detailed contact survey data and the pronounced socioeconomic differences between the two countries, which we expect to be reflected in demography and contact behaviour. The POLYMOD study provides contact data for the UK [10] and the study conducted by Kiti et al. [11] provides contact data from the region of coastal Kenya covered by the Kilifi Health and Demographic Surveillance System [27]. The demographic transition terms (QDemo) are parametrized directly from empirical data [28–32], without recourse to any fitting procedure.
Throughout, we study the dynamics of infection in the absence of vaccination or other forms of control. We focus on the early invasion dynamics and equilibrium distribution of infection, and how these are impacted by accounting for different transmission heterogeneities. The household size distribution at demographic equilibrium (the solution of equation (2.2) is illustrated for the UK-like and a Kenya-like populations in figure 2, and is partitioned into younger (blue) and older (red) households. The UK-like population is dominated by older households without children as a result of its relatively long life expectancy and low birth rate, in part a reflection of the high proportion of adults (approx. 17% [29]) not having any children at all. The Kenya-like population features a much higher proportion of households with children, resulting from a combination of a lower life expectancy, a higher average number of children, and comparable inter-birth interval to the UK, which causes adults to spend a much higher proportion of their life living with children compared the UK-like population. These demographic equilibrium solutions form the underlying population structure to which invading and endemic infections are added. Figure 2. Household size and age distributions for populations with UK-like and Kenya-like parameters. Blue bars correspond to the first two phases of the demographic process outlined in figure 1 (prior to the eldest child leaving home), red bars correspond to the third and fourth phases (after the eldest child has left), so that moving along the x axis can be thought of as moving through successive stages in a household’s life.
Initially, we focus on a disease with mumps-like epidemiological parameters, such that the rate of transmission across a contact is relatively low (τmumps = 0.4118 per contact per day), and consider the short- and long-term dynamics in a UK-like population (table 2 and figure 3). We compare the full model (which includes both age-structured and household-structured transmission) to the homogeneous, purely age-structured and purely household-structured sub-models. To fairly compare the four models, the transmission rate τ in the three sub-models is re-scaled to achieve the same initial growth rate r in all cases—conceptualized as matching the four models to the same early epidemic data and then making predictions about the long-term dynamics. Figure 3. Percentage distribution of cases per household conditional on the size and age of the household, under the four different transmission models. These results are for a mumps-like infection in a UK-like population. For purposes of clarity, we do not plot the percentage of disease-free households since this is several orders of magnitude larger than the other percentages combined. As in figure 2, blue bars correspond to younger households (the first two rows in figure 1) and red to older households (the second two rows in figure 1). The pink open bars correspond to the results from the homogeneous mixing model, which are shown for ease of comparison; the open circles show the total amount of infection in the households accounting for multiple infections.
Parameter values capturing early invasion dynamics into a naive population, and those describing long-term endemicity are listed in table 2. r defines the asymptotic rate of early growth, such that Cases ∼ exp(rt); while R* (which is the household counterpoint of the basic reproductive ratio, R0) defines the average number of secondary households infected as a consequence of infection in an average household during the early epidemic [6,33]. Although the transmission parameter is chosen such that r is the same in all models, those with household structure allow the amplification of infection within the household, which naturally produces a larger R*. The equilibrium level of infection, I¯, is comparable between the four models, as for a fully immunizing infection prevalence is largely determined by the birth-rate generating new susceptibles. This birth rate provides an upper bound on the equilibrium incidence of infection, with lower values only occurring when individuals escape infection their entire lives. The prevalence is marginally higher for the age-structured model (compared to the other three) as this has greater transmission between school-age children (where the majority of the infection is maintained) but does not suffer from the within-household depletion of susceptible individuals. Finally, the probability of infection during childhood, PR, is calculated the chance of an individual being infected before they leave home and is increased slightly by the addition of either household or age structure which focuses infection into children, with a further increase when these two structures are combined. Figure 3 shows distribution of cases per household (conditional on household size) for the four transmission models. In the homogeneous-mixing model, due to the lack of any structure, the probability of having at least one case increases (almost) linearly with the number of individuals; this idealized pattern is plotted as an open pink bar in the other models. It is the departure from this idealized linear case that informs about the actions of age and household structure. For the age-structured model, the greater mixing between school-children means that larger households (with more children) have a disproportionately higher risk of infection (the open circles correspond to the total expected amount of infection in households). However, for older households (red) which are likely to contain older children, the chance of infection is less than in the homogeneous model as there is a greater chance that these older children are immune from past infection. In both the homogeneous and age-structured model, the chance of observing multiple infections within the same household is vanishingly rare as within household transmission has been ignored. The household-structured model allows for local outbreaks such that multiple members of the same household are infected at the same time. This is visualised in the bottom two subplots of figure 3 using colour-coded stacked bars, with the height of each segment corresponding to the proportion of households containing a given number of cases. These local outbreaks lead to higher total levels of infection (as shown by the open circles) concentrated in the large households where sibling-to-sibling transmission is likely. However, household outbreaks are generally limited in scale due to levels of immunity in the household from historic infections. The full model, which includes both household and age structure, produces equilibrium distributions with a more pronounced difference between older and younger households than models with either one of these transmission structures in isolation. Large households in the earlier demographic stages (blue bars) are likely to contain school-age children who expose the household to age-structured transmission, while also containing younger children who lack previous exposure and so are susceptible to infection via these school-age siblings. While, in general, we expect age-structured mixing to boost infection (due to assortativity) and household-structured mixing to impede it (due to susceptible depletion), these results demonstrate that the two structures can act synergistically to amplify the concentration of infection into large young households. Having gained an understanding of the effects of model structure, we now restrict our attention to the full model (with both age and household structure) and consider the impact of epidemiological and demographic parameters. The epidemiological parameters are those listed in table 1, such that measles has a greater rate of transmission across a contact, but a slightly shorter infectious period than mumps. UK- and Kenya-like demographic parameters are used as exemplars of a high-income country with relatively low population growth and a stationary population-age pyramid (similar numbers of individuals across most ages) and a low-income country with relatively high population growth and an expansive population-age pyramid (with higher numbers of younger ages). The main demographic differences between the UK and Kenya are captured by a much higher number of children per woman in Kenya (leading to a higher birth-rate) and a slightly shorter life expectancy; while Kenyan age-structured mixing is less assortative than the UK ([10,11], electronic supplementary material). To illustrate the effects of epidemiological and demographic parameters on the full age- and household-structured model, we first calculated early growth dynamics and equilibrium distributions for measles- and mumps-like diseases in UK- and Kenya-like populations (table 3). As expected, we find that measles has greater growth than mumps in a naive population (as characterized by both r and R*) and that these measures of early behaviour are higher in Kenya than the UK. Such findings are attributable to the greater transmission rate of measles across a contact compared to mumps, and the larger average family size (and hence greater number of close contacts) in the Kenyan population compared to the UK. When considering endemic quantities, the higher birth-rate in the Kenya-like population (due to larger family sizes) is the main factor determining the endemic prevalence of infection (I¯), with relatively little difference between measles and mumps. The population birth rate, and hence the rate that new susceptible individuals are produced, generates an upper bound on the equilibrium level of infection—lower values of equilibrium prevalence only occur when individuals escape infection for their entire lives. By contrast, the proportion of children infected before they leave home (PR) is strongly influenced by both the epidemiological characteristics and the demography of the population; almost all children are predicted to catch measles in a Kenyan-like population, but less than half predicted to catch mumps in a UK-like population.
Figure 4. Percentage distribution of cases per household stratified by demographic state under UK- and Kenya-like demographic parameters for measles- and mumps-like infectious parameters. For purposes of clarity, we do not plot the percentage of disease-free households. As in figure 2, blue bars correspond to younger households (the first two rows in figure 1) and red to older households (the second two rows in figure 1). The pink open bars correspond to the results from the homogeneous mixing model with the same early growth rate (r) to demonstrate the effect of stratifying contact behaviour; the open circles show the total amount of infection in the households accounting for multiple infections. All plots are at the same scale to improve comparison, and we note that the data in figure4a is the same as that in figure 3d.
The histograms of infection levels in different household types provide more details of the underlying dynamical behaviour (figure 4, again the frequency of 1, 2, 3 or more cases conditional on the size and phase of the household is shown; the electronic supplementary material contains plots that show the absolute frequency, such that the lower density of the largest households is reflected). As before, the equilibrium distribution from a homogeneous model with the same early growth rate (pink bars) shows the simplest assumption where heterogeneities are only driven by the size and age of the household, households containing older individuals are already likely to have been infected experience less infection leading to nonlinear infection probability with household size. We compare our full model predictions to this homogeneous ideal. For mumps in both the UK and Kenya, the action of age and household structure is to exaggerate the differences between older and younger households, and between small and large households (cf pink bars and open circles)—due to the greater concentration of infection in school-age children, and the concentration of school-age children in larger younger households. For mumps in Kenya, we observe a slight saturating effect, with larger households less likely to experience infection than in the homogeneous model (cf pink and solid bars); although the scale of the outbreaks within the infected households more than compensates for this effect. When considering a highly transmissible infection like measles, this saturation is far greater as fewer children escape early infection and therefore reservoirs of susceptibility cannot build-up within households. In this way, young households of size 5 in the UK are predicted to contain more measles than households of size 6. In Kenya, these nonlinear saturating effects are even more pronounced: infection is concentrated in households with young children and these young households in general only ever have single measles cases; two or more cases within the household is relatively uncommon despite the strong rate of within-household transmission. This saturation is amplified by age and household structure, as age-structured transmission concentrates infection in school-aged children and household structure leads to rapid infection of any (younger) susceptible siblings. For measles-like infection in a Kenya-like population, households of size 10 show a substantially elevated level of infection in both the full and homogeneous model; this is because (under our demographic assumptions) such households always contain very young children who are the most likely to become infectious. The strong interactions within households and between individuals of similar ages dominates epidemiological transmission dynamics for a range of infectious diseases. The POLYMOD study [10] was revolutionary in quantifying our understanding of age-structured mixing, and shows strong diagonal and off-diagonal elements within the mixing matrix corresponding to connections between similar age cohorts and within families. This work has led to a number of diary-based studies that have improved our knowledge of age-structured mixing [11,34,35], and this wealth of new data has been pivotal in helping to produce accurate models of multiple infectious diseases [36,37]. However, in such models, there is no distinction between the repeated close-contact within the home environment and a ‘random’ contact; the repeated nature of contacts within the household means that this pool of susceptibles is rapidly depleted reducing the transmission potential compared to the frequently used homogeneous mixing assumption. By contrast, household models [6,13] explicitly recognize both the strong within-household transmission and the rapid depletion of susceptibles within the household environment, but generally do not capture the structure of between household mixing and ignore births into the population, restricting their application to epidemic scenarios [14,15]. Here, we have combined these two approaches to generate simple mechanistic models that capture the impact of household and age-structured mixing on the spread and distribution of endemic infectious diseases. A range of infectious and demographic settings can be simulated by choosing suitable parameters. While our model specifically describes infection in a population of nuclear households, our basic approach of coupling infectious and demographic dynamics can be applied to a more diverse set of demographic settings by coupling to a more complex set of demographic events. Although high-dimensional, our model can be formulated from mechanistic principles and can be expressed as a series of coupled ODEs. These equations can be re-written in terms of matrix operations, with nonlinear transmission terms acting to re-scale elements of these matrices; this allows us to exploit computationally efficient methods to calculate early growth rates of outbreaks as well as endemic equilibria. The emergent dynamics are highly complex and are the result of four interacting processes: (i) age structure acts to focus transmission within school-age children (due to their greater social mixing) which in turn disperses infection between households; (ii) in contrast with age structure, household structure concentrates infection within family groups and hence leads to transmission between distinct age classes; (iii) exposure over time means that an individual’s probability of immunity increases with age so that older households tend to contain a higher density of immune individuals and thus experience less infection, (iv) the continuous circulation means that newborns often find themselves in households which have already experienced infection, increasing the local density of immune individuals and sheltering newborns from infection, making large local outbreaks relatively rare. The tension between these four factors is determined by the demographic and epidemiological parameters. Our comparison between model structures includes the important finding that while age structure and household structure have opposite effects on the population-level growth of infection, they act in tandem to concentrate infection in younger households. This demonstrates the importance of more detailed outputs in understanding the behaviour of complex models. From a public-health perspective our results suggest that household-based control (such as prophylaxis [14] or cocoon vaccination [38,39]) is most likely to be effective for weakly transmitted or non-immunizing infections where many older individuals remain susceptible and hence household outbreaks are common. Although our model incorporates only a single level of spatial structure (that of the household), it can easily be adapted to incorporate more detailed location-based contact data. Contact studies including the POLYMOD study stratify contacts by location according to categories including school, work and transport [10], allowing us to express our age-structured contact matrix as a sum of location-based components. By considering this extra level of structure, we can then model interventions such as school closure by scaling down or removing the appropriate contact rates. All of the code used to create the tables and figures in this paper and its supplement is available at github.com/JBHilton/HiltonKeeling_EndemicDiseases. We declare we have no competing interests. This work was partly funded by the EPSRC, with grant reference no. EP/P511079/1. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4582511. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 23To successfully locate mates, find food or escape predators or unfavourable environments, most animals need to travel along a given bearing in a relevant direction. For this, two main sources of directional information can be used; (i) information given by internal (proprioceptive) cues, for instance, by body rotations or leg movements [1–3], and (ii) information derived from external compass cues, such as the sun or the earth's magnetic field [4–8]. In practice, if an animal relies on internal mechanosensory information alone, it is not able to travel any greater distance from its current position (following a Brownian search where the distance travelled is proportional to the square root of the number of steps taken). This is clearly modelled by Cheung et al. [9]: an agent moving forward using only proprioceptive cues will, to some extent, with each successive step, deviate from the angular direction of the previous step taken. This is due to the accumulation of noise in the motor and sensory systems that will unavoidably result in the loss of ability to maintain the desired direction, subsequently making straight-line orientation impossible. This has been demonstrated behaviourally in animals as diverse as humans, spiders and beetles [10–12]. In most navigating and migrating animals, directional guidance is acquired from multiple sources of information [13–16], originating from the movement of the animal itself (internal cues) [1–3] and/or from its surroundings (external cues) [17–19]. Experimental studies from species as diverse as ants [16], butterflies [20], elk [21] and grey seals [22] have considered how animals may balance these two sources of directional information to navigate within their local environment. In ants, different sources of directional information are weighted relative to their respective certainty, where the cue conveying the highest certainty is afforded the highest weight in the directional output from the compass [16,23,24]. For instance, if the visual scenery fails to provide the information needed, the ant will rely more strongly on the directional information provided by its path integrator to complete its navigational task [16]. In contrast to most migrating and homing animals, a ball-rolling dung beetle simply needs to move along one single bearing for the duration of its current travels. At the dung pat—where the journey of a ball-rolling beetle begins—competition for dung can be fierce [25]. In order to obtain a sufficient amount of food, beetles gather and shape dung into balls and roll them away. To ensure the most efficient escape from the competitors at the pat, the beetles exit in all directions along paths as straight as the terrain allows [6,26–29]. In this way, with every step taken, they maximize the distance between themselves and their competitors. For diurnal beetles, the most prominent compass cue used to steer along this set bearing is the sun [6,27,28,30]. Together with the celestial polarization pattern, the skylight intensity gradient and the colour gradient across the sky, this forms a highly robust straight-line orientation system [30–32]. If these cues are eliminated from the ball-rolling dung beetle's field of view, the animal soon loses its ability to maintain a straight bearing [12]. Understanding how animals balance internal and external directional cues to maintain a straight bearing is still an open question in movement ecology. It is challenging to model behaviour in the ecological context of homing insects [16,20], elks [21] and seals [22] that forage within a familiar territory. This is because most theoretical models assume a straight-line optimal trajectory [33], while the actual trajectories travelled by these animals may be more tortuous. In contrast, ball-rolling dung beetles strive to move along straight paths from the start to the end of their journeys [6,12,27–30]. Here, we characterize the size of the directional error generated with each step in the presence or absence of external compass cues in two closely related species of dung beetle, that differ greatly in size. This allows us to (i) estimate the influence of step size on the precision of straight-line orientation [34,35] and (ii) model the weight given to external sky compass cues over internal proprioceptive cues for straight-line orientation in dung beetles. Two closely related species of diurnal dung beetles, Scarabaeus (Scarabaeus) ambiguus, and Scarabaeus (Kheper) lamarcki were collected within the elephant park Adventures with Elephants (27.95°E, 24.78°S) and Stonehenge game farm (24.32°E, 26.39°S), respectively, in South Africa with the aid of dung-baited pitfall traps. Experiments were performed at Stonehenge game farm and Thornwood lodge (28.02°E, 24.77°S), during February 2017. All experiments (apart from the studies of orientation performance in the absence of visual cues which were conducted in complete darkness in a light-tight indoor laboratory) were performed outdoors under clear skies at solar elevations between 45 and 60°. Each beetle was marked individually with a number on its thorax using a white marker (Tipp-Ex®). An overhead Sony Handycam HDR-CX730E (fitted with a 0.42× wide angle lens to extend the field of view when required) was used to record dung beetle rolling trajectories and exit bearings. Individual beetles were allowed to roll their dung ball across a flat, sand-coated (Dried Ochre, granular paint, Fired Earth™), 50 cm radius, outdoor arena (figure 1a). Step size was determined as the distance from the point at which the front foreleg (left or right) was stable on the arena surface, to the point at which the same limb was again stable on the surface. Image processing software, ImageJ1© (National Institutes of Health, Bethesda, MD, USA), was used to extract the x and y coordinates of the start and endpoint of each step from the overhead videos. From this, the length of the step was calculated and converted to true length by using a millimetre scale present in the frame for reference. The step length for each species was determined by calculating the average length of 10 strides per beetle for each species (N = 10). Figure 1. Description of the experimental arenas and beetles. Individuals of Scarabaeus ambiguus (left) and S. lamarcki (right) are depicted side-by-side for size comparison. Photo: Christopher Collingridge (a). For all treatments, a beetle was placed alongside a ball in the centre of a circular, sand-coated arena (b) and filmed with an overhead camera (c). The beetle was allowed to roll its ball to the perimeter of the arena, where the exit angle was noted (0° = magnetic North). Depending on the experimental treatment, the beetle either rolled only once, or was repeatedly placed back in the arena centre beside its ball until it had exited the arena 20 times. The ball rolled was either a natural dung ball or a standard putty ball (shown in a). For the experiment where the sun was mirrored, a 75 cm radius arena was used (not depicted in the figure). For all other experiments conducted, three differently sized arenas were used depending on the species of beetle (b): 50 cm (S. ambiguus and S. lamarcki, black solid line), 33 cm (S. ambiguus, red dotted inner circle), 52 cm (S. lamarcki, red solid outer circle) radius. (Online version in colour.)
