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

The hybrid bio-robotic swarm as a powerful tool for collective motion research: a perspective

Swarming or collective motion is ubiquitous in natural systems, and instrumental in many technological applications. Accordingly, research interest in this phenomenon is crossing discipline boundaries. A common major question is that of the intricate interactions between the individual, the group, and the environment. There are, however, major gaps in our understanding of swarming systems, very often due to the theoretical difficulty of relating embodied properties to the physical agents-individual animals or robots. Recently, there has been much progress in exploiting the complementary nature of the two disciplines: biology and robotics. This, unfortunately, is still uncommon in swarm research. Specifically, there are very few examples of joint research programs that investigate multiple biological and synthetic agents concomitantly. Here we present a novel research tool, enabling a unique, tightly integrated, bio-inspired, and robot-assisted study of major questions in swarm collective motion. Utilizing a quintessential model of collective behavior-locust nymphs and our recently developed Nymbots (locust-inspired robots)-we focus on fundamental questions and gaps in the scientific understanding of swarms, providing novel interdisciplinary insights and sharing ideas disciplines. The Nymbot-Locust bio-hybrid swarm enables the investigation of biology hypotheses that would be otherwise difficult, or even impossible to test, and to discover technological insights that might otherwise remain hidden from view.

Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis

Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.

Leadership Through Influence: What Mechanisms Allow Leaders to Steer a Swarm?

Collective migration of cells and animals often relies on a specialised set of “leaders”, whose role is to steer a population of naive followers towards some target. We formulate a continuous model to understand the dynamics and structure of such groups, splitting a population into separate follower and leader types with distinct orientation responses. We incorporate leader influence via three principal mechanisms: a bias in the orientation of leaders towards the destination (orientation-bias), a faster movement of leaders when moving towards the target (speed-bias), and leaders making themselves more clear to followers when moving towards the target (conspicuousness-bias). Analysis and numerical computation are used to assess the extent to which the swarm is successfully shepherded towards the target. We find that successful leadership can occur for each of these three mechanisms across a broad region of parameter space, with conspicuousness-bias emerging as the most robust. However, outside this parameter space we also find various forms of unsuccessful leadership. Forms of excessive influence can result in either swarm-splitting, where the leaders break free and followers are left rudderless, or a loss of swarm cohesion that leads to its eventual dispersal. Forms of low influence, on the other hand, can even generate swarms that move away from the target direction. Leadership must therefore be carefully managed to steer the swarm correctly.

Mathematical models for cell migration: a non-local perspective

We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Multi-scale analysis and modelling of collective migration in biological systems

Collective migration has become a paradigm for emergent behaviour in systems of moving and interacting individual units resulting in coherent motion. In biology, these units are cells or organisms. Collective cell migration is important in embryonic development, where it underlies tissue and organ formation, as well as pathological processes, such as cancer invasion and metastasis. In animal groups, collective movements may enhance individuals' decisions and facilitate navigation through complex environments and access to food resources. Mathematical models can extract unifying principles behind the diverse manifestations of collective migration. In biology, with a few exceptions, collective migration typically occurs at a 'mesoscopic scale' where the number of units ranges from only a few dozen to a few thousands, in contrast to the large systems treated by statistical mechanics. Recent developments in multi-scale analysis have allowed linkage of mesoscopic to micro- and macroscopic scales, and for different biological systems. The articles in this theme issue on 'Multi-scale analysis and modelling of collective migration' compile a range of mathematical modelling ideas and multi-scale methods for the analysis of collective migration. These approaches (i) uncover new unifying organization principles of collective behaviour, (ii) shed light on the transition from single to collective migration, and (iii) allow us to define similarities and differences of collective behaviour in groups of cells and organisms. As a common theme, self-organized collective migration is the result of ecological and evolutionary constraints both at the cell and organismic levels. Thereby, the rules governing physiological collective behaviours also underlie pathological processes, albeit with different upstream inputs and consequences for the group. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

Correlated random walks inside a cell: actin branching and microtubule dynamics

Correlated random walks (CRW) have been explored in many settings, most notably in the motion of individuals in a swarm or flock. But some subcellular systems such as growth or disassembly of bio-polymers can also be described with similar models and understood using related mathematical methods. Here we consider two examples of growing cytoskeletal elements, actin and microtubules. We use CRW or generalized CRW-like PDEs to model their spatial distributions. In each case, the linear models can be reduced to a Telegrapher's equation. A combination of explicit solutions (in one case) and numerical solutions (in the other) demonstrates that the approach to steady state can be accompanied by (decaying) waves.

Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.

