Non-local Parabolic and Hyperbolic Models for Cell Polarisation in Heterogeneous Cancer Cell Populations

Tumours consist of heterogeneous populations of cells. The sub-populations can have different features, including cell motility, proliferation and metastatic potential. The interactions between clonal sub-populations are complex, from stable coexistence to dominant behaviours. The cell–cell interactions, i.e. attraction, repulsion and alignment, processes critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this study, we develop a mathematical model describing cancer cell invasion and movement for two polarised cancer cell populations with different levels of mutation. We consider a system of non-local hyperbolic equations that incorporate cell–cell interactions in the speed and the turning behaviour of cancer cells, and take a formal parabolic limit to transform this model into a non-local parabolic model. We then investigate the possibility of aggregations to form, and perform numerical simulations for both hyperbolic and parabolic models, comparing the patterns obtained for these models.     

Modeling Group Formation and Activity Patterns in Self-Organizing Collectives of Individuals

We construct and analyze a nonlocal continuum model for group formation with application to self-organizing collectives of animals in homogeneous environments. The model consists of a hyperbolic system of conservation laws, describing individual movement as a correlated random walk. The turning rates depend on three types of social forces: attraction toward other organisms, repulsion from them, and a tendency to align with neighbors. Linear analysis is used to study the role of the social interaction forces and their ranges in group formation. We demonstrate that the model can generate a wide range of patterns, including stationary pulses, traveling pulses, traveling trains, and a new type of solution that we call zigzag pulses. Moreover, numerical simulations suggest that all three social forces are required to account for the complex patterns observed in biological systems. We then use the model to study the transitions between daily animal activities that can be described by these different patterns.     

Hyperbolic and kinetic models for self-organized biological aggregations and movement: a brief review

We briefly review hyperbolic and kinetic models for self-organized biological aggregations and traffic-like movement. We begin with the simplest models described by an advection-reaction equation in one spatial dimension. We then increase the complexity of models in steps. To this end, we begin investigating local hyperbolic systems of conservation laws with constant velocity. Next, we proceed to investigate local hyperbolic systems with density-dependent speed, systems that consider population dynamics (i.e., birth and death processes), and nonlocal hyperbolic systems. We conclude by discussing kinetic models in two spatial dimensions and their limiting hyperbolic models. This structural approach allows us to discuss the complexity of the biological problems investigated, and the necessity for deriving complex mathematical models that would explain the observed spatial and spatiotemporal group patterns.     

Weakly nonlinear analysis of a hyperbolic model for animal group formation

We consider an one-dimensional nonlocal hyperbolic model for group formation with application to self-organizing collectives of animals in homogeneous environments. Previous studies have shown that this model displays at least four complex spatial and spatiotemporal group patterns. Here, we use weakly nonlinear analysis to better understand the mechanisms involved in the formation of two of these patterns, namely stationary pulses and traveling trains. We show that both patterns arise through subcritical bifurcations from spatially homogeneous steady states. We then use these results to investigate the effect of two social interactions (attraction and alignment) on the structure of stationary and moving animal groups. While attraction makes the groups more compact, alignment has a dual effect, depending on whether the groups are stationary or moving. More precisely, increasing alignment makes the stationary groups compact, and the moving groups more elongated. Also, the results show the existence of a threshold for the total group density, above which, coordinated behaviors described by stationary and moving groups persist for a long time.      

Metastable spiking networks in the replica-mean-field limit

Characterizing metastable neural dynamics in finite-size spiking networks remains a daunting challenge. We propose to address this challenge in the recently introduced replica-mean-field (RMF) limit. In this limit, networks are made of infinitely many replicas of the finite network of interest, but with randomized interactions across replicas. Such randomization renders certain excitatory networks fully tractable at the cost of neglecting activity correlations, but with explicit dependence on the finite size of the neural constituents. However, metastable dynamics typically unfold in networks with mixed inhibition and excitation. Here, we extend the RMF computational framework to point-process-based neural network models with exponential stochastic intensities, allowing for mixed excitation and inhibition. Within this setting, we show that metastable finite-size networks admit multistable RMF limits, which are fully characterized by stationary firing rates. Technically, these stationary rates are determined as the solutions of a set of delayed differential equations under certain regularity conditions that any physical solutions shall satisfy. We solve this original problem by combining the resolvent formalism and singular-perturbation theory. Importantly, we find that these rates specify probabilistic pseudo-equilibria which accurately capture the neural variability observed in the original finite-size network. We also discuss the emergence of metastability as a stochastic bifurcation, which can be interpreted as a static phase transition in the RMF limits. In turn, we expect to leverage the static picture of RMF limits to infer purely dynamical features of metastable finite-size networks, such as the transition rates between pseudo-equilibria.     