Beetles were placed individually alongside their dung ball in the centre of a flat, circular, sand-coated, wooden, outdoor arena, measuring 75 cm in radius, and allowed to roll the ball to the edge, where the exit bearing was noted. The beetle was then removed from its ball and placed back in the centre of the arena alongside its ball. At the same time, the sun's apparent position was changed by 180°, using a mirror (30 × 30 cm), while simultaneously concealing the real sun from the beetle's field of view with a wooden board (100 × 75 cm). Again, the beetle was allowed to roll to the edge of the arena and its second exit bearing was noted. A third trial, with an un-manipulated sun position as in the first trial, was performed as a control to verify that the beetle strived to adhere to approximately the same bearing throughout the experiment. This held true for all beetles tested. In total, 45 individuals per species rolled from the centre to the edge of the arena three times. Angular change was calculated as the difference in exit bearing between two exits from the arena (figure 2). Directional statistics were obtained from Oriana 3.0 (Kovach Computing Services, Anglesey, UK). Figure 2. The role of the sun in the celestial compass system of two beetle species. The response to a mirrored sun while rolling outdoors, under a clear sky, was tested in two closely related, but differently sized beetle species. A schematic diagram of the experiment is presented in (a). Forty-five individuals of Scarabaeus ambiguus (b) and S. lamarcki (c), respectively, were individually placed under the natural sky, alongside a dung ball in the centre of a 75 cm radius, circular sand-covered arena. The beetles were allowed to roll to the perimeter of the arena where their exit angles were noted. From here, the beetles were placed back in the centre to exit a second time, either under (i) the same natural sky as during the previous roll (control, grey circles), or (ii) a manipulated sky where the apparent position of the sun is changed by 180° by the use of a mirror (test, yellow circles). The difference between two exit angles was calculated and used to define the mean change in bearing (control, dotted grey lines; test, solid red lines). Error bars represent one circular standard deviation. When allowed to roll twice under the sun, individuals of both species showed no significant change in bearing between consecutive rolls (dotted grey line). Under the mirrored sun, both species responded to this treatment by a change in exit bearing approaching 180° (solid red lines). (Online version in colour.)
To eliminate any influence of size or shape of beetle-made dung balls on the orientation performance of the different beetles, ‘standard balls’ were made from dung infused Play-Doh® (Hasbro, Pawtucket, RI, USA). These balls had a set size of 1.7 cm diameter for Scarabaeus ambiguus and 3.5 cm diameter for S. lamarcki (figure 1a). These dimensions were determined from the average diameter of beetle crafted balls for each species (S. ambiguus: 1.74 ± 0.13 cm (mean ± s.e.m.), S. lamarcki: 3.54 ± 0.60 cm) (N = 10). In these experiments, beetles were individually placed beside a ‘standard ball’ of the size associated with their species (figure 1b), at the centre of a flat, circular, sand-coated, wooden, arena, measuring 50 cm in radius in the complete darkness. From here, the beetle rolled the ball to the perimeter of the arena. This marked the end of the trial (figure 3). To record the beetles' trajectories in the dark, the overhead camera was fitted with an infrared lamp, and individuals were marked with high-gain reflective paint (Soppec©: Technima Nordic AB, Mölndal, Sweden) on their thorax. In total, 10 individuals per species were tested. Figure 3. Rolling trajectories in the absence of external visual cues. The two closely related, but differently sized beetles, were allowed to roll a dung ball from the centre of a flat, sand-coated arena, in complete darkness. The full trajectories of 10 randomly chosen beetles of each species are shown. On the 50 cm radius arena (black perimeter) S. ambiguus (a), obtained a significantly lower straightness index (higher tortuosity) compared to the larger S. lamarcki (p = 0.02, N = 15) (b). When analysed over a radial distance corresponding to 20 steps for each species respectively (32 cm for S. ambiguus and 52 cm for S. lamarcki) (a, inner red perimeter; b, outer red perimeter) no significant difference in straightness was recorded (p = 0.08, N = 15). (Online version in colour.)
The beetle's position in each video frame was determined using custom-made tracking software in Matlab R2016a (Mathworks Inc., Natwick, MA, USA, courtesy of Dr Jochen Smolka). Camera calibration software in Matlab was used to correct for optical distortion and true distances were obtained from a calibration pattern (3 × 3 cm black and white squares) placed on the surface of the arena. Path length of each roll was calculated by summing the two-dimensional distance travelled between successive frames. To determine how straight the beetle's trajectories were, a straightness index (S) was calculated as D/W [36], where D is the distance from the starting point to the perimeter of the arena, and W is the length of the path taken. In order to determine the angular error generated by each step in the absence of external visual cues, 15 individuals of each species were filmed at close range, with an overhead camera fitted with an infrared lamp. Using a custom-made tracking software in Matlab 2017b (Mathworks Inc. Natwick, MA, USA), the angular error generated by each step of the beetle, was determined. For this, we defined a step as the instance of foreleg-surface contact. The position of two consecutive surface contacts by the same foreleg was tracked and a vector between these two consecutive points was drawn to determine the bearing direction of one step. From this, angular error per step was calculated as the absolute difference in bearing direction between two consecutive vectors (figure 4a). Figure 4. Estimation of motor errors, compass errors, and their balance. A model of a beetle performing a random walk where θi is the direction of movement of the previous step and ΔXi, ΔYi are the distance travelled in step i along the x and y directions, respectively (defined in equations (2.1) and (2.2)) (a). A flow diagram describing the process of estimating pairs of w and θ*BRW for two beetle species that differ in size (b). Step 1: θ*CRW is estimated from the width of the angular errors of a beetle orienting in the absence of visual cues. Step 2: Two sets of BCRW trajectories are illustrated; one at the limit of pure CRW (w = 0) and one at the limit of pure BRW (w = 1). To generate these trajectory examples, we chose arbitrarily θ*BRW = θ*CRW = 5° (these values were arbitrarily chosen for the purpose of illustrating the model). Each trajectory is shown in a different colour. Step 3: Mean vector length (R) for each species is generated from the simulation and compared to the experimentally measured values (shown as red dotted line on the colour bar and on the heat-map). Step 4: The extracted w and θ*BRW for each species is shown. (Online version in colour.)
To determine the orientation performance of the two beetle species under an open sky, each beetle, together with a species specific ‘standard ball’, was repeatedly placed in the centre of a flat, circular, sand-coated, wooden, outdoor arena, until each beetle had rolled its ball to the edge of the arena 20 times. Two different sized arenas were used (figure 1c); (i) one with the effective radius set to a distance equivalent to the length of 20 steps for the species tested (S. ambiguus: 32.38 cm, S. lamarcki: 51.79 cm) and (ii) one with a radius of 50 cm. The exit bearings of 20 rolls performed by each beetle were again noted and all trajectories were recorded from above. In total, 20 individuals per species were tested. Orientation performance of each individual was determined by the mean resultant vector length (R) calculated in Oriana 3.0 (Kovach Computing Services, Anglesey, UK) from the 20 exit bearings for one individual (figure 5). This value is used to describe the concentration of unimodal circular distribution, and ranges from a value of 0 (random distribution of angles) to a value of 1 (no dispersion in distribution of angles). The better oriented the beetle, the closer the exit bearings cluster around one direction, and the closer the mean resultant vector is to unity. In total, the orientation performance of 20 individuals per species for the treatments described above were recorded. Figure 5. Measuring orientation performance in the presence of external visual cues. As a measure of orientation performance (a), the mean vector length for each beetle was calculated from 20 tracks over a radius equivalent of 50 cm, as well as of a radius equivalent of 20 step lengths of the corresponding species (32 cm for Scarabaeus ambiguus and 52 cm for S. lamarcki) (white circle, mean value for S. ambiguus; black circle, mean value for S. lamarcki; red solid line, median value for S. ambiguus and S. lamarcki). An R-value of 1 indicates that the beetles maintained the same direction over 20 rolls. When rolling over a radius of 50 cm, the smaller species, S. ambiguus (N = 20), showed a significantly shorter resultant vector length compared to the larger species (N = 20) (RS. ambiguus: 0.88 ± 0.02; RS. lamarcki: 0.92 ± 0.01, p = 0.028, N = 20). However, no significant difference was seen when both species rolled over a distance equivalent to 20 steps (RS. ambiguus: 0.91 ± 0.015; RS. lamarcki: 0.91 ± 0.02, p = 0.42, N = 20). Paths travelled by four individuals for each species and radial distance (b) are shown (from left: S. ambiguus (50 cm); S. lamarcki (50 cm); S. ambiguus (32 cm); S. lamarcki (52 cm)). Each colour represents 20 trajectories of one individual. * = p < 0.05, n.s. = p > 0.05. There was no difference in the straightness of the 20th exit path compared to the 1st exit path performed by the same individual in any of the conditions (p50 cm(S. ambiguus) = 0.16, p50 cm(S. lamarcki) = 0.16; pstep length(S. ambiguus) = 0.30, pstep length(S. lamarcki) = 0.16, N = 20). (Online version in colour.)
The integration of proprioceptive cues in the orientation system of an agent generates noise (termed motor error) affecting the motoric output of the agent. Similarly, the detection and integration of external compass cues by the orientation system generates noise (termed compass error). Both of these sources of noise can be expected to affect the motoric output of an agent exercising straight-line orientation. The biased correlated random walk model [35] was used to estimate the compass error of external cues and determine how much weight is given to external visual cues over internal proprioceptive cues for straight-line orientation in the beetles. In both biased and correlated random walks, the agent's goal is to walk in a straight line in an arbitrary direction θ0 (figure 4a). In a biased random walk (BRW) the instantaneous angular error, θi, arises from noise in external visual cue acquisition (compass error), however, in a correlated random walk (CRW) it arises from accumulated noise in motoric execution (motor error). The biased correlated random walk model combines the two errors in the following manner: ΔXi+1=l[wcos(θ0+θiBRW)+(1−w)cos(θi+θiCRW)]2.1 andΔYi+1=l[wsin(θ0+θiBRW)+(1−w)sin(θi+θiCRW)]2.2 where l is the step length of the current step, θi is the direction of movement of the previous step, θiCRW is an motor error term, θiBRW is an compass error term, and w ∈ [0, 1] is the weighting given to external cues. We assume that θiCRW and θiBRW are random angles drawn from a von Mises distribution with a zero-mean and standard deviation θ*CRW and θ*BRW, respectively. Table 1 summarizes the model parameters and their experimental equivalents. θ*CRW was estimated for each species from the angular errors experimentally measured in the absence of visual cues and used as input parameters to the model (figure 4b, step 1). BCRW trajectories with the estimated value of θ*CRW were generated numerically (N = 500) with a range of values for w and θ*BRW (figure 4b, step 2). The resulting mean vector length, R, was compared against the experimentally measured R (figure 4b, step 3) to estimate pairs of w and θ*BRW for the two different sized beetle species (figure 4b, step 4). Due to rotational symmetry, R is independent of the direction θ0, hence we arbitrarily set its value to zero.
Ten individuals of each species were allowed to sculpt and roll a dung ball from a pat of 1 l cow dung, placed in the savannah on a sunlit day (figure 6). Trials alternated between the two species. The paths of the beetles were recorded by a hand-held video camera (GoPro® HERO5 Black), held approximately 1 m above the beetle as it rolled, until the beetle started to bury its ball. Bearing direction and distance from the centre of the dung pat to the site of burial were then measured. Positional information for the individual segments of the path trajectories from beetles rolling across this natural terrain was obtained using a tailor-made analysis tool [37] (courtesy of Dr Benjamin Risse, University of Münster). To extract shape and total distance travelled for each trajectory, a custom-made Matlab script was used. Figure 6. Orientation performance in a natural environment. The smaller Scarabaeus ambiguus and the larger S. lamarcki were allowed to form a dung ball and roll it away from a dung pat (marked with a star) in their natural environment (N, north; E, east; S, south; W, west) (a). Their trajectories (blue lines, S. ambiguus; orange lines, S. lamarcki), were recorded until they started to bury their balls (blue circles, S. ambiguus; orange circles, S. lamarcki (b)). This marked the end of the trial and the radial distance from the dung pat to the site of burial was measured. The dotted concentric circles indicate radial distances from the dung pat in 5 m increments. In total, trajectories of 10 individuals per species were recorded. The smaller species, S. ambiguus, rolled a significantly shorter radial distance from the pat before burying its ball when compared to the larger species (S. ambiguus: 7.56 m ± 1.05 m, S. lamarcki: 12.45 m ± 1.28 m, N = 10) (p = 0.001, N = 10). (Online version in colour.)