Traveling pulse emerges from coupled intermittent walks: A case study in sheep

Monitoring small groups of sheep in spontaneous evolution in the field, we decipher behavioural rules that sheep follow at the individual scale in order to sustain collective motion. Individuals alternate grazing mode at null speed and moving mode at walking speed, so cohesive motion stems from synchronising when they decide to switch between the two modes. We propose a model for the individual decision making process, based on switching rates between stopped / walking states that depend on behind / ahead locations and states of the others. We parametrize this model from data. Next, we translate this (microscopic) individual-based model into its density-flow (macroscopic) equations counterpart. Numerical solving these equations display a traveling pulse propagating at constant speed even though each individual is at any moment either stopped or walking. Considering the minimal model embedded in these equations, we derive analytically the steady shape of the pulse (sech square). The parameters of the pulse (shape and speed) are expressed as functions of individual parameters. This pulse emerges from the non linear coupling of start/stop individual decisions which compensate exactly for diffusion and promotes a steady ratio of walking / stopped individuals, which in turn determines the traveling speed of the pulse. The system seems to converge to this pulse from any initial condition, and to recover the pulse after perturbation. This gives a high robustness to this coordination mechanism.

The impact of short- and long-range perception on population movements

Navigation of cells and organisms is typically achieved by detecting and processing orienteering cues. Occasionally, a cue may be assessed over a much larger range than the individual's body size, as in visual scanning for landmarks. In this paper we formulate models that account for orientation in response to short- or long-range cue evaluation. Starting from an underlying random walk movement model, where a generic cue is evaluated locally or nonlocally to determine a preferred direction, we state corresponding macroscopic partial differential equations to describe population movements. Under certain approximations, these models reduce to well-known local and nonlocal biological transport equations, including those of Keller-Segel type. We consider a case-study application: "hilltopping" in Lepidoptera and other insects, a phenomenon in which populations accumulate at summits to improve encounter/mating rates. Nonlocal responses are shown to efficiently filter out the natural noisiness (or roughness) of typical landscapes and allow the population to preferentially accumulate at a subset of hilltopping locations, in line with field studies. Moreover, according to the timescale of movement, optimal responses may occur for different perceptual ranges.

Counter-propagating wave patterns in a swarm model with memory

Hyperbolic transport-reaction equations are abundant in the description of movement of motile organisms. Here, we focus on a system of four coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. By modeling how the behavior depends on the time since the last reversal, we introduce a memory effect. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies. Stability analysis reveals conditions for the wave formation as well as pulsating-in-time spatially constant solutions.

Direction-dependent interaction rules enrich pattern formation in an individual-based model of collective behavior

Direction-dependent interaction rules are incorporated into a one-dimensional discrete-time stochastic individual-based model (IBM) of collective behavior to compare pattern formation with an existing partial differential equation (PDE) model. The IBM is formulated in terms of three social interaction forces: repulsion, alignment, and attraction, and includes information regarding conspecifics’ direction of travel. The IBM produces a variety of spatial patterns which qualitatively match patterns observed in a PDE model. The addition of direction-dependent interaction rules exemplifies how directional information transfer within a group of individuals can result in enriched pattern formation. Our individual-based modelling framework reveals the influence that direction-dependent interaction rules such as biological communication can have upon individual movement trajectories and how these trajectories combine to form group patterns.

The epidemiological models of Karl-Peter Hadeler

The most frequently cited articles out of KP Hadeler's 45 papers with epidemiological applications are summarized. Parasitic diseases which increase the death rate of the hosts proportional to the integer number of parasites present were described by integral equations for the generating function of the age- and time-dependent number of parasites. A model was derived for a population structured by the continuous level of parasitic infection.

Stimulated by the spread of AIDS a new class of epidemic models was developed which take into account explicitly the formation and separation of pairs. For predator-prey populations with parasitic infections threshold conditions for the persistence of the predator were derived. The interaction of epidemics and demography was analysed. Several epidemiological conditions led to backward bifurcations associated with multiple infective stationary states.

Kinetic Models for Topological Nearest-Neighbor Interactions

We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system consists of a finite number of particles characterized by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of its closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys 163:41-60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.

Pattern formation in a nonlocal mathematical model for the multiple roles of the TGF-β pathway in tumour dynamics

The growth and invasion of cancer cells are very complex processes, which can be regulated by the cross-talk between various signalling pathways, or by single signalling pathways that can control multiple aspects of cell behaviour. TGF-β is one of the most investigated signalling pathways in oncology, since it can regulate multiple aspects of cell behaviour: cell proliferation and apoptosis, cell-cell adhesion and epithelial-to-mesenchimal transition via loss of cell adhesion. In this study, we use a mathematical modelling approach to investigate the complex roles of TGF-β signalling pathways on the inhibition and growth of tumours, as well as on the epithelial-to-mesenchimal transition involved in the metastasis of tumour cells. We show that the nonlocal mathematical model derived here to describe repulsive and adhesive cell-cell interactions can explain the formation of new tumour cell aggregations at positions in space that are further away from the main aggregation. Moreover, we show that the increase in cell-cell adhesion leads to fewer but larger aggregations, and the increase in TGF-β molecules - whose late-stage effect is to decrease cell adhesion - leads to many small cellular aggregations. Finally, we perform a sensitivity analysis on some parameters associated with TGF-β dynamics, and use it to investigate the relation between the tumour size and its metastatic spread.