Phase Transitions in a Logistic Metapopulation Model with Nonlocal Interactions

The presence of one or more species at some spatial locations but not others is a central matter in ecology. This phenomenon is related to ecological pattern formation. Nonlocal interactions can be considered as one of the mechanisms causing such a phenomenon. We propose a single-species, continuous time metapopulation model taking nonlocal interactions into account. Discrete probability kernels are used to model these interactions in a patchy environment. A linear stability analysis of the model shows that solutions to this equation exhibit pattern formation if the dispersal rate of the species is sufficiently small and the discrete interaction kernel satisfies certain conditions. We numerically observe that traveling and stationary wave-type patterns arise near critical dispersal rate. We use weakly nonlinear analysis to better understand the behavior of formed patterns. We show that observed patterns arise through both supercritical and subcritical bifurcations from spatially homogeneous steady state. Moreover, we observe that as the dispersal rate decreases, amplitude of the patterns increases. For discontinuous transitions to instability, we also show that there exists a threshold for the amplitude of the initial condition, above which pattern formation is observed.     

Dynamics of Social Behavior in Fruit Fly Larvae

We quantified the extent and dynamics of social interactions among fruit fly larvae over time. Both a wild-type laboratory population and a recently-caught strain of larvae spontaneously formed social foraging groups. Levels of aggregation initially increased during larval development and then declined with the wandering stage before pupation. We show that larvae aggregated more on hard than soft food, and more at sites where we had previously broken the surface of the food. Groups of larvae initiated burrowing sooner than solitary individuals, indicating that one potential benefit of larval aggregations is an improved ability to dig and burrow into the food substrate. We also show that two closely related species, D. melanogaster and D. simulans, differ in their tendency to aggregate, which may reflect different evolutionary histories. Our protocol for quantifying social behavior in larvae uncovered robust social aggregations in this simple model, which is highly amenable to neurogenetic analyses, and can serve for future research into the mechanisms and evolution of social behavior.     

A nonlocal kinetic model for predator–prey interactions

We extend the aggregation model from Fetecau (2011) by adding a field of vision to individuals and by including a second species. The two species, assumed to have a predator–prey relationship, have dynamics governed by nonlocal kinetic equations that include advection and turning. The latter is the main mechanism for aggregation and orientation, which results from interactions among individuals of the same species as well as predator–prey relationships. We illustrate numerically a diverse set of predator–prey behaviors that can be captured by this model. We show that a prey’s escape outcome depends on the social interactions between its group members, the prey’s field of vision and the sophistication of the predator’s hunting strategies.     

KIN‐BASED RECOGNITION AND SOCIAL AGGREGATION IN A CILIATE

Aggregative groups entail costs that must be overcome for the evolution of complex social interactions. Understanding the mechanisms that allow aggregations to form and restrict costs of cheating can provide a resolution to the instability of social evolution. Aggregation in Tetrahymena thermophila is associated with costs of reduced growth and benefits of improved survival through “growth factor” exchange. We investigated what mechanisms contribute to stable cooperative aggregation in the face of potential exploitation by less‐cooperative lines using experimental microcosms. We found that kin recognition modulates aggregative behavior to exclude cheaters from social interactions. Long‐distance kin recognition across patches modulates social structure by allowing recruitment of kin in aggregative lines and repulsion in asocial lines. Although previous studies have shown a clear benefit to social aggregation at low population densities, we found that social aggregation has very different effects at higher densities. Lower growth rates are a cost of aggregation, but also present potential benefits when restricted to kin aggregations: slow growth and crowd tolerance allow aggregations to form and permit longer persistence on ephemeral resources. Thus in highly dynamic metapopulations, kin recognition plays an important role in the formation and stability of social groups that increase persistence through cooperative consumptive restraint.     