The two closely related, ball-rolling species of South African dung beetles, Scarabaeus ambiguus and S. lamarcki differ significantly in size with a pronotum width of 1.09 ± 0.01 cm and 2.07 ± 0.03 cm and a body length (tip of abdomen to tip of pronotum) of 1.52 ± 0.01 cm and 2.86 ± 0.04 cm (mean ± s.e.m. N = 10) respectively (Wilcoxon rank-sum test; pPronotum < 0.001, pBody length < 0.001, ZPronotum = −3.75, ZBody length = −3.76). Not surprisingly, the average step size for the smaller S. ambiguus (1.69 ± 0.09 cm, N = 20), is significantly shorter than that of the larger S. lamarcki (2.89 ± 0.08 cm, N = 20) (Wilcoxon rank-sum test; p < 0.001). To investigate the role of the sun in the compass system of the two species, the response of an orienting beetle in terms of change in roll bearing, was tested under an un-manipulated sky as well as under a sky with the position of the sun changed by 180° by the use of a mirror (and the real sun simultaneously shielded from view of the beetle). When the position of the sun was artificially changed by 180° (test) on the second roll, both species changed their headings in response to this manipulation, with the same order of magnitude (μS. ambiguus = 152.37° ± 105.29°, μS. lamarcki = 139.39° ± 117.45°, Mardia–Watson–Wheeler test; p = 0.92, N = 45, W = 0.16) (figure 2). No significant change in direction (μ) was seen in individuals of either of the two species of beetles between two rolls made under an unobscured sky (control) (μS. ambiguus = −3.61° ± 53.56° (mean ± s.d.), μS. lamarcki = 4.43° ± 38.61°, V-test (with the expected mean of 0°); pS. ambiguus < 0.001, pS. lamarcki < 0.001, N = 45, VS. ambiguus = 0.65, VS. lamarcki = 0.79). This demonstrates that both species orient using a sun compass. When rolling over a radial distance equivalent to the length of 20 steps of the respective species (S. ambiguus: 32.38 cm, S. lamarcki: 51.79 cm, see Methods, figure 1) there was no significant difference between the straightness of the trajectories travelled by the two species (SS. ambiguus = 0.48 ± 0.12; SS. lamarcki = 0.62 ± 0.16, Wilcoxon rank-sum test; p = 0.08, Z = −0.79) (figure 3). When instead analysing the straightness of tracks across a radial distance of 50 cm, the smaller species, Scarabaeus ambiguus had a lower straightness index (S) compared to the larger S. lamarcki (SS. ambiguus = 0.45 ± 0.12; SS. lamarcki = 0.65 ± 0.17 (mean ± s.e.m., N = 15), Wilcoxon rank-sum test; p = 0.02, Z = 2.32) (figure 3). This indicates that the size of the angular error generated by each step, θ*CRW, does not differ between the two species of beetles. This was further confirmed from the direct comparison between species (S. ambiguus: 33.11° ± 5.12°, N = 5; S. lamarcki: 29.41° ± 9.92°, N = 10, Wilcoxon rank-sum test; p = 0.86, U = 42). When rolling 20 times across an arena with a radius proportional to 20 step lengths of the two species (S. ambiguus: 32.38 cm, S. lamarcki: 51.79 cm), we found no significant difference in mean resultant vector length (i.e. spread of exit bearings) between the different sized beetles (RS. ambiguus: 0.91 ± 0.015; RS. lamarcki: 0.91 ± 0.02, Wilcoxon rank-sum test; p = 0.42, N = 20, Z = −0.83) (figure 5). When instead testing the orientation performance over a radius of 50 cm, Scarabaeus ambiguus showed a significantly shorter mean resultant vector length compared to that of S. lamarcki (RS. ambiguus: 0.88 ± 0.02; RS. lamarcki: 0.92 ± 0.01, Wilcoxon rank-sum test; p = 0.028, N = 20, Z = −2.20) (figure 5). This indicates that S. ambiguus, the smaller of the two species, is less able to maintain its bearing over a given distance compared to its larger relative when rolling under a natural sky. We found no difference in the straightness of the 20th exit path compared to the 1st exit path performed by the same individual (Wilcoxon rank-sum test; p50 cm(S. ambiguus) = 0.16, Z = 1.42; p50 cm(S. lamarcki) = 0.16, Z = 1.42; pstep length(S. ambiguus) = 0.30, Z = 1.04; pstep length(S. lamarcki) = 0.16, Z = 1.42, N = 20). This suggests that the error generated with each step does not change in size with distance rolled but remains the same regardless of the number of steps taken. The angular error generated by each step in the absence of external visual compass cues was introduced as an estimation of motor error (θ*CRW) in the biased correlated random walk model [36] (figure 4b, step 1). This allowed us to compare the resulting mean vector length, R, of the modelled data against the experimentally obtained R values (figure 4b, step 3) to estimate pairs of w (the balance between CRW and BRW) and θ*BRW (standard deviation of compass error) for the two species (figure 4b, step 4). From this, no significant difference was found in the balance between CRW and BRW, between the two species (wS. ambiguus: 0.84 ± 0.09, wS. lamarcki: 0.83 ± 0.08; Wilcoxon rank-sum test; p = 0.54, N = 13, U = 183). This also held true for the mean compass error θ*BRW (θS. ambiguus∗BRW: 5.95∘ ± 2.97∘,θS. lamarcki∗BRW: 6.34∘ ± 3.01∘, Wilcoxon rank-sum test; p = 0.65, N = 13, U = 205.5). This suggests that the relative balance between internal and external compass cues for straight-line orientation in beetles is not affected by differences in stride length. This finding is consistent with the model hypothesis. When allowed to form a ball and roll it away from a dung pat under a clear, sunlit sky in their natural habitat (figure 6a), the smaller species, Scarabaeus ambiguus, buried their dung balls 7.56 m ± 1.05 m (N = 10) away from the pat. This is significantly closer to the pat than the average radial distance travelled by the larger species before burial (S. lamarcki: 12.45 m ± 1.28 m, N = 10) (Wilcoxon rank-sum test; p = 0.010, Z = 2.57) (figure 6b). Interestingly, the average total distances rolled to reach these burial spots did not differ between the two species (S. ambiguus: 20.43 m ± 4.54 m, N = 10, S. lamarcki: 18.66 m ± 1.94 m, N = 10) (Wilcoxon rank-sum test; p = 0.85, Z = 0.19). This suggests that there is a behavioural mechanism to compensate for the increase in tortuosity that the smaller beetles unavoidably seem to experience (figure 6). When deprived of all visually mediated compass cues, the dung beetles failed to maintain a straightforward course and instead moved along tortuous paths (figure 3). Unsurprisingly, spiders, amphibians and humans are also unable to move along a given bearing in the dark [10,38,39]. Under these circumstances, these animals are expected to travel by means of a correlated random walk (CRW) [33], where each step is intended to point in the same direction as the previous one. This is also the case for the dung beetle, as captured by the biased correlated random walk model in the limit w = 0 (i.e. with no external compass input, and thus only the second term in equations (2.1) and (2.2) contributes to the accumulated error). Assuming that only internal sensory information was available to these beetles when orienting in the dark, a directional error, most likely caused by mechanosensory noise in the muscles of their moving limbs, accumulated with every step taken. Thus, the instantaneous angular error arising from the accumulated noise in motoric execution is determined as equivalent to the motor error. A direct analysis of the direction of each subsequent step when rolling in complete darkness further reveals that this motor error lies at around 30° per step, irrespective of species (S. ambiguus: 33.11° ± 5.12°, S. lamarcki: 29.41° ± 9.92°). As can be expected, these findings showed that over the same radial distance (50 cm) the smaller S. ambiguus (with more steps taken per distance travelled) rolled its ball along a significantly more tortuous trajectory compared to the larger, S. lamarcki (figure 3). Tortuous paths can also be observed outdoors, under overcast skies, or when the beetle is prevented from seeing the sky by the use of a cap [12]. Under these conditions, just as in the dark, the beetle cannot access any external visual compass cues to correct for the accumulation of errors in its orientation system. From the trajectories presented in figure 5 it is evident that, under the open sky, both species of dung beetles orient along straight paths in a given direction, presenting a clear contrast to the more tortuous trajectories taken in the absence of external visual cues (figure 3). The large change in roll bearing recorded for Scarabaeus ambiguus and S. lamarcki in response to a 180° displacement of the sun (figure 2) demonstrates the common use of a sun compass in these species during straight-line orientation. These results are well in line with previous studies of the celestial compass system of S. lamarcki [6,27–30]. When rolling repeatedly to the edge of the 50 cm diameter arena, under an open sky, the larger S. lamarcki was significantly better oriented than its smaller relative S. ambiguus (figure 5a). This difference in performance no longer prevailed when the orientation performance over a radius proportional to 20 steps of the two species (S. ambiguus: 32.38 cm, S. lamarcki: 51.79 cm) was considered. This again suggests that both species gain a similar sized error with every step taken. Interestingly, the error that is accumulated while rolling seems not to accumulate over the course of 20 consecutive rolls, as no difference in straightness was found between the first and last roll for either of the species while rolling under an open sky (figure 5). This clearly demonstrates that the error generated by each step taken, while the beetle is rolling its ball, remains the same size irrespective of the number of steps taken. To understand the effect of step size on straight-line orientation [34,35], and to model how much weight is given to external sky compass cues over internal proprioceptive cues for straight-line orientation in dung beetles, we chose to connect a vector-weighted biased correlated random walk model for directed movement, where external cues are balanced with internal ones [34,35]. The model assumes that the beetle intends to move in a straight line, which is what we have observed in this and many earlier studies of dung beetle orientation (figure 5b) [6,12,26–32]. The values for the angular error generated in the absence of visual cues, determined as equivalent to the motor error (33° for S. ambiguus and 29° for S. lamarcki), were used as input parameters to the biased correlated walk model, allowing us to estimate the balance between a biased random walk (BRW) and a CRW employed by a beetle when orienting outdoors. From this model, we can also describe the compass error generated with each step in the two species of dung beetles. When the beetles are rolling outdoors, under the full view of a natural sky, the weight given to external cues over internal cues is significantly shifted towards external cues (wS. ambiguus = 0.84 and wS. lamarcki = 0.83), revealing that the paths of the beetles, irrespective of beetle size, are primarily dictated by a BRW. This balance did not differ between the species. Interestingly, our model further reveals that the compass error is remarkably smaller than the motor error (compass error: S. ambiguus: 5.94° and S. lamarcki: 6.34° versus motor error: S. ambiguus: 33.11° and S. lamarcki: 29.41°) with no significant difference in the size of the error between the two species. This difference in angular error, or ‘noise’, generated by the two sources of information (motor and compass) provide a possible explanation for the shift towards external cues by the compass when orienting outdoors. In parallel to the weighting of external visual cues over internal proprioceptive cues by the dung beetle, ants also seem to rely more heavily on the cue that currently provides the more precise directional information [16]. If directional information from the path integrator (PI) of the ant and the visual scene are set in conflict, the weighting towards the PI will increase as the ants move further from their nest, and their PI vector becomes increasingly longer. In summary, our results suggest that (i) the analysis systems of the compass cues (visual system and neuronal system) of the smaller beetle are as precise as that of the larger beetles and that (ii) the compass system of the smaller beetles (as in the balance between a CRW and a BRW) is not specifically evolved to compensate for the directional challenges that arise due to differences in stride length. A recent study on ants shows that the ability to orient using compass information is the same across ants that differ in size by a factor of three, but similar to this study, the accumulation of errors increases with the number of steps taken [40]. Consequently, smaller ants, just like the beetles, can be observed to move over more tortuous paths than their larger relatives. Together, these studies nicely demonstrate that the ‘step size error’ has an effect on the ability of these insects to maintain a straight bearing even under an open sky, suggesting that regardless of the availability of an external visual compass cue, this noise cannot be fully compensated for and will work to the disadvantage of the smaller species. The recorded accumulation of error with distance rolled in the beetles is most likely partly due to mechanosensory noise in the motor system when executing each step, and partly due to noise in the acquisition and processing of the celestial compass cues that direct the steps taken [41]. Smaller insects with smaller steps risk travelling along more tortuous, and thus more energetically costly paths, compared to their larger relatives. But, this is only a disadvantage if the orienting insects—regardless of size—aim to travel the same distance. For the beetles, this is not the case. The smaller beetles tended to bury their balls closer to the dung pat, compared to the larger beetles within the same terrain (figure 6b). The phenomenon of smaller sized individuals travelling shorter distances than their larger peers, is not uncommon [42–45], on the contrary, there is a strong positive correlation between the distance an animal travels and its body size. In the case of the beetle, it is still unknown how it measures the distance travelled, but the impact of the relative speed at which angular errors accumulate might play a role in this behavioural outcome, resulting in smaller beetle species reaching shorter effective distances with their balls of dung. This will be the focus of future studies. Our results show that for an orienting ball-rolling beetle, an angular error accumulates over each step in the absence as well as in the presence of external visual compass cues. Consequently, smaller insects, with proportionally shorter legs, will produce a larger directional error over the same distance travelled. Our results further imply that the nature of the compass systems of different sized insects is not specifically evolved to compensate for the size (step size) of the animal. All experiments in this study were performed in accordance with the regulations referred by the South African and Swedish guidelines for animal experiments. The datasets supporting this article are available as electronic supplementary material. L.K., O.P., C.T., M.B. and M.D. conducted experiments; L.K., O.P., L.M. and M.D. designed experiments; L.K., O.P. and M.D. analysed the data; L.K. drafted the manuscript; O.P., L.M., M.B. and M.D. revised the manuscript. All authors are accountable for the presented work and approved the final version of the manuscript for publication. We declare we have no competing interests. Funding was provided by the Swedish Research Council. We thank Jochen Smolka and Benjamin Risse for the track analysis scripts, Therese Reber for assistance in the field, and James Foster for helpful comments on the manuscript. We also thank Adventures with Elephants for allowing us to work on their reserve, and the Harvey family for allowing us to work on their farm. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4577855. © 2019 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. References
Page 24Animals that possess potent weapons largely participate in ritualized fighting to resolve conflicts over resources without serious injury or fatality [1]. When ritualized behaviour escalates from displays to physical combat, the availability of adequate protective armour could be indispensable. For animals such as red deer, which physically interlock their antlers during ritualized combat, the antlers function effectively as both weapon and impact-resistant armour [2,3]. Such multifunctional structures are beneficial because the costs of developing specialized or elaborate weaponry can limit the development of other morphological traits [4], making it potentially difficult for animals to evolve both robust weapons and armour. Examples of distinct armour for ritualized fighting are rare, but mantis shrimp present an interesting case in which their unique form of ritualized combat, termed telson sparring [5], may have coevolved with specialized morphological armour. During telson sparring, mantis shrimp take turns firing their ballistic weapons (i.e. raptorial appendages) against the telson, or tailplate, of their opponent. Individuals direct nearly every strike to the telson, and this is facilitated by the recipient assuming a coiled position referred to as a ‘telson coil’ [5,6]. This behaviour imparts the telson with a fundamental role in ritualized combat, for which it must be mechanically robust to the repeated strikes of conspecifics. Fascination with the raptorial weapon and its prospects for bioinspiration have led to extensive study of the appendage and strike mechanics [7–13], greatly overshadowing the nature and potential of telson armour. Using a spring-loaded system, smashing mantis shrimp unfurl their raptorial appendages with extreme accelerations that cause cavitation and produce impact forces as great as 1500 N [8]. Predatory strikes produce enough force to break hard mollusc shells, but may require anywhere from a single strike to hundreds of strikes to do so [14]. Sparring strikes contain more energy than predatory ones [15], yet the telson withstands multiple strikes during a single sparring match and likely numerous strikes from conspecifics during the long intermoult period (over three months [16]). Intraspecific fighting is common because burrows and cavities are used for shelter and mating, but are often a limited resource that mantis shrimp aggressively defend [17], and telson sparring is a critical element in this defence. It is essential for mantis shrimp to maintain the structural integrity of both weapon and armour. Damaged or lost appendages, and even moulting, causes mantis shrimp to concede contests more readily [6,18,19]. A damaged telson may likewise reduce an animal's ability to defend its burrow, placing selective pressure on impact-resistance. Telson morphology is unsurprisingly diverse among the more than 500 species of mantis shrimp [20,21], though there are some generally consistent features. Most telsons, for instance, are dome-shaped with 1 to 3 carinae, which are raised protuberances of relatively thick and heavily calcified cuticle (figure 1). When multiple carinae are present, their spacing leaves insufficient width for the dactyl heel to contact the cuticle between them, thereby restricting strikes to the carinae. Just as the dactyl heel must be sufficiently hard to resist penetration during impact, so must be the carinae of the telson. In Neogonodactylus wennerae, the carinae are heavily calcified, but the surrounding cuticle is generally thinner and less calcified, imparting flexibility to the telson [22]. Thus the telson as a structure functions like engineered impact resistant armour, where the hard carinae resist penetration and the surrounding cuticle deforms to allow energy dissipation. Figure 1. Telson diversity of mantis shrimp species used in this study. (a) Upper two rows are sparring species. (b) Bottom row is non-sparring species. Scale bar, 10 mm. All telson images, except that of S. empusa, modified from and courtesy of Claverie & Patek [20]. (Online version in colour.)
The impact response of the telson defines its ability to withstand strikes, but it may also provide size- or performance- based information to both individuals during sparring. Energy exchange occurs during each impact and several parameters have the potential for assessment by opponents. Impact parameters such as the coefficient of restitution (COR), impulse, and duration of contact all correlated with body size in the smasher N. wennerae [22]. Any of these parameters could indicate an animal's mechanical potency, thereby providing relevant information on aggression or endurance, which are valuable for assessment. Body size is generally a good predictor of contest winners [5,23], so information extracted from telson impacts may be especially helpful when one individual remains obscured within the burrow entrance. If so, the morphology of the telson may contribute to ritualized fighting in multiple ways, as armour and a source of information for assessment. Telson sparring is described as common among mantis shrimp with smasher type appendages (Gonodactylidae), but is not known to occur in species with spearer appendages [17,24]. Spearer appendages are distinct from those of smashers in morphology (the dactyls are armed with spines rather than a bulbous heel), strike kinematics (they achieve lower accelerations and impact forces) [25], and function (they are primarily used to capture fast-moving or soft-bodied prey) [17]. Despite these differences, spearing appendages are still deadly and spearer species will either strike other mantis shrimp with a closed dactyl to limit damage or simply impale them, causing significant injury [17]. Telson sparring has not been observed in any spearer species, but if it were present, their reduced impact forces may not require as robust impact-resistant armour as smasher species. Superficially, telsons appear to be more robust in smashers that participate in telson sparring and it has therefore been hypothesized that telson armour coevolved with ritualized fighting [17]. The morphological variation in mantis shrimp telsons and the confined presence of telson sparring behaviour to smasher species posit interesting possibilities about its development as both impact-resistant armour and an assessment tool, and ultimately its role in the evolution of ritualized fighting. We examined these possibilities by testing the hypotheses that the telsons of sparring species (i) are consistently specialized for impact-resistance, (ii) are more impact-resistant than those of non-sparring species, and (iii) have impact parameters that correlate with body size. Meaningful assessment of impact mechanics in biological systems is challenging, and we took the approach of using collision energetics as established in previous work on the mantis shrimp telson impact response [22]. The exchange of kinetic energy during impact is defined by the structures involved and is informative of their mechanical behaviour under realistic interactions. Ball drop tests were used to measure and compare the telson impact mechanics of 15 species of mantis shrimp, encompassing smashers, spearers, and an undifferentiated form [9]. Key aspects were mapped onto a phylogeny for evolutionary context and detailed analysis of telson morphology and mechanical properties of a smasher and a spearer species provided further structural insights. A total of 91 individuals from the 15 chosen mantis shrimp species (Crustacea: Stomatopoda) were either collected from the field or purchased from commercial suppliers (table 1). Telson morphology is diverse among these species (figure 1). The large species, Hemisquilla californiensis and Lysiosquillina maculata, were kept in individual 20 l tanks filled with recirculating artificial seawater (salinity: 32–36 ppt, 22°C). All other species were kept in individual 2 l plastic cups filled with artificial seawater that was changed twice weekly (salinity: 32–36 ppt, 22°C). Animals were fed both fresh and frozen grass shrimp twice weekly.
Body mass and sex were determined for each animal prior to testing. Animals were checked for moult stage by examining a pleopod under the microscope [16]. Only intermoult animals were used for this study. Males and females were combined for analyses due to limited availability of animals and the fact that both sexes are known to participate in telson sparring. None of the impact response parameters differed between sexes for G. chiragra and N. festae (T-tests; all p ≥ 0.43), or for N. wennerae from a previous study [22]. Immediately prior to each impact test, individuals were anaesthetized and euthanized by placement in a −20°C freezer until dead but not frozen. Data for Neogonodactylus wennerae were gleaned from a previous study [22]. The impact response of the mantis shrimp telson was determined through impact tests, as described in detail in a previous study [22]. Whole mantis shrimp were positioned horizontally and secured on top of a 2.5 cm thick steel countertop slab. The telson rested on a small 3.0 mm thick Plexiglass strip glued to the slab, which allowed positioning for a direct, collinear impact. To prevent dislodgement of the animal at impact, a small drop of cyanoacrylate glue was placed on the tip and base of each uropod, and at the base of both sides of the fourth abdominal tergite. A small 440C stainless steel ball (6.33 mm diameter; 1.022 g; Rockwell C: 58-65; Small Parts, Miramar, FL, USA) was dropped through the air, without spin, from an electromagnet (model E-66-100-34, Magnetic Sensor Systems, Van Nuys, CA, USA) that was attached to a ring stand. The electromagnet was positioned approximately 100 mm above each animal, giving an impact velocity of 1.67 m s−1. The kinetic energy of this collision was calculated using 12mv2, where m is ball mass and v is velocity at impact, and was determined to be equivalent to a smasher mantis shrimp (body mass of 1.0 g) striking at 15 m s−1. This energy is thus comparable either to a small animal striking with high velocity or to a large animal striking with low velocity, which is reflective of the negative scaling of strike velocity with body mass across species of smashing and spearing mantis shrimp [12]. To facilitate comparison among species, we used the same impact energy for all drop tests. It is important to acknowledge that our tests do not encompass the range of strike velocities observed in different mantis shrimp species, which limits the scope of our results because the COR is sensitive to impact velocity [26].For each animal, the ball was dropped 10 times onto each of two targets: the centre carina of the telson and the centre of the fifth abdominal tergite (see [22]). The order of testing on either the telson or abdomen was randomized to control for test order. For the two large species, H. californiensis and L. maculata, additional ball drop tests were performed in between or adjacent to the carinae. Animals were covered by a seawater-soaked paper towel in between ball drops to prevent dehydration. Each ball drop test was recorded with a high-speed digital video camera (APX-RS, Photron, San Diego, CA, USA or Phantom Miro 310, Vision Research, Wayne, NJ, USA) at 15 000 frames s−1, 0.067 ms shutter duration, and 256 × 512 pixel resolution. A 10 mm × 10 mm grid was placed in the camera's field of view for calibration in addition to the ball. A coarse examination of impact strength and failure using ball drop tests was performed on two species: a smasher, Neogonodactylus bredeni (N = 5, mass = 0.95 ± 0.63 g), and a spearer, Pseudosquilla ciliata (N = 5, mass = 3.12 ± 1.7 g). For these tests, animals were prepared and tests were performed as described above, but drop tests were conducted with a series of four steel balls in order of increasing size: (i) 6.33 mm diameter, 1.022 g, (ii) 9.53 mm diameter, 3.466 g, (iii) 15.89 mm diameter, 15.09 g, and (iv) 19.05 mm diameter, 27.85 g. All balls were 440C and of the same Rockwell C hardness (58-65; Small Parts, Miramar, FL, USA). This series of ball drops produced impact energies estimated to be equivalent to: (i) a 1.0 g animal striking at 15 m s−1, (ii) a 2.0 g animal striking at 20 m s−1, (iii) a 5.0 g animal striking at 20 m s−1, and (iv) a 25 g animal striking at 15 m s−1. Each ball was dropped on the telson once, in order of increasing size. After each impact the telson was examined for cracks and tests were ended once cracks were detected. Impact strength was estimated as the greatest impact energy that did not induce any visible cracks. The COR was calculated by measuring the velocity before and after ball impact, using the 10 frames preceding contact and the 10 frames following separation, respectively. The first and last frames in which the ball was in contact with the specimen were determined (Irfanview v. 4.20, Irfan Skiljan, Austria) and then ball displacement over 10 frames was measured (SigmaScan Pro 5.00, SPSS, Chicago, IL, USA). From these distances and changes in time, impact and separation velocities were determined and used to calculate the coefficient of restitution, e: e=vfvi, where vf is the velocity at separation and vi is the velocity at impact [26,27]. The amount of energy lost during impact is calculated as1−e2, where e is the coefficient of restitution [28,29].Impact was also modelled as a collision between two springs, and spring stiffness, k, was calculated using k=mπ2t2, where m is ball mass and t is impact duration [28,30].Impact duration was calculated as the time between first contact and separation. Deformation was estimated as the displacement of the ball during the compression phase of impact. Finally, impulse, I, was calculated using the change in momentum I=mΔv, where m is ball mass and Δv is the change in ball velocity (i.e. vi + vf).Impact parameters were calculated for each drop test and then averaged for each animal. Materials testing was conducted on the telson of one smasher species, Neogonodactylus bredeni (N = 8), and one spearer species, Squilla empusa (N = 8). Telsons were excised from the animals and cleaned of internal tissue. Dissection scissors were used to carefully cut around the base of the central carina and then the ventral surface of the carina was sanded with sandpaper until flat. The ventral side of the prepared carina was then secured to an aluminium block with cyanoacrylate, exposing the dorsal surface of the carina for indentation. A series of five indents were made along the midline of the carina, penetrating the epicuticle layer, using a nanoindentation materials testing machine (Nano Hardness Tester, Nanovea, Irvine, CA, USA) equipped with a Berkovich diamond indenter tip. Indents were carried out with a load of 40 mN and loading and unloading rates of 80 mN min−1. This load limited penetration to the outer region of the cuticle (epi- and exocuticle layers). Hardness and stiffness values of indentations were averaged for each specimen. Telsons from one Neogonodactylus bredeni and one Squilla empusa were examined using scanning electron microscopy (SEM). The telsons were excised, cleaned, and cut transversely across the dorsal surface. Samples were placed in a critical point drier (AutoSamdri 815 Series A, Tousimis, Rockville, MD, USA), secured to a double 90° SEM mount revealing the cross section, and sputter coated with iridium. Cross sections were examined with an ultra-high resolution SEM equipped with energy dispersive X-ray (EDX) (XL30 SFEG, FEI, Hillsboro, OR, USA; Oxford X-MAX 80 EDS detector, Concord, MA, USA) at a 20 kV acceleration voltage. EDX elemental mapping was carried out at 10 kV and used to visualize the distribution and density of key elements (Ca, Mg, and P) in the dorsal region of the carinae. The sampling of Stomatopoda for this study arguably spans the known phylogeny of the group [31]. To assess possible covariation of telson armour with ritualized fighting, a phylogeny was constructed for the 15 mantis shrimp species used in this study. DNA sequences for mitochondrial cytochrome oxidase subunit I (COI), 12S rRNA (12S) and 16S rRNA (16S) and nuclear 28S rRNA (28S) and 18S rRNA (18S) were sourced from GenBank for 14 of the terminals and new sequences were generated for Neogonodactylus festae (table 2). The COI sequences were aligned using MUSCLE [32], while rDNA sequences (12S, 16S, 18S and 28S) were separately aligned using MAFFT [33], with the Q-INS-I option and default gap open and extension parameters. The third codon positions of COI were excluded based on the evidence of saturation found by Van der Wal et al. [31]. The five gene partitions were concatenated and analysed using RAxML 8 [34] using the GTR+G model separately for each partition. Clade support was assessed via 100 bootstrap pseudoreplicates using the same model. The tree was rooted using Hemisquilla californiensis, based on a previous phylogenetic study of Stomatopoda [31].
Impact parameters were compared between the telson and the abdomen of each species using either paired t-tests or Mann–Whitney and were correlated with body mass using least-squares linear regression. To facilitate comparisons across species, those known to spar were coded as a binary character, ‘Sparring’; those with smashing appendages and known to spar were coded as sparrers, while those with spearer appendages were coded as ‘non-sparring’ (= L. maculata, P. ciliata and S. empusa). Smasher species with no documentation of sparring behaviour were coded as ‘unknown’. The undifferentiated form, H. californiensis, does not telson spar and was coded as a ‘non-sparring’ species. Impact parameters were compared across species using ANCOVA with body mass as a covariate followed by post hoc Tukey tests when appropriate. Tukey adjusted p-values were used to account for multiple pairwise comparisons (105 tests) and are reported as ranges of p-values. All statistics were performed using R (v3.0.2). Results are represented as mean ± s.d. To consider the evolution of telson impact features within the phylogeny, the mean COR and spring stiffness for each species were mapped as continuous characters onto the maximum-likelihood phylogeny after being scored in Mesquite 3.6 [35]. Transformations were estimated using maximum parsimony. The transformation for Sparring was then mirrored against transformations for COR and spring stiffness to visually assess any possible covariability. The maximum-likelihood phylogenetic analysis gave the best tree (figure 2) (log-likelihood = −18784.2) that was largely congruent, given the taxon sampling, with previous results [31]. The main exception was the placement of the gonodactyloid Haptosquilla, which did not group with other members of this taxon. This Haptosquilla clade showed a relatively long branch compared to other stomatopods. Also, Pseudosquilla formed a clade with Squilla rather than with other gonodactyloids. There was overall low bootstrap support for these relationships, as is also apparent in Van der Wal et al. [31] and it would appear that much more data is required to properly infer stomatopod phylogeny. The transformations for sparring evolution, spring stiffness and COR are shown in figure 3 on the maximum-likelihood topology. The placement of Haptosquilla means that sparring either evolved twice or that the Pseudosquilla/Squilla clade has lost sparring. Both spring stiffness and COR showed marked homoplasy and neither showed any obvious patterns relative to the occurrence of sparring behaviour (figure 3). Figure 2. Maximum-likelihood tree from the concatenated 5 gene (COI, 12S, 16S, 18S and 28S) dataset. The tree is rooted with the unusual smashing mantis shrimp H. californiensis based on previous studies. Bootstrap scores are at the nodes.
Figure 3. Most parsimonious reconstructions on the maximum-likelihood topology for the character ‘Sparring’ mirrored against continuous characters (a) COR and (b) spring stiffness. (Online version in colour.)
The COR of the telson was significantly lower than that of the abdominal tergite for only three species of smashers (N. bredeni, N. wennerae, and H. glyptocercus) and the primitive smasher H. californiensis (table 3). All other species had similar CORs for both telson and abdominal tergite (table 3). All but two species had the same telson COR (ANCOVA, F14,60 = 10.03, p ≪ 0.001) and there was no effect of mass (F14,60 = 2.73, p = 0.10) or interaction between species and mass (F14,60 = 2.73, p = 0.14) (figure 4a). The species that differed were H. californiensis, which had a COR lower than all species (adj p ≪ 0.001 to 0.007) except S. empusa (adj p = 0.21) and S. empusa, which had a COR lower than seven other species (adj p ≪ 0.001 to 0.02). The telson COR does not reveal any evolutionary pattern related to sparring and non-sparring behaviour among species (figure 3a). Figure 4. Telson impact properties. (a) Mean COR does not differ between sparring (white) and non-sparring (grey) species. (b) Mean spring constant does not differ between sparring (white) and non-sparring (grey) species. Smasher species with ‘unknown’ sparring denoted with asterisk. Box boundaries: 25th and 75th percentile; error bars: 10th and 90th percentile; solid line: median.
Spring stiffness of the telson was consistently greater than that of the abdomen for all smasher species (table 3) and was only statistically greater than the abdomen for one species of spearer (L. maculata). In alignment with the spring constant, most telsons experienced significantly less deformation during impact than the abdomens, at least for the smasher species (table 3). All species, regardless of sparring behaviour and body mass, had the same telson spring stiffness (ANCOVA, F14,60 = 1.28, p = 0.25; mass covariate F14,60 = 0.67, p = 0.42) (figure 4b). Telson deformation differed between some species (ANCOVA, F14,60 = 10.80, p ≪ 0.001) with no effect of mass (F14,60 = 3.25, p = 0.08), but an interaction between species and mass (F14,60 = 3.59, p ≪ 0.001). Only three species differed in telson deformation: H. trisponosa and O. latirostris each had significantly greater telson deformation than 10 other species (adj p ≪ 0.001–0.01), while G. espinosus had significantly less telson deformation than 5 other species (adj p ≪ 0.001–0.03). Duration of impact was significantly less in the telson than the abdominal tergite for the majority of smashers and one of the spearer species (S. empusa) (table 3). Telson impact duration was only significantly different between O. latirostris and G. chiragra (ANCOVA, F14,60 = 3.40, adj p = 0.04) and between O. latirostris and G. smithii (adj p = 0.04). All other species had the same impact duration regardless of sparring behaviour (table 3). The impulses experienced by the telson and the abdominal tergite were the same for most species, except N. festae, N. oerstidii, and H. glyptocercus (table 3). There was a significant difference in telson impulse between species (ANCOVA, F14,60 = 14.35, p ≪ 0.001), with no effect of mass (F14,60 = 2.27, p = 0.14) or interaction between species and mass (F14,60 = 1.53, p = 0.13). Hemisquilla californiensis had a significantly lower telson impulse than 12 of the other species (adj p ≪ 0.001–0.008), S. empusa had a lower impulse than nine of the species (adj p ≪ 0.001–0.04), and N. festae had a lower impulse than 10 other species (adj p ≪ 0.001–0.008). Other than these few species, mantis shrimp had the same telson impulse regardless of sparring behaviour (table 3). For H. californiensis (unusual smasher), the COR of the region adjacent to the carina was 0.37 ± 0.03, which was not statistically different from that of the abdomen (paired t-test, t = 1.257, N = 3, p = 0.34). On the other hand, in L. maculata, the COR of the region adjacent to the carina was 0.66 ± 0.07 and significantly higher than that of the abdomen (paired t-test, t = −4.581, N = 4, p = 0.02). The telsons of both N. bredeni and P. ciliata exhibited similar failure behaviour. Cracks formed in the thinner cuticle regions surrounding the carinae, which showed no visible damage. During impacts from ball 2, the telsons compressed nearly completely and sprung back to the original state, demonstrating significant elasticity and no visible damage. Three of the five telsons from each species incurred cracks, some major, with impacts from ball 3 and not all returned fully to their original state. Remaining telsons from N. bredeni failed catastrophically during impacts with ball 4, whereas those of P. ciliata suffered major cracks and permanent deformation. Interestingly, impacts from ball 4 produce an impact energy of 0.04 J, which is above the average strike energy of N. bredeni, but within their range as reported by Green et al. [15]. Telson impact parameters generally showed no correlation with body mass among the species measured (table 4), though small sample sizes resulted in low statistical power for some species. Yet, correlations were detected for some of the parameters in smasher and spearer species. These include the COR for N. wennerae and G. smithii; spring constant for N. wennerae, H. californiensis, P. ciliata, S. empusa; impulse for N. wennerae; and impact duration for N. wennerae, H. californiensis, and S. empusa.
The smasher N. bredeni has three carinae on the telson compared to the spearer S. empusa that has only one (figure 1). The carinae of both species have the same bulbous shape with thickened cuticle. Cuticle thickness of the carina is 0.32 mm in N. bredeni and 0.35 mm in S. empusa, meaning that the smaller N. bredeni has a thicker carina relative to body size. The adjacent cuticle is also similar: 0.19 mm and 0.21 mm, respectively. Both species have relatively uniform distributions of Ca and Mg across the cuticle. Phosphorus also occurs throughout, with greater density in the exocuticle region (figure 5). Figure 5. Elemental mapping of key elements in a smasher (N. bredeni) and a spearer (S. empusa) carinae cuticle using EDX. Ca and Mg are uniformly distributed, but P shows greater density in the outer exocuticle layer. (Online version in colour.)
The mean hardness of the contact surface of the carina of N. bredeni (0.63 ± 0.38 GPa) was not statistically different from that of S. empusa (hardness: 0.56 ± 0.21 GPa) (t-test, t = 0.442, N = 8, p = 0.67) (figure 6). Neither was carina stiffness different between the two species (N. bredeni: 14.05 ± 9.68 GPa; S. empusa: 20.19 ± 16.43 GPa) (t-test, t = −0.911, N = 8, p = 0.38) (figure 6). Whereas the hardness of N. bredeni's carina decreased with increased body mass (LSR, slope = −0.66, d.f. = 6, R2 = 0.69, F = 13.2, p = 0.01), stiffness did not (LSR, slope = −5.41, d.f. = 6, R2 = 0.07, F = 0.48, p = 0.52). Neither hardness (LSR, slope = 2.1 × 10−4, d.f. = 6, R2 = 4.0 × 10−5, F = 2.4 × 10−4, p = 0.99) nor stiffness (LSR, slope = −0.26, d.f. = 6, R2 = 0.01, F = 0.07, p = 0.81) correlated with body mass for S. empusa. Figure 6. Mechanical properties of telson carinae in a smasher (N. bredeni) and a spearer (S. empusa) species. Neither hardness (white) nor stiffness (grey) differed between the species. Box boundaries: 25th and 75th percentile; error bars: 10th and 90th percentile; solid line: median; dashed line: mean.