Consumer Aggregations Drive Nutrient Dynamics and Ecosystem Metabolism in Nutrient-Limited Systems

Differences in animal distributions and metabolic demands can influence energy and nutrient flow in an ecosystem. Through taxa-specific nutrient consumption, storage, and remineralization, animals may influence energy and nutrient pathways in an ecosystem. Here we show these taxa-specific traits can drive biogeochemical cycles of nutrients and alter ecosystem primary production and metabolism, using riverine systems that support heterogeneous freshwater mussel aggregations. Freshwater unionid mussels occur as distinct, spatially heterogeneous, dense aggregations in rivers. They may influence rates of production and respiration because their activities are spatially concentrated within given stream reaches. Previous work indicates that mussels influence nutrient limitation patterns, algal species composition, and producer and primary consumer biomass. Here, we integrate measures of organismal rates, stoichiometry, community-scaled rates, and ecosystem rates, to determine the relative source–sink nutrient dynamics of mussel aggregations and their influence on net ecosystem processes. We studied areas with and without mussel aggregations in three nitrogen-limited rivers in southeastern Oklahoma, USA. We measured respiration and excretion rates of mussels and collected a subset of samples for tissue chemistry and for thin sectioning of the shell to determine growth rates at each site. This allowed us to assess nutrient remineralization and nutrient sequestration by mussels. These rates were scaled to the community. We also measured stream metabolism at three sites with and without mussels. We demonstrated that mussel species have distinct stoichiometric traits, vary in their respiration rates, and that mussel aggregations influence nutrient cycling and productivity. Across all mussel aggregations, we found that mussels excreted more nitrogen than they sequestered into tissue and excreted more phosphorus than they sequestered except at one site. Furthermore, gross primary productivity was significantly greater at reaches with mussels. Collectively, our results indicate that mussels have ecosystem-level impacts on nutrient availability and production in nutrient-limited rivers. Within these streams, mussels are affecting the movement of nutrients and altering nutrient spiralling.     

The Walking Behaviour of Pedestrian Social Groups and Its Impact on Crowd Dynamics

Human crowd motion is mainly driven by self-organized processes based on local interactions among pedestrians. While most studies of crowd behaviour consider only interactions among isolated individuals, it turns out that up to 70% of people in a crowd are actually moving in groups, such as friends, couples, or families walking together. These groups constitute medium-scale aggregated structures and their impact on crowd dynamics is still largely unknown. In this work, we analyze the motion of approximately 1500 pedestrian groups under natural condition, and show that social interactions among group members generate typical group walking patterns that influence crowd dynamics. At low density, group members tend to walk side by side, forming a line perpendicular to the walking direction. As the density increases, however, the linear walking formation is bent forward, turning it into a V-like pattern. These spatial patterns can be well described by a model based on social communication between group members. We show that the V-like walking pattern facilitates social interactions within the group, but reduces the flow because of its “non-aerodynamic” shape. Therefore, when crowd density increases, the group organization results from a trade-off between walking faster and facilitating social exchange. These insights demonstrate that crowd dynamics is not only determined by physical constraints induced by other pedestrians and the environment, but also significantly by communicative, social interactions among individuals.     

Developmental plasticity and variability in the formation of egg-spots,a pigmentation ornament in the cichlid Astatotilapia calliptera