Evolving impact-resistant armour is essential if sparring mantis shrimp are to withstand the forceful strikes of conspecifics during telson sparring. The telson is undoubtedly an effective shield against impacts, but our data do not support the hypothesis that sparring mantis shrimp evolved telsons that are mechanically more robust to impact than non-sparring species. Neither energy dissipation (low COR) nor spring stiffness are correlated with the occurrence of sparring within the phylogeny (figure 3). All of the species examined in this study have telsons that exhibit similar impact behaviour, regardless of appendage type and telson morphology. The telson is a relatively stiff spring that dissipates most of the impact energy. Furthermore, telson impact parameters tend not to be correlated with body size, rendering its broad use for size assessment during combat limited. Telson armour, as characterized by impact response, does not appear to have coevolved with ritualized fighting in mantis shrimp. From a simplified point of view, crustacean cuticle is akin to bicycle helmets, and other engineered impact-resistant armour, that use a hard outer shell and a compliant inner liner to effectively reduce peak accelerations and minimize internal damage [36]. The heavily calcified outer layers of the cuticle (epicuticle and exocuticle) generally provide hardness while the inner layer (endocuticle) confers toughness [37,38]. The composite nature and layered, helicoidal organization of fibres provide multiple mechanisms to control crack propagation, and imbue the cuticle with toughness against impact that surpasses that of model tough materials, such as abalone shells [39]. While this is a general feature of crustacean cuticle, fine-tuning of composition and arrangement at multiple levels of organization can yield enhanced impact resistance. The dactyl heel is a striking example of this specialization; it has a modified structure and composition that prevents significant wear and damage from frequent forceful impacts [11,40]. Specifically, the dactyl heel is enhanced by thickened cuticle with distinct regions that vary in fibre organization and the amount, orientation, and crystallinity of key minerals (Ca, Mg and P) [11]. Increased calcium phosphate in the relatively thick impact region raises hardness and stiffness, which facilitates the transfer of kinetic energy to prey or conspecifics, whereas inner regions provide fracture toughness to minimize damage [11,40]. It is logical to expect that the telson would share some similar modifications to withstand impact. The carina of the telson, which is struck by the dactyl heel during telson sparring, imitates the structure of the dactyl heel to some extent. In both of the two species examined for morphology, N. bredeni and S. empusa, the telson carinae are also rounded in cross section and characterized by thickened cuticle, with an outer impact region (epi- and exocuticle layers) that accounts for approximately 21 and 30% of total cuticle thickness, respectively, and is comparable to the estimated 29% for the dactyl heel [11] and unspecialized crustacean cuticle [41]. Based on EDX maps, Ca and Mg are relatively uniform across the cuticle, but P appears to be in higher density in the outer impact region, which may serve to harden the contact surface similarly to the dactyl heel. Despite the similarities in morphology though, the impact surface of the telson carina has approximately 6× lower hardness and 4× lower stiffness than the impact region of the dactyl heel [11]. These differences in mechanical properties between the dactyl heel and telson result in greater impact energy being transferred to the telson during a sparring strike. Telsons accommodate significant impact energy through their macroscale structure as well. In a departure from the specialized dactyl heel and the non-specialized abdominal tergites, the telson combines both stiff and compliant regions of cuticle that together prevent penetration, spread the impact force, and dissipate energy. Rather than a uniformly hard and stiff structure that would increase brittleness and incur more costs to construct and carry, hard material is restricted to the carinae, which are surrounded by compliant cuticle. Impacts produce consistent behaviour among diverse telsons, where the carinae do not deform but the compliant region permits compression of the entire dome-like dorsal surface. As a whole, telsons generally have greater spring stiffness compared to non-specialized abdominal tergites, resulting in less deformation and shorter contact times. This would be expected to increase the impact force and potential damage by focusing the impact energy over a shorter time scale, a strategy avoided in engineered designs, such as vehicles that use crumpling to absorb energy and minimize damage. However, in live interactions this is mitigated by the telson coil position that allows significant abdominal flexion and displacement of the telson. Under our testing conditions, collisions with the telson and abdomen have similar COR values for most species, meaning that on average 71% of the energy gets dissipated during impact. The main difference between these segments is that the abdomen dissipates energy through significant deformation, which may be damaging to internal tissues. Telson impact mechanics are generally consistent among species, regardless of their appendage type and sparring behaviour. Contrary to our predictions that sparring mantis shrimp would have telsons with enhanced impact resistance, none of the impact parameters significantly differed between sparring and non-sparring species. In particular, we expected that sparring species evolved low CORs and high spring constants because these measures reflect greater energy dissipation and stiffness during impacts, and because they characterized a smasher species from a previous study [22]. Yet, when these parameters were analysed against the phylogeny, neither measure was correlated with the presence of telson sparring. Our data do not support the hypothesis that sparring species coevolved more robust telson armour. While a larger assortment of species studied could yield a broader pattern between armour and behaviour, our data show clearly that several sparring and non-sparring species are not distinguishable in their telson response to impact forces. This inspires the question of whether the generalization of impact-resistant telson armour reflects an adaptive value for non-sparring species or is simply an inherent feature of telson structure. Telson structure, particularly the carinae, varies not only between sparring and non-sparring mantis shrimp, but also extensively within each of these groups. Among the smashers used in this study, telsons typically possess three carinae, but their shapes are highly variable (figure 1). The Neogonodactylus species tend to have broad, bulbous carinae, but others have narrow carinae (O. latirostris) or even flat, rounded carinae (H. glyptocercus and H. trisponosa). Of the spearers, both L. maculata and S. empusa have one wide carina in the middle, but P. ciliata is similar to the smasher O. latirostris that has three narrow carinae. Concomitant with these variations in structure are variations in the proportions of stiff and compliant elements, but these differences, and the differences in carina shape, do not emerge in the structural responses of telsons to biologically relevant impact energies. It appears that impact-resistance is inherent across a range of telson structural templates. Consistent impact behaviour across diverse telsons poses interesting questions about the integration across scale (from chemical to structural) for impact resistance, and also about the function of the telson. Telson carinae and other attributes, such as spines, may be important for ecological factors other than ritualized fighting. For example, telsons are sometimes used to physically block burrow entrances and species will attempt to overtake burrows by striking the telson of the inhabitant in a non-ritualized manner [17,42]. In this scenario, impact resistance could be of importance for non-sparring species that might engage with interspecific aggressors. Mantis shrimp also use their telsons in other aggressive ways, such as to stab or push an opponent [17,43], actions that also require a stiff structure. There are thus a variety of behaviours other than sparring that may pose selective pressures on telson morphology. Our approach to characterizing the impact-resistance of telsons using ball drop tests and energy dynamics has the advantage of observing responses under biologically relevant conditions, but does not address all possible factors that may be important. For example, species may differ in telson strength and mode of failure, which could be correlated with the impact forces that they generate during a strike. When spearers strike, they do so with significantly lower accelerations and impact forces than smashers [25]. Our coarse comparison of the telson impact strength and mode of failure between the sparring N. bredeni and the non-sparring P. ciliata revealed that both failed in the same manner and withstood similar impact forces despite having different carina structure. Both species withstood impact energies estimated to be comparable to that of a medium-sized smasher's (2.0 g) strike at 20 m s−1, but failed at impact energies comparable to a larger animal (5.0 g) striking at that same impact velocity and even some of the higher energy strikes of N. bredeni [15]. We did not test impact strength broadly enough to determine if it is a potential factor evolutionarily correlated with sparring behaviour, but it is evident that this parameter is similar among at least some spearer and smasher species. For our analysis, we linked the occurrence of telson sparring with appendage type and grouped all smashers known to spar as sparring species and smasher species not documented as sparring as ‘unknown’. This approach was used because multiple previous studies have declared that sparring is only carried out by smasher species [17,24]. Spearers, in contrast, tend to strike other parts of the body during conflicts, but can modulate the damage inflicted by striking with either a closed or open dactyl [17]. Within the smashing gonodactylids, and even within populations of species, there is a range in levels of aggressiveness and the likelihood of telson sparring [17]. If telson armour is correlated with sparring behaviour, then it would be expected that telson armour varies within the smashers based on fighting proclivity. Aggression levels during sparring are not available for most species in our study, so it is not possible to test for correlations between species aggression and telson impact properties. Yet, the similar telson impact properties of N. bredeni, which is known to be an aggressive sparrer [17], and G. chiragra, which is less likely to spar [6], lessens the likelihood that telson impact response correlates with species aggression. Our cumulative data showing consistent impact response across multiple smasher species further diminish this assertion. Body size is generally a good indicator of winners in contests and opportunities for size assessment arise from multiple displays that comprise ritualized fighting. Physical contact during telson sparring can provide an honest signal for assessment by both participants because each individual experiences the energy dynamics of an impact, which are governed by the laws of physics. How much energy is imparted, dissipated, and returned depends greatly on the properties of each of the colliding structures, and there are several components of an impact that may correlate with body size. A previous study on N. wennerae found that the COR, impulse, and impact duration all correlated with body mass [22]. Yet in this study, the relationships between impact parameters and body size were diffuse, with only some species displaying correlations and with generally only one parameter. Based on our data, different telson templates can confer size-based information during sparring, but its broad utility as a size assessment tool appears to be limited. We cannot reject the possibility, however, that different scaling relationships may emerge at higher or lower impact energies relevant to species-specific strikes. Once ritualized fighting has escalated to combat, other indicators of fighting ability, such as animal condition, aggression, or endurance, become more important to assess than body size. In contests between the mantis shrimp N. bredeni, winners are determined by the number of strikes rather than strike force [5], implicating that endurance is an important signal to assess. Both physiological and mechanical endurance can be qualified through telson impacts, either by fatigue in strike force or telson structure. Mechanical fatigue in the telson may emerge in some impact parameters over multiple strikes and yield information about telson condition. For example, plastic deformation of the contact area and residual stresses due to repeated impacts can cause changes to the COR [44]. Telsons are not as hard as dactyl heels and are thus more likely to experience fatigue, which could be communicated through changes in the COR and acted upon by either individual in their decision to continue the fight or concede. We know from a previous study that N. wennerae exhibited no fatigue in telson COR over 100 sequential impacts [22]. This is far more impacts than the typical number of strikes exchanged during a sparring interaction, but it is reasonable that higher energy impacts would induce material fatigue at a faster rate and that fatigue may manifest in other impact parameters like spring constant. We conducted a comparative study of telson impact mechanics among smasher and spearer mantis shrimp species and found no evidence that smasher telsons are more robust to impact than spearers, nor that telson armour is correlated with the occurrence and evolution of telson sparring. Rather, it appears that telsons are mechanically similar over a range of morphologies due to the fundamental integration of stiff and compliant material and structure. This architecture makes telsons uniquely resistant to high impact forces and a source for bioinspiration. Our unique and integrative approach to studying impacts in biological systems has helped to define the limited functional role that the telson played in the evolution of ritualized fighting in mantis shrimp and more generally in the coevolution of specialized weapons and armour. The datasets supporting this article have been uploaded as part of the electronic supplementary material. J.R.A.T. conceived of the study, carried out the impact analyses, and wrote the manuscript, N.I.S. carried out mechanical testing, and G.W.R. constructed the phylogenetic tree, carried out the phylogenetic analysis, and edited the manuscript. We declare we have no competing interests. N.I.S. was supported by a Scripps Undergraduate Research Fellowship through a NSF-OCE funded REU site grant to the Scripps Institution of Oceanography (NSF award no. 1659793). This work was supported by a National Science Foundation postdoctoral fellowship and a Hellman fellowship to J.R.A.T. R. Caldwell inspired the question and J. Vincent inspired the approach. S.N. Patek, T. Claverie, M. deVries, J. Christy, R. Heard and E. Staaterman collected and made available mantis shrimp specimens, S.N. Patek provided material and intellectual support, and M. Porter provided the gene sequences for N. festae. Footnotes†Present address: University of the Virgin Islands, St Thomas, US Virgin Islands 00802, USA. Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608437. References
Page 25A ubiquitous feature of biological motility is the combination of stereotyped movements in seemingly random sequences. Capturing the essential characteristics of motion thus requires a statistical description, in close analogy to the random-walk formulation of Brownian motion in physics. A canonical example is the ‘run-and-tumble’ behaviour of E. coli bacteria, in which relatively straight paths (runs) are interspersed by rapid and random reorientation events (tumbles) [1]. The random walk of E. coli can thus be characterized by two random variables (run length and tumble angle) and two constant parameters (swimming speed and rotational diffusion coefficient), and detailed studies over decades have yielded mechanistic models that link these key behavioural parameters to the underlying anatomy and physiology [2–5]. Random-walk theory has been fruitfully applied also to studies of eukaryotic cell migration in both two [6–8] and three [9] dimensions. Can a similar top-down approach be fruitfully applied to more complex organisms—for example, an animal controlled by a neural network? Animal behaviour is both astonishing in its diversity and daunting in its complexity, given the inherently high-dimensional space of possible anatomical, physiological and environmental configurations. It is, therefore, essential to identify appropriate models and parametrizations to succinctly represent the complex space of behaviours—a non-trivial task that has traditionally relied on the insights of expert biologists. In this study, we ask if one can achieve a similar synthesis by an alternative, physically motivated approach [10]. We seek a quantitative model with predictive power over behavioural statistics, and yet a parametrization that is simple enough to permit meaningful interpretations of phenotypes in a reduced space of variables. As an example, we focus on the motile behaviour of nematodes, which explore space using a combination of random and directed motility driven by undulatory propulsion. The nematode Caenorhabditis elegans has long been a model organism for the genetics of neural systems [11,12], and recent advances in imaging have made it feasible to record a large fraction of the worm’s nervous system activity at single-cell resolution [13–15]. These developments raise the compelling possibility of elucidating the neural basis of behavioural control at the organism scale, but such endeavours will require unambiguous definitions of neural circuit outputs and functional performance. The worm’s behavioural repertoire [16,17] is commonly characterized in terms of forward motion occasionally interrupted by brief reversals [18–20], during which the undulatory body wave that drives its movement [21] switches direction. In addition, worms reorient with a combination of gradual curves in the trajectory (weathervaning) [22,23] and sharp changes in body orientation (omega-turns [19] and delta-turns [24]). These elementary behaviours are combined in exploring an environment [22,25]. Environmental cues such as chemical, mechanical or thermal stimuli [26] lead to a biasing of these behaviours, guiding the worm in favourable directions [22,25,27]. Finally, in practical terms, the worm’s small size (approx. 1 mm in length), moderate propulsive speed (approx. 100 μm s−1) and short generation time (approx. 2 days) allow a considerable fraction of its behavioural repertoire to be efficiently sampled in the laboratory [18,28]. An influential example of such an analysis is the ‘pirouette’ model proposed by Pierce-Shimomura & Lockery [25] which describes the worm’s exploratory behaviour as long runs interrupted occasionally by bursts of reversals and omega turns that reorient the worm, in close analogy to the run-and-tumble model of bacterial random walks [1]. Later work by Iino et al. identified that worms also navigate by smoother modulations of their direction during long runs (weathervaning) [22], and Calhoun et al. have suggested that C. elegans may track the information content of environmental statistics in searching for food [29], a motile strategy that has been termed ‘infotaxis’ [30]. A recent study by Roberts et al. [20] analysed high (submicrometre)-resolution kinematics of C. elegans locomotion and developed a stochastic model of forward–reverse switching dynamics that include the short-lived (approx. 0.1 s) pause states that were identified between forward and reverse runs. Importantly, while these previous studies have illuminated different modes of behavioural control, they were not designed to obtain a predictive model of the trajectory statistics and thus a succinct parametrization of C. elegans motility remains an important open problem. A quantitative parametrization capturing the repertoire of C. elegans’ behavioural phenotypes would facilitate data-driven investigations of behavioural strategies: for example, whether worms demonstrate distinct modes of motility (characterized by correlated changes in parameters) over time, or in response to changes in environmental conditions [28,31–33]. Variation in the obtained parameters among individuals can inform on the distribution of behavioural phenotypes within a population, and reveal evolutionary constraints and trade-offs between strategies represented by distinct parameter sets [34]. Caenorhabditis elegans is a member of the Nematoda phylum, one of the largest and most diverse phylogenetic groups of species [35,36]. Despite the diversity of ecological niches these animals inhabit [35], comparisons of nematode body plans have revealed a remarkable degree of conservation, even down to the level of individual neurons [37]. This combination of highly conserved anatomy and ecological diversity makes nematode motility a compelling case for studies of behavioural phenotypes. Anatomical conservation suggests it might be possible to describe the behaviour of diverse nematodes by a common model, and identifying the manner in which existing natural variation is distributed across the parameter space of the model could reveal distinct motility strategies resulting from optimization under different environmental conditions. In this study, we develop a simple random walk model describing the translational movements of a diverse collection of nematode species, freely moving on a two-dimensional agar surface. In addition to providing a quantitative and predictive measure of trajectory dynamics, the parameters of our model define a space of possible behaviours. Variation within such a space can occur due to changes in individual behaviour over time (reflecting temporal variation in the underlying sensorimotor physiology, or ‘mood’), differences in behaviour among individuals (reflecting stable differences in physiology, or ‘personality’) and differences between strains and species (reflecting cumulative effects of natural selection). By quantitative analyses of such patterns of variation, we seek to identify simple, organizing principles underlying behaviour. In order to identify conserved and divergent aspects of motility strategies, we sampled motile behaviour over a broad evolutionary range. We selected a phylogenetically diverse collection of nematodes with an increased sampling density closer to the laboratory strain C. elegans (figure 1a; electronic supplementary material). To sample individual variation, we recorded the motility of up to 20 well-fed individuals per strain and each individual for 30 min on a food-free agar plate at 11.5 Hz with a resolution of 12.5 μm px−1 (see electronic supplementary material). Figure 1. Nematodes perform random walks off-food with a mean speed and effective diffusivity that varies across strains. (a) Phylogenetic tree with the strains used in this study. The bold numbers are the major clades of Nematoda. The grey box indicates genetically distinct wild isolates of C. elegans. A representative worm image and 30 min trajectory are shown to the right. (b) The average mean-squared displacement, MSD, across N2 individuals is shown in black. For comparison, we show the MSD expected from ballistic (blue) and diffusive (red) dynamics. The motility transitions from a ballistic to diffusive regime within a time scale of tens of seconds. Shaded regions indicate a 95% confidence interval. (c) Mean speed 〈s〉 and effective diffusivity Deff (mean and 95% confidence intervals) for each strain, calculated from fits of the MSD as in (b). Across strains, both 〈s〉 and Deff vary by orders of magnitude. (Online version in colour.)
We measured the centroid position x(t) and calculated the centroid velocity v(t), using image analysis techniques (electronic supplementary material, figure S11). We chose the centroid as the measure of the worm’s position because it effectively filters out most of the dynamics of the propulsive body wave. There was considerable variation in the spatial extent and degree of turning visible in the trajectories both within and across strains (figure 1a; electronic supplementary material, S2). As previously seen in C. elegans [38], the measured mean-squared displacement, ⟨[Δx(τ)]2⟩≡⟨|x(t+τ)−x(t)|2⟩,2.1 revealed a transition from ballistic to diffusive motion within a 100 s time scale (figure 1b; electronic supplementary material, S3). Over short times, the worm’s path was relatively straight, with the mean-squared displacement scaling quadratically with the time lag τ and speed s as 〈s2〉τ2 (i.e. a log–log slope of 2). Over longer times, the slope decreased with τ reflecting the randomization of orientation characteristic of diffusion, and an effective diffusivity Deff was extracted by fits to 〈[Δx(τ)]2〉 = 4Deffτ (see electronic supplementary material). On the time scales at which the worms start encountering the walls of the observation arena, the slope of the mean-squared displacement decreased yet further, which has been shown to be a property of confined random walks [39]. Nonetheless, we have confirmed that the decay of the velocity autocorrelation function is not significantly affected by the confinement, and is consistent with a ballistic to diffusive transition (electronic supplementary material, figure S1). This analysis revealed that the visible differences in the spatial extent of these 30 min trajectories stem from variation by nearly an order of magnitude in speed and two orders of magnitude in diffusivity (figure 1c; electronic supplementary material, tables S1 and S2).The broad range of observed speeds and diffusivities suggest that these diverse nematodes have evolved a variety of strategies for spatial exploration. To gain further insights into the manner in which such contrasting behaviours are implemented by each strain, we sought to extract a minimal model of the nematodes’ random walk by further decomposing the trajectory statistics of all nine measured strains. In this and the following three subsections, we illustrate our analysis and model development with data from three contrasting strains: CB4856 and PS312, which demonstrated two of the most extreme phenotypes, and the canonical laboratory strain N2 (see electronic supplementary material for equivalent data for all strains). The translational motion of the worm can be described by the time-varying centroid velocity v(t) which can in turn be decomposed into speed s(t) and direction of motion (hereafter referred to as its ‘bearing’) ϕ(t): v(t)=dx(t)dt=s(t) [cosϕ(t),sinϕ(t)].2.2 To account for head–tail asymmetry in the worm’s anatomy, we additionally define the body orientation (ψ(t); hereafter referred to simply as ‘orientation’) by the angle of the vector connecting the worm’s centroid to the head (figure 2a). The centroid bearing is related to this orientation of the worm by ϕ(t)=ψ(t)+Δψ(t),2.3 where the difference Δψ(t) is a measure of the alignment of the direction of movement with the worm’s body orientation (hereafter referred to simply as ‘alignment’). We found for all strains that the distribution of Δψ(t) was bimodal with peaks at 0° and 180° (figure 2c; electronic supplementary material, S7A). These match the forward and reverse states of motion described in C. elegans [18,19].Figure 2. The random walk of nematodes is composed of speed, turning and reversal dynamics. (a) We describe the motility of the worm by the time-varying quantities s(t) (speed; black), ψ(t) (orientation; red) and Δψ(t) (alignment; green) which measures the difference between the alignment of the velocity ϕ(t) (blue) and ψ(t). (b) One-minute examples of speed, orientation and velocity alignment time series for individuals from three exemplar strains. (c) The probability distribution of Δψ(t) reveals bimodality corresponding to forward and reverse motion. Shaded regions indicate a 95% confidence interval. (Online version in colour.)
Each of the three components of the worm’s motility (speed, orientation and alignment) varied considerably over time and in qualitatively different ways between strains (figure 2b). For example, the three strains shown in figure 2b differed not only in their average speed but also in the amplitude and time scale of fluctuations about the average speed. Similarly, the statistics of orientation fluctuations about the drifting mean also differed visibly between strains. Finally, transitions between forward and reverse runs were far more frequent in PS312 when compared with N2 and CB4856. Given the apparently random manner in which these motility components varied over time, we proceeded to analyse the dynamics of each of these three components as a stochastic process. Speed control has not been extensively studied in C. elegans, but it is known that worms move with a characteristic speed that is influenced by stimuli [26]. When intervals corresponding to transitions between forward and reverse runs were excluded from the time series, we found that the autocorrelation in speed fluctuations decayed exponentially over a few seconds (figure 3a; electronic supplementary material, S5A), a time scale similar to the period of the propulsive body wave. These dynamics are naturally captured by an Ornstein–Uhlenbeck process [40], which describes random fluctuations arising from white noise (increments of a diffusive Wiener process, dWt [40]) with magnitude 2Ds that relax with time scale τs back to an average value, μs = 〈s〉: ds(t)=τs−1[μs−s(t)] dt+2DsdWt.2.4 Numerical integration of this equation closely reproduced the observed speed distributions during runs (electronic supplementary material, figure S5B).Figure 3. Statistical characterization of the motility dynamics. (a) The autocorrelation of the speed indicated that fluctuations decayed exponentially over a few seconds. (a, inset) Speed distributions for three exemplar strains. (b) The mean-squared angular displacement (MSAD) increased quadratically. (b, inset) The orientation autocorrelation function did not decay exponentially, with some worms demonstrating significant undershoots below zero. (c) The velocity alignment autocorrelation decayed exponentially over tens of seconds to a positive constant. In each plot, the ensemble averages for all individuals from the strains are shown with solid lines and trends are shown with dashed lines. Shaded regions indicate a 95% confidence interval. (Online version in colour.)
The orientation ψ(t) captures turning dynamics that are independent of abrupt changes in bearing ϕ(t) due to reversals. To change orientation, C. elegans executes a combination of large, ventrally biased [41] sharp turns [18,24] and gradual ‘weathervaning’ [22], both of which contribute to randomization of orientation over time. This random walk in orientation was not purely diffusive: the orientation correlation Cψ(τ)=⟨cos[ψ(t+τ)−ψ(t)]⟩ does not decay exponentially (figure 3b inset; electronic supplementary material, S6B), and the mean-squared angular displacement, MSAD(τ) = 〈[ψ(t + τ) − ψ(t)]2〉, increases nonlinearly with time (figure 3b; electronic supplementary material, S6A). We found that this nonlinear MSAD of ψ(t) could be well fitted by a quadratic function of the time delay τ: MSAD(τ)=kψrms2τ2+2Dψτ, corresponding to a diffusion-and-drift model with root-mean-square (rms) drift magnitude kψrms and angular diffusion coefficient Dψ (see electronic supplementary material for derivation). A non-zero drift magnitude kψrms≠0 indicates that in addition to purely random (diffusive) changes in orientation, there is an underlying bias (i.e. directional persistence) in the worms’ turning over 100 s windows, consistent with previous studies in larger arenas [23]. These observations lead to a simple model for the orientation dynamics that combines drift (approximated as a deterministic linear process over a 100 s window) with stochastic diffusion: dψ(t)=kψ dt+2Dψ dWt,2.5 where we set the drift magnitude kψ=kψrms and dWt represents increments of a Wiener process [40].We note that while this model described well the orientation dynamics within 100 s windows, over longer time scales additional dynamics may be relevant. The magnitude of kψ in our data (approx. 1° s−1) was similar to that of weathervaning excursions reported for C. elegans navigating in salt gradients [22]. The observation that motion during runs switched abruptly between forward and reverse states (with Δψ ≈ {0°, 180°}, respectively; figure 2b,c; electronic supplementary material, S7A) suggested that reversals could be described as a discrete stochastic process. The manner in which reversals contribute to randomization of bearing over a time lag τ is captured by the autocorrelation function of Δψ(t), CΔψ(τ) ≡ 〈cos (Δψ(t + τ) − Δψ(t))〉. We found that CΔψ(τ) decayed nearly exponentially to a non-zero baseline (figure 3c; electronic supplementary material, figure S7C). This is the predicted behaviour for the autocorrelation function of the simplest of two-state processes (a ‘random telegraph process’): P(Tfwd>t)=exp(−tτfwd)2.6 andP(Trev>t)=exp(−tτrev),2.7 in which the distributions of forward and reverse run intervals (Tfwd and Trev) are completely determined by a single time constant (τfwd and τrev, respectively). The random telegraph process yields an autocorrelation function that decays exponentially as CΔψ(τ)=CΔψ(∞)+(1−CΔψ(∞))e−τ/τRT to a minimum value CΔψ(∞) ≡ ((τfwd − τrev)/(τrev + τfwd))2 with a time scale τRT≡(τfwd−1+τrev−1)−1 [42]. Results obtained from fitting the autocorrelation function are consistent with those obtained from the distribution of time intervals between detected switching events (electronic supplementary material, figure S7). In principle, the forward and reverse states could be characterized by differences in motility parameters of our model other than these transition times, as forward and reverse motion are driven by distinct command interneurons in C. elegans [43,44]. However, we found that run speeds were nearly identical between forward and reverse runs (electronic supplementary material, figure S8). While we expect that this symmetry will be broken under some specific conditions, such as the escape response [45], the strong speed correlation between the two states motivates the assumption, adopted in our model, that reversals change only the bearing (by 180°) and the propensity to reverse direction, represented in our model by the time constants τfwd and τrev.Given that the dynamics of the worm’s speed, turning and reversals could be described as simple stochastic processes, we asked whether combining them as independent components in a model of the worms’ random walk could sufficiently describe the observed motility statistics (figure 4a). We simulated trajectories of worms by numerically integrating equations (2.4)–(2.7) for the speed, orientation and reversal dynamics, respectively, which yields the worm’s velocity dynamics through equations (2.2) and (2.3), with Δψ(t) equal to 0° during forward runs and 180° during reverse runs. Simulations of this model using parameters fitted to individual worms produced trajectories that qualitatively resembled real trajectories and varied considerably in their spatial extent (figure 4b). Figure 4. A model consisting of independent speed (Ornstein–Uhlenbeck process), turning (drift and diffusion) and reversal dynamics (random telegraph process) quantitatively captures nematode motility. (a) Summary of the model. (b) Simulated trajectories for the three exemplar strains. (c) Statistical comparison of the data (black) and simulations (red), ensemble averaged across individuals for each strain. (c, top) The mean-squared displacement (MSD) was closely reproduced in all cases. (c, middle) The normalized velocity autocorrelation, Cv(τ)/Cv(0) (VACF), was less well captured. (c, bottom) The relatively small errors in the simulated VACF (red) can be traced to the assumption of independence in the dynamics of the speed, orientation, and velocity alignment (blue). Shaded regions indicate a 95% confidence interval. (Online version in colour.)
Next, we quantitatively assessed the performance of the model in reproducing the statistics of the observed trajectories over the time scale of 100 s, within which all strains completed the transition from ballistic to diffusive motion (figure 4c). We found that the model based on independent speed, turning and reversal dynamics closely reproduced not only the diffusivity of each strain but also the time evolution of the mean-squared displacement (〈[Δx(τ)]2〉) across the ballistic-to-diffusive transition (figure 4c, top). A closer inspection of the dynamics across this transition is possible by examining the velocity autocorrelation function (Cv(τ)), the time integral of which determines the slope of the mean-squared displacement through (d/dt)⟨[Δx(τ)]2⟩=2∫0τdτ′Cv(τ′), a variant of the Green–Kubo relation [46,47]. The transition from ballistic to diffusive motion is characterized by the manner in which the normalized velocity autocorrelation Cv(τ)/Cv(0) decays over the time lag τ from unity (at τ = 0) to zero (as τ → ∞). We found that Cv(τ) varied considerably across strains, not only in the overall ballistic-to-diffusive transition time, but also in the more detailed dynamics of the autocorrelation decay over time (figure 4c, middle). Salient features, such as the transition time, of the measured velocity autocorrelation functions Cv,obs were reproduced closely by the simulated velocity autocorrelation functions Cv,model, but there were also subtle deviations in the detailed dynamics for a number of strains. Given our model’s simplifying assumption that dynamics for s(t), ψ(t) and Δψ(t) are independent stochastic processes, we asked whether the remaining discrepancies between the simulated and measured velocity autocorrelation dynamics could be explained by violations of this assumption of independence. As a model-free assessment of the degree of non-independence, we first calculated the predicted velocity autocorrelation for the case that the dynamics of all three components are independent, Cv,indep(τ) = Cs(τ)Cψ(τ)CΔψ(τ), where Cs(τ), Cψ(τ) and CΔψ(τ) are the autocorrelation functions of the measured data for each of the components (see electronic supplementary material for derivation). We then compared the differences Cv,obs − Cv,indep (blue curve in figure 4c, bottom) and Cv,obs − Cv,model (red curve in figure 4c, bottom). Indeed, there were subtle differences both on shorter (approx. 1 s) and longer time scales (approx. 10 s). However, these errors for the simulated model were very similar to, or less than, those for the model-free prediction from the data under the assumption of independence (i.e. Cv,obs−Cv,model≲Cv,obs−Cv,indep). These results demonstrate that modelling s(t), ψ(t) and Δψ(t) as independent stochastic processes provides a very good approximation to trajectory statistics across the ballistic-to-diffusive transition. The relatively subtle differences between the data and model arise primarily in instances where this assumption of independence between the three motility components breaks down. Consistent with these conclusions, inspection of cross-correlation functions computed from the data revealed that correlations between s(t), ψ(t) and Δψ(t) are largely absent, with only weak correlations between speed (s) and reversals (Δψ) in a subset of strains (electronic supplementary material, figure S9). The results presented in the previous sections demonstrate that a random-walk model with seven parameters describing independent speed, turning and reversal dynamics provides a good approximation of the worms’ motile behaviour over the approximately 100 s time scale spanning the ballistic-to-diffusive transition. The model parameters thus define a seven-dimensional space of motility phenotypes in which behavioural variation across strains and species can be examined. If components of behaviour were physiologically regulated or evolutionarily selected for in a coordinated manner, we would expect to find correlated patterns in the variation of these traits. We fitted our model to the trajectory statistics of each individual worm and built a phenotype matrix of 106 worms×7 behavioural parameters (summarized in electronic supplementary material, tables S2–S4). The correlation matrix for these seven parameters, figure 5a, demonstrates that the strongest correlation were the forward and reverse state lifetimes (τfwd, τrev), followed by those describing speed and forward state life times (μs, τfwd). More broadly, there were extensive correlations among the model parameters, not only within the parameters of each motility component (speed, orientation and reversals) but also between those of different components. Figure 5. Motility parameters co-vary along an axis controlling exploratory behaviour. (a) Correlation matrix of the behavioural parameters across the whole dataset. (b, left) Fraction of variance captured by each mode and the amount expected for an uncorrelated dataset (red line). (b, right) The components of the top eigenvector. (c) The effective diffusivity (top) and a 30 min trajectory (bottom, colours match points on graph) from simulations in which the projection onto the top eigenvector was varied; the principal mode can be used as an effective phenotype from a more dwelling to a more roaming behaviour. The projections and effective diffusivity of the measured trajectories are shown as black points, and the average of each strain is shown as a square. Mode projections are obtained by the dot product of the seven-dimensional vector of parameters obtained for each trajectory, and the principal eigenmode. (Online version in colour.)