Vertebrate pigmentation patterns are highly diverse, yet we have a limited understanding of how evolutionary changes to genetic, cellular, and developmental mechanisms generate variation. To address this, we examine the formation of a sexually-selected male ornament exhibiting inter- and intraspecific variation, the egg-spot pattern, consisting of circular yellow-orange markings on the male anal fins of haplochromine cichlid fishes. We focus on Astatotilapia calliptera, the ancestor-type species of the Malawi cichlid adaptive radiation of over 850 species. We identify a key role for iridophores in initializing egg-spot aggregations composed of iridophore-xanthophore associations. Despite adult sexual dimorphism, aggregations initially form in both males and females, with development only diverging between the sexes at later stages. Unexpectedly, we found that the timing of egg-spot initialization is plastic. The earlier individuals are socially isolated, the earlier the aggregations form, with iridophores being the cell type that responds to changes to the social environment. Furthermore, we observe apparent competitive interactions between adjacent egg-spot aggregations, which strongly suggests that egg-spot patterning results mostly from cell-autonomous cellular interactions. Together, these results demonstrate that A. calliptera egg-spot development is an exciting model for investigating pigment pattern formation at the cellular level in a system with developmental plasticity, sexual dimorphism, and intraspecific variation. As A. calliptera represents the ancestral bauplan for egg-spots, these findings provide a baseline for informed comparisons across the incredibly diverse Malawi cichlid radiation.     

Deciphering Interactions in Moving Animal Groups

Collective motion phenomena in large groups of social organisms have long fascinated the observer, especially in cases, such as bird flocks or fish schools, where large-scale highly coordinated actions emerge in the absence of obvious leaders. However, the mechanisms involved in this self-organized behavior are still poorly understood, because the individual-level interactions underlying them remain elusive. Here, we demonstrate the power of a bottom-up methodology to build models for animal group motion from data gathered at the individual scale. Using video tracks of fish shoal in a tank, we show how a careful, incremental analysis at the local scale allows for the determination of the stimulus/response function governing an individual''s moving decisions. We find in particular that both positional and orientational effects are present, act upon the fish turning speed, and depend on the swimming speed, yielding a novel schooling model whose parameters are all estimated from data. Our approach also leads to identify a density-dependent effect that results in a behavioral change for the largest groups considered. This suggests that, in confined environment, the behavioral state of fish and their reaction patterns change with group size. We debate the applicability, beyond the particular case studied here, of this novel framework for deciphering interactions in moving animal groups.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Analyzing fish movement as a persistent turning walker

The trajectories of Kuhlia mugil fish swimming freely in a tank are analyzed in order to develop a model of spontaneous fish movement. The data show that K. mugil displacement is best described by turning speed and its auto-correlation. The continuous-time process governing this new kind of displacement is modelled by a stochastic differential equation of Ornstein–Uhlenbeck family: the persistent turning walker. The associated diffusive dynamics are compared to the standard persistent random walker model and we show that the resulting diffusion coefficient scales non-linearly with linear swimming speed. In order to illustrate how interactions with other fish or the environment can be added to this spontaneous movement model we quantify the effect of tank walls on the turning speed and adequately reproduce the characteristics of the observed fish trajectories.     

Linking rates of diffusion and consumption in relation to resources

The functional response is a fundamental model of the relationship between consumer intake rate and resource abundance. The random walk is a fundamental model of animal movement and is well approximated by simple diffusion. Both models are central to our understanding of numerous ecological processes but are rarely linked in ecological theory. To derive a synthetic model, we draw on the common logical premise underlying these models and show how the diffusion and consumption rates of consumers depend on elementary attributes of naturally occurring consumer-resource interactions: the abundance, spatial aggregation, and traveling speed of resources as well as consumer handling time and directional persistence. We show that resource aggregation may lead to increased consumer diffusion and, in the case of mobile resources, reduced consumption rate. Resource-dependent movement patterns have traditionally been attributed to area-restricted search, reflecting adaptive decision making by the consumer. Our synthesis provides a simple alternative hypothesis that such patterns could also arise as a by-product of statistical movement mechanics.     

Effect of immersion at low tide on distribution and movement of the mud snail, Ilyanassa obsoleta (Say), in the upper Bay of Fundy, eastern Canada