We looked for dominant patterns in the correlations using principal component analysis [48] (figure 5b), uncovering a single dominant mode of correlated variation (figure 5b, left). Dominant modes are obtained by diagonalizing the correlation matrix. The eigenvalues of the correlation matrix capture the amount of variance of the variables that can be accounted by linear correlation, and therefore the magnitude of these eigenvalues organize the eigenvectors in terms of explained variance. For more details of the principal component analysis, see electronic supplementary material. This principal mode (mode 1), capturing nearly 40% of the total variation, described significant correlations among all the parameters except for Ds and Dψ (figure 5b, right; electronic supplementary material, table S5). We did not attempt to interpret higher modes since, individually, they either did not significantly exceed the captured variance under a randomization test (mode 3 and higher; see electronic supplementary material, and figure 5b, left) or were found upon closer inspection to be dominated by parameter correlations arising from fitting uncertainties (mode 2). We used numerical simulations to determine the effects on motile behaviour of varying parameters along the principal mode. The measured trajectory phenotypes projected onto this mode fall in the range {− 4, 2}: the mode projections are not evenly distributed around the average phenotype at the origin. We performed simulations for parameter sets evenly sampled along this range, which largely reproduced the observed variation in the measured diffusivities Deff as a function of the projection along the first mode. The agreement was particularly good at higher values (>−1) of the mode projection, but at lower values we noted a tendency for the Deff from simulations to exceed that of the data. The latter discrepancy can be explained by elements of behaviour not captured by our model (see Discussion). Nevertheless, as illustrated by simulated trajectories (figure 5c, bottom), trajectories became more expansive as the mode projection increased, as did Deff by nearly two orders of magnitude over the tested range. This suggested that the principal mode indicates exploratory propensity (figure 5c), and we confirmed that it is indeed more strongly associated with changes in Deff than expected for randomly generated parameter sets (electronic supplementary material, figure S10). Interestingly, this mode of variation we found across individual phenotypes is reminiscent of ‘roaming’ and ‘dwelling’ behavioural variability that has been shown within individuals across time, in C. elegans [28,32] as well as other organisms [49,50]. The principal behavioural mode discussed in the preceding section was identified by analysing variation across all individual worms measured in this study, coming from diverse strains and species that differ in their average behaviour (see electronic supplementary material, tables S2–S4). How does the variability among individuals of a given strain compare to differences between the average phenotypes of strains/species? On the one hand, each strain might be highly ‘specialized’, with relatively small variation within strains as compared to that across strains. On the other hand, strains might implement ‘diversified’ strategies in which genetically identical worms vary strongly in their behaviour. To address these two possibilities, we analysed the distribution of individual phenotypes within each strain, as well as that of the set of averaged species phenotypes. For each measured individual, we computed the projection of its motility parameter set along the principal behavioural mode and estimated strain-specific distributions of this reduced phenotype (figure 6; electronic supplementary material, table S6). In principle, any detail in the shape of these distributions could be relevant for evolutionary fitness, but here we focused our analysis on the mean and standard deviation, given the moderate sampling density (less than or equal to 20 individuals per strain). Furthermore, we computed the principal-mode projection of the average phenotype of each species to define an interspecies phenotype distribution (figure 6). Figure 6. Variation of model parameters reveal specialized and diversified behavioural strategies across strains. (a) Distribution of the average phenotype for each species (interspecies variation, blue) or individuals within a strain (red). Note that the variation of individual phenotypes in some strains (e.g. QX1211, PS1159) is comparable in magnitude to that of interspecies variation. Observations are indicated with coloured ticks. (b) Comparison of the width of the phenotype distributions (coloured bars), quantified as the bootstrapped standard deviation of the data points in (a), with the uncertainty in the determination of the phenotype (black bars), quantified as the standard deviation of individual phenotype determinations over bootstrapped 100 s time windows. Error bars correspond to 95% confidence intervals across bootstrap samples. (Online version in colour.)
Strains varied considerably in both the position and breadth of their phenotypic distributions along the principal behavioural mode. Remarkably, variation across individuals within each strain was comparable in magnitude to that for the set of average phenotypes across species (figure 6). Some strains were specialized towards roaming or dwelling behaviour, such as CB4856 and PS312, respectively, with a strong bias in their behaviour and comparatively low individual variability. Others, such as QX1211 and PS1159, appeared more diversified with an intermediate average phenotype and higher individual variability. These considerable differences in phenotype distributions across strains reveal the evolutionary flexibility of population-level heterogeneity in nematodes, and suggest a possible bet-hedging mechanism for achieving optimal fitness in variable environments [51,52]. In assessing such variability of phenotypes, it is essential to ask how uncertainty in the determined parameters (obtained from model fits) contributes to the observed variability in phenotypes. We, therefore, computed the contribution of uncertainties in the individual phenotype determination by bootstrap resampling of the 100 s windows of each individual’s recorded trajectory (see electronic supplementary material). The uncertainties thus computed reflect contributions from both parameter uncertainties in curve fitting of data, as well as temporal variability in an individual’s parameters over time scales longer than the window size (100 s). With the exception of two strains (sjh2 and CB4856), this measure of uncertainty accounted for less than half of the individual variation within each strain (figure 6b). These findings support the view that the phenotypic variation estimated in the current analysis largely represented stable differences in individual behaviour. We have presented a comparative quantitative analysis of motile behaviour across a broad range of strains and species of the nematode phylum, ranging from the laboratory strain C. elegans N2 to Plectus sjh2 at the base of the chromadorean nematode lineage. Despite the vast evolutionary distances spanned by strains in this collection [53], we found that a behavioural model described by only seven parameters could account for much of the diversity of the worms’ translational movement across the approximately 100 s time scale spanning the ballistic-to-diffusive transition. This simple model provides a basis for future studies aiming to capture more detailed aspects of nematode behaviour, or to connect sensory modulation of behaviour to the underlying physiology. More generally, our results demonstrate how quantitative comparisons of behavioural dynamics across species can provide insights regarding the design of behavioural strategies. We focused on a high-level output of behaviour—translational and orientational trajectory dynamics—and sought to build the simplest possible quantitative model that could capture the observed behavioural statistics. We found that a model with only three independent components—(1) speed fluctuations that relax to a set point on a time scale of a few seconds, (2) orientation fluctuations with drift and (3) stochastic switching between forward and reverse states of motion—describes well, overall, the trajectory statistics of all tested nematode species across the ballistic-to-diffusive transition (figure 4). Notably, we have not included explicit representations of some reorientation mechanisms that have been studied in the past, such as the deep turns (omega- and delta-turns) [18,24], or the combination of such turns with reversals (pirouettes) [25]. In our data, we find that the timings of the initiation and termination of reversals, which would both count as runs in the pirouette description, follow exponential distributions with similar time constants as previously reported for the pirouette run distribution. While omega and delta turns must indeed be mechanistically distinct from gradual turns, we have chosen here not to explicitly model their occurrence since orientation changes in our trajectory data were adequately described by a continuous diffusion–drift process (figures 3b; electronic supplementary material, S6A). It is possible, however, that explicit representations of pirouettes and/or omega turns would be important in other experimental scenarios, e.g. those that include navigation in the presence of gradient stimuli. In our model, ‘roaming’ and ‘dwelling’ were not assigned discrete behavioural states (as was done e.g. in [28,31,32]), but instead emerged as a continuous pattern of variation among motility parameters describing the worm’s random walk. However, robust extraction of motility parameters required pre-filtering of trajectory data that likely biased them towards more ‘roaming’ phenotypes (see electronic supplementary material), which we believe account for the noted tendency of model simulations to overestimate Deff that was more pronounced for trajectories at the ‘dwelling’ end of the spectrum (figure 5c). In its current form, our simple model does not account for possible correlations between the dynamics of the three motility components (speed, orientation and reversals). Indeed, at least weak correlations do exist between the components (electronic supplementary material, figure S9). Comparisons of simulated versus measured trajectories demonstrated that the effects of such correlations on the motility statistics are small but detectable (figure 4c). The differences were most significant for the velocity-autocorrelation dynamics on an approximately 10 s time scale, and were similar to those for model-free predictions obtained by combining component-wise correlation functions under the assumption of independence. Discrepancies on this intermediate time scale occurred most often in fast-moving strains that frequently approached the repellent boundary. Therefore, we suspect that the discrepancy arises from a stereotyped sequence, such as the escape response [45], that introduces temporal correlations between speed changes, turning and reversals. While here we have focused on the transition to diffusive motion, some recent experiments suggest that C. elegans might engage in superdiffusive behaviour on time scales longer than 100 s [23,33]. Superdiffusive behaviour could arise from non-stationarities in motile behaviour, such as the roaming/dwelling transitions on time scales of several minutes [32]. Another mechanism for superdiffusion is directed motility [23] in response to external stimuli such as chemical or thermal gradients. In such environments, nematodes are known to use at least two distinct mechanisms for navigation [22,25] and the model here could be extended by studying the dependence of motility parameters on environmental statistics. Information about the body shape can be incorporated to build a more complete behavioural model that also includes dynamics hidden by centroid behaviour [38,54]. Indeed, work by Brown et al. showed that a rich repertoire of dynamics can be identified as temporal ‘motifs’ in the postural time series of C. elegans and used to classify mutants with high discriminatory power [55]. We have found that all of the species tested here can also be described with a common set of postural modes (not shown), suggesting future directions on the evolutionary space of postural dynamics. While we found that a single behavioural model could be used to characterize nematode motility across the chromadorean lineage, the parameters of the model varied extensively from strain to strain. Quantitatively, about 37% of the variation corresponded to a correlated change in the parameters underlying the timing of forward and reverse runs and the dynamics controlling speed and turning (figure 5b). We find that this principal mode of variation is associated with strong changes in exploratory propensity, as characterized by Deff (figure 5c). This pattern of parameter variation drove a change from low-speed short runs to high-speed long runs, resembling the canonical descriptions of roaming and dwelling in C. elegans [32]. Roaming and dwelling are thought to represent fundamental foraging strategies reflecting the trade-off between global exploration and local exploitation of environmental resources [56]. Recent work has suggested that such archetypal strategies can be recovered by quantitatively analysing the geometry of phenotypic distributions in parameter space [31,34]. The motility phenotypes we found in the present study were biased along one principal dimension, with the extremes corresponding to roaming and dwelling behaviours. This observation compels us to suggest that an exploration–exploitation trade-off is the primary driver of phenotypic diversification in the motility of chromadorean nematodes in the absence of stimuli. Interestingly, a recent study on the motility of a very different class of organisms (ciliates) yielded a similar conclusion [50]: across two species and different environments, the diversity of motility phenotypes was found to be distributed principally along an axis corresponding to roaming and dwelling phenotypes. The emergence of roaming/dwelling as the principal mode of variation in such disparate species underscores the idea that the exploration–exploitation trade-off is a fundamental constraint on biological motility strategies. A surprising finding in our study was that, for a majority of strains, the extent of behavioural variability across individuals within a strain was comparable to that for variation of phenotypes across species (figure 6). In slowly changing environments, the most evolutionarily successful species are those that consistently perform well in that environment. This can be achieved by evolving a specialized, high fitness phenotype that varies little among individuals (such as with PS312 and sjh2). However, increased phenotypic variability among individuals can improve fitness in more variable environments if some individuals perform much better in each condition—a so-called bet-hedging strategy [51,52]. The large variability we observed among individual phenotypes within each strain might reflect such a bet-hedging strategy in nematode exploratory behaviour. The observation that the variation among genetically identical individuals can be comparable to that between disparate species raises the intriguing possibility that there exist conserved molecular and/or physiological pathways driving diversification of spatial exploration strategies. Analogous variation in exploratory behaviour was also detected in an analysis of non-stationarity in the behaviour of wild-type and mutant C. elegans under various nutritional conditions [31]. Physiologically, protein kinase G (PKG) signalling and DAF-7 (TGF-β) signalling from the ASI neuron are thought to be major mechanisms controlling roaming and dwelling in C. elegans [28,31]. PKG signalling is also involved in controlling foraging in Drosophila and other insects as well as many aspects of mammalian behaviour [57,58]. Flavell et al. also elucidated a neuromodulatory pathway involving serotonin and the neuropeptide pigment dispersing factor controlling the initiation and duration of roaming and dwelling states [32]. Perturbations to the molecular parameters of such pathways underlying global behavioural changes might provide a mechanism for the observed correlated variations at the individual, intra- and inter-species levels. The identification of such conserved pathways affecting many phenotypic parameters is of fundamental interest also from an evolutionary perspective, as they have been proposed to bias the outcome of random mutations towards favourable evolutionary outcomes [59,60]. Our simple model provides a basis for future investigations to uncover conserved mechanisms that generate behavioural variability, by defining a succinct parametrization of behaviour that can be combined with genetic and physiological methods. Trajectory data and scripts used to process, analyse and build the figures can be obtained from https://figshare.com/articles/Helms2019/7770953. We declare we have no competing interests. This work was supported by NWO/FOM and the Paul G. Allen Family Foundation. We thank Massimo Vergassola, Vasily Zaburdaev, Alon Zaslaver and Jeroen van Zon for helpful suggestions and critical reading of the manuscript, Will Ryu, Aravi Samuel and Andre Brown for inspiration and encouragement, and members of the Shimizu laboratory for discussions. Casper Quist and Hans Helder of Wageningen University provided wild nematodes isolated from soil and useful information regarding the ecology of nematodes. Footnotes†These authors contributed equally to this work. ‡Present address: Department of Applied Mathematics, University of Waterloo, Waterloo, Canada. Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4608476. References
Page 26Unlike animals, individual plants, in general, lack mobility, usually rooting at certain sites in the course of their entire lives. Therefore, their activities that involve dynamic mobilities, such as dispersal of seeds, fruits or pollen, often rely on foreign vectors/agents including wind and animals. Due to the action of natural selection, the interactions between plants and their vectors are evolutionally optimized. The interactions between plants and their vectors also play an essential role in agriculture for our food production. Without insect pollination, many flowering plants would fail to reproduce sexually or decrease the quality of their seeds and fruits, and we would lose significant amount of food supply and other plant products [1]. Contrary to an increasing human population, declines in pollinators, which can lead to potential failure of food production, have been reported from every continent except Antarctica [2]. Considering the process of insect pollination, during the journey of pollen from a male part of a plant toward a receptive female part of another plant, pollen undergoes numerous cycles of detachment and attachment events at the interfaces of floral parts and insect body parts. Each step requires specific properties of both surfaces in contact. For example, the pollen grains in Asteraceae, first presented on the surfaces of styles, require resistance against sheer forces caused by the wind and some regulation of the amount of grains taken by individual pollinators [3]. The grains should be securely fixated on pollinators that are flying in the wind and causing vibrations. Finally, the grains fixated on pollinators should be easily removed and captured by the stigmatic surfaces of another individual plant. Therefore, the insights into adhesive properties of pollen on corresponding surfaces of plants and their pollinators are key to better understanding insect pollination. Additionally, finding out the mechanisms that achieve the desired manipulation of microparticles through proper interactions at interfaces can be potentially further applied to various fields, for instance, coating [4], printing [5], drug delivery [6] and respiratory medicine [7]. Pollen adhesion on stigmatic surfaces was first measured with liquid assay with magnetic stirrer [8], with centrifugation [9] or chemical treatments [10]. The above-mentioned methods, counting released grains from stigmas under aqueous environment, are useful to investigate the pollen–stigma adhesion arising from physiological phenomena, such as pollen tube penetration into stigmatic tissues. However, immersing stigmas into liquid makes it difficult to correctly measure the initial adhesion events that are dominated by physical interactions. Since Zinkl & Preuss [10] conceived an alternative method capable of measuring pollen–stigma adhesion in a dry environment [10,11], there have been almost no studies in this topic except the very recent one from Lin et al. [12]. Despite the vast diversity in morphologies and secreted liquid of stigmatic surfaces and pollen [13], to the best of our knowledge, the quantitative adhesion measurements of dry pollen have been done only on stigmatic surfaces of a few plant species: Arabidopsis thaliana [10,11] and Helianthus annuus [12]. Other plant species and other floral parts that play crucial roles in pollination, for instance, anthers and styles, have been previously not studied. We have recently reported adhesive properties of fresh and aged pollen of Hypochaeris radicata on artificial substrates using atomic force microscopy (AFM) [14]. The study has suggested that capillary bridges formed at the interface between pollen grains and substrates play crucial roles in pollen adhesion, whereas the pollenkitt, traditionally regarded as a kind of pollen adhesive liquid, not necessarily enhances, but rather decreases the pollen adhesion due to weakened water capillarity in some conditions. In all species in Asteraceae family, pollination is initiated with opening anthers inwards, releasing pollen grains into anther tubes. As the style elongates within the anther tube, it pushes or brushes the pollen out of the anther tube, exposing the pollen at the hairy surfaces of the style [15]. The pollen grains presented on the stylar surfaces require detachment to be transported by pollinators until they finally land on stigmatic surfaces. The initial adhesion on the stigmatic surfaces is dominated by physical attractions, such as van der Waals forces and capillarity [9,10,13], but later a cascade of physiological events occurs, leading to fertilization [16]. In this study, we focused on the pollen adhesion at the stage of physical attraction, aiming to provide the first comparative insights of pollen adhesion on the most important floral parts in pollination: the style as the departure site and the stigma as the destination site. Flowering stems of Hypochaeris raticata were collected and put into water to permit the youngest ray flowers to push the styles through the anther tubes and expose the fresh pollen grains on the stylar surfaces (figure 1). The newly opened ray flowers were picked up from capitula (flower heads), and then an individual fresh grains were quickly collected using human eyelashes and finally glued upon the tip of AFM tipless cantilevers (Nanosensors, Neuchatel, Switzerland) using a slight amount of epoxy glue (figure 2c). The epoxy glue was cured for 15 min before the onset of AFM adhesion measurements for the pollen in the fresh state. After the cycles of adhesion measurements, the pollen–cantilever tips were stored under controlled laboratory conditions (temperature: 20–25°C, RH: 40–60%). After several days, the adhesion of the pollen grains was tested again for the comparison between pollen in fresh and aged state. Figure 1. Floral structures of Hypochoeris radicata at three different scales: a whole flower head composed of a number of florets (left), a matured floret (middle) presenting pollen grains on its style, and bifurcated ends of a style (right) that possesses stigma at inner surfaces. (Online version in colour.)