Habitat heterogeneity often affects movement behaviours of animals, and consequently their spatial distribution. We evaluated the effect of immersion at low tide on the distribution, fine-scale movement patterns and daily movement patterns of the mud snail Ilyanassa obsoleta on a mudflat in the upper Bay of Fundy, Canada. Mud snails migrate onto intertidal mudflats in the summer, and our field survey showed that their density was higher inside tide pools relative to adjacent areas that are exposed at low tide. Using time-lapse videography, we evaluated the effect of snail size, snail density, and immersion at low tide on fine-scale movement patterns of I. obsoleta. Time until snails stopped moving and burrowed was unaffected by snail size, but snails at low and high densities burrowed somewhat faster than those at intermediate densities. Snail size and snail density had no detectable effect on displacement speed or linearity of displacement. Immersion affected snail movement: snails within tide pools delayed burrowing and traveled in more convoluted paths compared to those on exposed mud. Snails increased their turning angles within tide pools, which is probably the mechanism by which aggregations are formed. We also performed a mark-recapture experiment to compare daily movement patterns of snails released inside and outside tide pools. Snails released in tide pools moved shorter distances, but did not orient themselves differently than snails released outside tide pools. Both groups exhibited significant directionality, moving against the mean water current direction over 24 h. In sum, immersion at low tide affected the behaviour and spatial distribution of snails, resulting in snail aggregations within tide pools. These snail aggregations, in turn, may be a major factor influencing spatial dynamics on mudflats, including causing changes in distribution patterns of the burrowing amphipod Corophium volutator, a dominant inhabitant and key species in the food web of mudflats.     

Microtubule Patterning in the Presence of Stationary Motor Distributions

In this paper, we construct a novel nonlocal transport model that describes the evolution of microtubules (MTs) as they interact with stationary distributions of motor proteins. An advection term accounts for directed MT transport (sliding due to motor protein action), and an integral term accounts for reorientation of MTs due to their interactions with cross-linking motor proteins. Simulations of our model show how MT patterns depend on boundary constraints, as well as model parameters that represent motor speed, cross-linking capability (motor activity), and directionality. In large domains, and using motor parameter values consistent with experimentally-derived values, we find that patterns such as asters, vortices, and bundles are able to persist. In vivo, MTs take on aster patterns during interphase and they form bundles in neurons and polarized epithelial cells. Vortex patterns have not been observed in vivo, however, are found in in vitro experiments. In constrained domains, we find that similar patterns form (asters, bundles, and vortices). However, we also find that when two opposing motors are present, anti-parallel bundles are able to form, resembling the mitotic spindle during cell division. This model demonstrates how MT sliding and MT reorientation are sufficient to produce experimentally observed patterns.     

Analysis of foraging movements of Atlantic bluefin tuna (<Emphasis Type="Italic">Thunnus thynnus</Emphasis>): individuals switch between two modes of search behaviour

We investigate the application of quantitative techniques for distinguishing adaptive search behaviour in Atlantic bluefin tuna (Thunnus thynnus). The analysis demonstrates the application of a novel spectral analysis technique for resolving and measuring periodicity in animal behaviour patterns. Two different search strategies are identified that include regulation of turning (klinokinesis) and speed (orthokinesis). Our results provide evidence that bluefin tuna attempt to optimize their searching efficiency through adjustments in the duration and timing of switching between these two searching strategies. Repetitive, diurnal deep dives were also found to coincide with switching of search behaviour. Additional tracking experiments with larger sample sizes are needed to better identify how individuals switch between the two search strategies and how such decisions may collectively improve the searching and foraging efficiency of their schools (synchrokinesis, social taxis) in response to changes in the size or composition of prey aggregations.     

Quantifying the interplay between environmental and social effects on aggregated-fish dynamics

Demonstrating and quantifying the respective roles of social interactions and external stimuli governing fish dynamics is key to understanding fish spatial distribution. If seminal studies have contributed to our understanding of fish spatial organization in schools, little experimental information is available on fish in their natural environment, where aggregations often occur in the presence of spatial heterogeneities. Here, we applied novel modeling approaches coupled to accurate acoustic tracking for studying the dynamics of a group of gregarious fish in a heterogeneous environment. To this purpose, we acoustically tracked with submeter resolution the positions of twelve small pelagic fish (Selar crumenophthalmus) in the presence of an anchored floating object, constituting a point of attraction for several fish species. We constructed a field-based model for aggregated-fish dynamics, deriving effective interactions for both social and external stimuli from experiments. We tuned the model parameters that best fit the experimental data and quantified the importance of social interactions in the aggregation, providing an explanation for the spatial structure of fish aggregations found around floating objects. Our results can be generalized to other gregarious species and contexts as long as it is possible to observe the fine-scale movements of a subset of individuals.