Figure 2. Adhesion force measurements. (a) Experimental set-up for AFM force measurements on a floral sample. (b) Retraction parts of typical force–distance curves in fresh Hypochoeris radicata pollen on the stylar and stigmatic surface. The arrows define pull-off forces. (c) SEM image of the pollen grain fixed on an AFM cantilever tip, after the AFM experiment. (Online version in colour.)
Stocks of Hypochaeris raticata were transplanted to containers to let their buds open indoors for reducing the risk of contamination or damage of the plant surfaces. The florets that opened a day before were carefully collected from the capitula and cleaned with a dust blower to remove pollen grains from the surfaces. A cleaned floret was fixed on a glass slide with a double-sided tape with either its stigma or upper end of style facing up for AFM force measurements. Together with the floret sample, water-soaked paper towel was wrapped in scotch tape, to avoid dehydration during further experimentation (figure 2a). The pollen adhesion was tested on either the stigmatic or stylar surfaces. We used AFM with the pollen–cantilever tips (see above) to measure pollen adhesion on the native stylar and stigmatic surfaces of the plant. The adhesion force was defined as the lowest measured pull-off force in the retraction part of the force–distance curve (figure 2b). We compared adhesion forces measured at three different times of the contact duration: 0, 30 and 180 s, using three cantilevers with three individual pollen grains. We performed 95 and 81 adhesion measurements using fresh pollen on samples of five stigmas and five styles, respectively. While we performed 71 and 82 adhesion measurements using aged pollen on samples of three stigmas and four styles, respectively. During the measurements, the preload force was 1 nN, and the piezo speed was 7.5 µm s−1. The measurements were performed on both stigmas and styles with three different contact times in a randomized order. After the measurements were finished, the pollen–cantilever tips were sputter coated with 10 nm Au–Pd and examined in a tabletop TM3000 scanning electron microscope (Hitachi Ltd, Tokyo, Japan). Cryo-scanning electron microscopy (SEM) was used to visualize plant surfaces in the native state. Newly opened florets were fixated on SEM stubs before being frozen in liquid nitrogen, and then they were transferred to a sputtering chamber, where samples were kept frozen at −140°C. Then they were sputter coated with 10 nm Au–Pd in the frozen state and observed in a Hitachi S 4800 scanning electron microscope (Hitachi Ltd, Tokyo, Japan) at −120°C. In figure 2b, typical force–distance curves taken on both a stigma and a style with a contact time of 180 s are demonstrated. The pollen adhesion on the stigma featured longer attraction regime after the contact breakage than on the stylar surface. Figure 3 shows the measured adhesion forces of both fresh and aged pollen on the stylar and stigmatic surfaces with different contact times. The adhesion force of fresh pollen on the stigmatic surfaces (figure 3a), compared with the instantaneous contact time (0 s), exhibited a dramatic increase over the contact time of 180 s by a factor of 11.9, while the adhesion force of fresh pollen on the stylar surfaces yielded an increase only by a factor of 2.7. For the adhesion force of fresh pollen on the stigmatic surfaces, the differences between these contact times are statistically significant in all the combinations (Dunn Kruskal–Wallis multiple comparison, 0 versus 30 s: Z = 2.52, p-value < 0.05; 180 versus 30 s: Z = −3.26, p-value < 0.005; 0 versus 180 s: Z = 5.90, p-value < 0.001), whereas a significantly different adhesion force for the fresh pollen on the stylar surfaces was only found in the comparison between 0 s and 30 s and between 0 s and 180 s (Dunn Kruskal–Wallis multiple comparison, 0 versus 30 s: Z = 2.95, p-value < 0.01; 180 versus 30 s: Z = 0.01, p-value = 0.99; 0–180 s: Z = 2.93, p-value < 0.01). For the comparison between the stylar and stigmatic surfaces, a significantly different adhesion force of fresh pollen was found, when contact time was 180 s (Kruskal–Wallis test, χ2 = 14.1, p-value < 0.001). Figure 3. Pull-off adhesion forces on the stylar and stigmatic surfaces with different contact times using pollen in the fresh (a) and aged state (b). The thick vertical bars are medians and the error bars are 75th and 25th quantiles. (Online version in colour.)
in the case of aged pollen (figure 3b), the adhesion force on the stigmatic surfaces increased by a factor of 6.8 during the contact time of 180 s, reaching its median value of 760 nN, and the differences between different contact times were statistically significant (Dunn Kruskal–Wallis multiple comparison, 0 versus 30 s: Z = 2.65, p-value < 0.05; 30 versus 180 s: Z = −2.60, p-value < 0.01; 0 versus 180 s: Z = 5.05, p-value < 0.001). Compared to the fresh pollen, the adhesion force of aged pollen on the stylar surfaces features a moderate force increase during the contact time of 180 s by a factor of 2: statistically significant difference was only found between 0 s and 180 s (Dunn Kruskal–Wallis multiple comparison, 0 versus 30 s: Z = 1.93, p-value = 0.08; 30 versus 180 s: Z = −0.68, p-value = 0.50; 0 versus 180 s: Z = 2.75, p-value < 0.05). For the comparison between the stylar and stigmatic surfaces, adhesion forces of aged pollen (figure 3b) on the stigmatic surfaces were found significantly higher than those obtained on the stylar surfaces for all the contact times (Kruskal–Wallis test, contact time = 0 s: χ2 = 4.83, p-value < 0.05; contact time = 30 s: χ2 = 10.6, p-value < 0.01; contact time = 180 s: χ2 = 26.2, p-value < 0.001). Compared with the fresh state, aged pollen yielded significantly higher adhesion only on the stigmatic surfaces with the contact time of 180 s (Kruskal–Wallis test, χ2 = 4.66, p-value < 0.05). However, no significant difference of the adhesion force on the stylar surfaces between fresh and aged pollen was found. Cryo-SEM observations were performed to investigate contact geometries of pollen grains on both the stylar and stigmatic surfaces in their native state. We observed that numerous conical papillae on the stigmatic surfaces are wet (figure 4a–c). The stigmatic papillae were flexible enough and therefore clumped with each other by the capillary forces caused by the surface liquid (figure 4a,b). The surface liquid was found to rise in the space between the stigmatic papillae, and to form capillary bridges that anchor pollen grains in the embracing stigmatic papillae (figure 4c). On the stylar surfaces, however, we observed distally tilted spines protruding from parallel aligned cuticular ridges (figure 4d,e). Unlike stigmatic surfaces, we hardly observed these stylar spikes being clumped by the capillarity. However, the tips of the stylar spikes were frequently found buckled (figure 4d,e). Figure 4. Cryo-SEM images of pollen grains on the stigmatic surfaces (a–c) and on the stylar surfaces (d–f) from different perspectives: top view (a,d) and lateral view at lower magnification (b,e), and at higher magnification (c,f). CB, capillary bridges; SL, bulk surface liquid; PD, plastically deformed structures.
We assume that pollen–stigma interaction in representatives of Asteraceae starts with an initial pollen capture on stigmatic surfaces, which is mediated by physico-chemical attractions (first stage). Then the pollenkitt release onto stigmatic surfaces results in the formation of an ‘attachment foot’ beneath each pollen grain (second stage). The water passes through the attachment foot leading to the pollen hydration (third stage). Finally, pollen tube germination occurs, resulting in its penetration through the stigmatic tissue (fourth stage) [16]. While the second and later adhesion stages involve physiological interactions, and some of the phenomena are irreversible once they occur, the adhesion force that arises from first stage adhesion can be measured repeatedly using pollen fixated on the AFM cantilevers. We assume that we measured the first stage adhesion on the stigmatic surfaces, based on the following facts: (1) the onset time of second stage adhesion on other species in Asteraceae is much longer than the contact time in our measurements [16], (2) the magnitude of adhesion force did not significantly change over the consecutive force measurements on the stigmatic surfaces, indicating absence of irreversible adhesion modes, and (3) any sign of the pollen tube germination was not observed in subsequent SEM studies after the force measurements (figure 2c), which means that this kind of adhesion/interlocking does not contribute to the forces measured in our experiments. The contact geometries of pollen–floral interfaces are species-specific since pollen surface structures and floral surfaces are incredibly diverse in size, shape, surface ornamentations and presence/absence of surface liquids [13,15–18]. Among the diversity in types of stigmas, Asteraceae family is known to possess ‘semi-dry’ stigma with some combination of characteristics of both dry and wet types of stigma [16]. The ‘semi-dry’ stigmas are characterized by a moderate amount of secreted water-based liquid from stigmatic surfaces, consisting of lipid, carbohydrates and proteins. The fact that the secreted liquid wets the whole stigmatic surfaces and clumps the stigmatic papillae into bundles (figure 4) indicates that (1) the stigmatic surfaces have high affinity to the secreted liquid and (2) the conical papillae are sufficiently flexible to be deflected by the capillary force. Unlike the plants with primary pollen presentation, where pollen is directly delivered to pollinators from anthers, all the species in Asteraceae family relocate pollen from anthers to other specialized floral structures. In the relocation of pollen before its dispersal to the pollinators, the morphology of the styles plays a crucial role. Diverse morphological features of styles and mechanisms of secondary pollen presentation in Asteraceae have been studied previously [15,19]. The subfamily Cichorioideae including H. radicata is known to possess brushing mechanisms [19]. Our cryo-SEM observation revealed that the stylar surfaces of H. radicata bear microspines tilted toward the bifurcated ends of the styles. When the styles elongate through the anther tubes, the tilted geometry of stylar spines could function to brush and drag pollen grains from the tubes to achieve pollen relocation and presentation on the stylar surfaces. The stylar spines presumably possess sufficient mechanical stiffness to overcome the frictional shear stress that the spines would undergo during the style elongation. Despite the surface liquid presence, which is likely to be the residue of the pollenkitt, we did not observe clumped stylar spines caused by the capillary force, such as that found in the stigmatic papillae. However, we frequently found plastically deformed tip ends of the stylar spines. The deformations are likely caused by the style elongation. The presence of plastic deformation indicates a poor flexibility of the stylar spines. Given that the effects of physiological interactions between the stigma and pollen grains are negligible in such short contact times, it is necessary to elucidate potential mechanisms behind pollen adhesion from the physico-chemical perspective. The measured adhesion force between pollen and a target surface is given as F=Fvdw+Fe+Fw,4.1 where Fvdw is van der Waals forces, Fe is electrostatic force, and Fw is the force originating from the capillary attraction. In the present study, Fe should be neglected, since we did not observe the long-range attraction in the force–distance curves during approach, which is an exclusive characteristic of electrostatic effect [14]. Therefore, it is reasonable to assume that Fvdw and/or Fw can increase during an extended contact with the stylar and stigmatic surfaces. From the retraction part of force–distance curves, we observed the more extended attraction regime after the breakage of solid–solid contact in pollen–stigma adhesion rather than in pollen–style adhesion (figure 2b). It is an indication that the pollen interacted with larger amount of liquid on the stigmatic surfaces rather than on the stylar surfaces during our force measurements. Cryo-SEM observation distinctively characterized morphology of both the styles and stigmas and the interaction between the surface liquid and the floral structures as well. On stigmatic surfaces, the surface liquid was observed to rise through the narrowed space between clumped conical papillae. The final height h of the liquid rise by capillary action is given ash= 2γcosθρgR,4.2 where γ is the surface tension of liquid, θ is the contact angle, ρ is the density of liquid, R is the capillary radius, and g is the gravitational acceleration. The smaller the value of R or the narrower the space is in which the liquid ascends, the higher the level that the liquid reaches. Therefore, the clumping of the conical papillae achieved by their flexibility narrows the space between them, leading to an enhancement of the liquid rise (figure 5). This could explain why we observed the liquid on the topmost part of the interface between the conical papillae in the cryo-SEM (figure 4b,c). The higher the liquid rises between the conical papillae, the higher the probability for the liquid to come into contact with the deposited pollen. Once the contact is achieved, a relatively large capillary bridge can be established beneath the pollen (figure 4c). Under these circumstances, the liquid is continuously supplied up to the capillary bridge, since the capillary bridge is connected to the bulk liquid on the basal stigmatic surfaces through the liquid path between the clumped conical papillae. The more liquid is supplied to the capillary bridge, the longer the three-phase contact line (TCL) becomes. Since the capillary force acts on the TCL, the increasing length of TCL over time enhances the capillary attraction to the stigma. Additionally, the stigmatic papillae that are in contact with the pollen would flexibly orient themselves toward the pollen with the aid of capillarity (figure 6). It would shorten the separation distance between the pollen and the stigmatic surface, leading to (1) an increasing area of solid–solid contact and (2) decreasing radius of the capillary bridge beneath the pollen (r2 < r1). Both effects lead to an increasing adhesion on the stigma, since Fvdw is proportional to the contact area, whereas Fw is inversely proportional to the radius of the capillary bridge. Therefore, the capillarity of surface liquid together with the flexibility of the stigmatic papillae would enable drastic increase in pollen adhesion over time. In the case of the style, however, it would be not possible to achieve such an increase in pollen adhesion due to the lack of flexibility of stylar spines and the absence of abundant surface liquid.Figure 5. Model of spontaneous liquid climbing mechanism of the stigma. The clumping of the flexible stigmatic papillae is caused by the capillary forces acting on three-phase contact line (red arrows). It narrows the space between them, leading to the enhancement of the capillary action of the surface liquid. (Online version in colour.)
Figure 6. Model of the spontaneous gripping system of the stigma. Once the stigmatic papillae come into contact with a pollen grain, they orient themselves toward the pollen with the help of their flexibility and capillarity in the presence of the stigmatic liquid. Capillary attractions due to smaller radius of a capillary bridge (r2 < r1) and van der Waals forces due to the larger solid–solid contact areas lead to the enhancement of adhesive interactions between the stigmatic surfaces and the pollen grain. (Online version in colour.)
Our comparison between fresh and aged pollen demonstrated the higher adhesion of aged pollen on the stigmatic surfaces at longer contact times (figure 3), whereas such significant difference was not found on the stylar surfaces. How can we explain this counterintuitive result? In a previous study, we have reported that fresh pollenkitt actually weakens adhesion on high surface energy surfaces under high humidity [14]. We reasoned this result with a water capillary model stating that pollenkitt, the major constituents of which are hydrophobic lipids, leads to an increase of the contact angle of condensed water on the pollen surface, resulting in the reduced water capillary attraction. This idea can be also applied to the present study, since the secreted liquid on the stigma mainly consists of water [20]. Unlike the stigmatic surfaces, the stylar surfaces do not secrete such an abundant liquid, and therefore it can be suggested that no considerable difference in adhesion on the stylar surfaces should occur between fresh and aged pollen. In the first step of insect pollination, the pollen grains presented on the styles are required to (1) sustain shear forces arising from both wind and oscillation of the flower and (2) be transferred to the pollinator surfaces following the breakage of existing contact to the stylar surfaces. That is why the pollen adhesion on the stylar surfaces should not drastically increase or decrease over a prolonged contact time: otherwise, it would fail to satisfy either of the above requirements. In the previous section, we discussed that adhesion of both fresh and aged pollen on the stylar surfaces had a tendency to increase with prolonged contact time, but the magnitude of the adhesion increase on the stylar surfaces is much lower than on the stigmatic surfaces. This result suggests that the pollen adhesion on the styles is well regulated to meet these requirements for the first step of pollination. In the later step of pollination, the pollen attached to the pollinator surface should be captured by the stigmatic surface. During this phase, pollen should remain fixated on the stigmatic surfaces until subsequent physiological events leading to successful fertilization take place. Our quantitative adhesion measurements along with cryo-SEM observations suggest that stigma of Hypochoeris radicata possesses a spontaneous gripping system drastically increasing pollen adhesion for a short period of time. Interestingly, this gripping system functions even more effectively for the aged pollen or pollen with less pollenkitt rather than for the fresh pollen. Each attachment/detachment cycle that pollen goes through would cause some loss of the pollenkitt. Therefore, the versatile mechanism of the gripping system in the stigma might provide an additional advantage for successful pollination and fertilization. In this study, we reported on the first quantitative comparative measurements of pollen adhesion on two most important floral parts used in pollination: the style and stigma. AFM force measurements revealed distinct adhesive properties of these floral parts depending on the contact time. The pollen adhesion on the stigmatic surfaces drastically increased over the contact time, whereas the increase of pollen adhesion on the stylar surfaces was rather moderate. Based on our cryo-SEM observations, we attribute this result to a spontaneous gripping system arising from unique morphological features of flexible stigmatic papillae that can enhance both capillary attraction and van der Waals forces. From the botanical perspective, an enhanced adhesion on the stigmatic surfaces and moderate adhesion on the stylar surfaces are of great advantage for successful pollination. The findings lead to a better understanding of adhesive interactions between pollen and various floral parts in the course of pollination. In addition, the working principles extracted from the pollination system of H. radicata can be applied to the biomimetic engineering of microgripping systems. The datasets supporting this article have been uploaded as part of the electronic supplementary material. S.I. and S.N.G. conceptualized the study. S.I. conducted the research and collected the data. S.I. analysed the data and wrote the manuscript. S.N.G. edited manuscript. All authors discussed the results and gave final approval for publication. The authors declare no competing or financial interests. This study was supported by ‘DAAD Research grant—doctoral programs in Germany’ to S.I. Authors are grateful to Esther Appel for technical supports. They are also grateful to Dr Hamed Rajabi, Halvor Tramsen, Dr Emre Kizilkan, Dr Yoko Matsumura and Prof. Agnieszka Kreitschitz for valuable discussions. FootnotesElectronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.4591628. References
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