Fronts under arrest: Nonlocal boundary dynamics in biology

We introduce a minimal geometric partial differential equation framework to understand pattern formation from interacting, counterpropagating fronts. Our approach concentrates on the interfaces between different states in a system, and relies on both nonlocal interactions and mean-curvature flow to track their evolution. As an illustration, we use this approach to describe a phenomenon in bacterial colony formation wherein sibling colonies can arrest each other's growth. This arrested motion leads to static separations between healthy, growing colonies. As our minimal model faithfully recovers the geometry of these competing colonies, it captures and elucidates the key leading-order mechanisms responsible for such patterned growth.

Mathematical analysis and validation of an exactly solvable model for upstream migration of fish schools in one-dimensional rivers

Upstream migration of fish schools in 1-D rivers as an optimal control problem is formulated where their swimming velocity and the horizontal oblateness are taken as control variables. The objective function to be maximized through a migration process consists of the biological and ecological profit to be gained at the upstream-end of a river, energetic cost of swimming against the flow, and conceptual cost of forming a school. Under simplified conditions where the flow is uniform in both space and time and the profit to be gained at the goal of migration is sufficiently large, the optimal control variables are determined from a system of algebraic equations that can be solved in a cascading manner. Mathematical analysis of the system reveals that the optimal controls are uniquely found and the model is exactly solvable under certain conditions on the functions and parameters, which turn out to be realistic and actually satisfied in experimental fish migration. Identification results of the functional shapes of the functions and the parameters with experimentally observed data of swimming schools of Plecoglossus altivelis (Ayu) validate the present mathematical model from both qualitative and quantitative viewpoints. The present model thus turns out to be consistent with the reality, showing its potential applicability to assessing fish migration in applications.

Steady and transient states in low-energy swarms: Stability and first-passage times

We investigate a class of agent-based models of self-propelled particles (SPP) that interact according to a Morse potential in the presence of friction, a class which was able to reproduce many of the intriguing patterns of collective motion observed in nature. Specifically, we compare two closely related SPP models in the literature that differ by their prescription of particle drag and self-propulsion. Writing both models in terms of nondimensional parameters allows us to show that the dynamics in the highly viscous regime is independent of the precise forms of drag and propulsion. In contrast to what is indicated in the literature both models yield the same low-energy self-organized states: the coherent flock and the rigid rotation states which are highly ordered in both the coordinate and the velocity spaces and a velocity-disordered droplet state where particles are confined to rings which pass through the lattice points of the underlying Lagrange configuration. In contrast to the first two states which are stable, the third state is found to be a long-lived transient. In the regime studied, relaxing to one of the ordered steady states is inevitable, but how and when the transition occurs and what is the probability of ending in one state rather than the other are functions of the model parameters. Two types of transitions are characterized and first passage times are computed. Eventually, the evolution of the order parameter is explored in the framework of a Langevin-type equation, and the possible metastability of the random droplet state is discussed.

Effective Rheological Properties in Semi-dilute Bacterial Suspensions

Interactions between swimming bacteria have led to remarkable experimentally observable macroscopic properties such as the reduction in the effective viscosity, enhanced mixing, and diffusion. In this work, we study an individual-based model for a suspension of interacting point dipoles representing bacteria in order to gain greater insight into the physical mechanisms responsible for the drastic reduction in the effective viscosity. In particular, asymptotic analysis is carried out on the corresponding kinetic equation governing the distribution of bacteria orientations. This allows one to derive an explicit asymptotic formula for the effective viscosity of the bacterial suspension in the limit of bacterium non-sphericity. The results show good qualitative agreement with numerical simulations and previous experimental observations. Finally, we justify our approach by proving existence, uniqueness, and regularity properties for this kinetic PDE model.

Locust Collective Motion and Its Modeling

Over the past decade, technological advances in experimental and animal tracking techniques have motivated a renewed theoretical interest in animal collective motion and, in particular, locust swarming. This review offers a comprehensive biological background followed by comparative analysis of recent models of locust collective motion, in particular locust marching, their settings, and underlying assumptions. We describe a wide range of recent modeling and simulation approaches, from discrete agent-based models of self-propelled particles to continuous models of integro-differential equations, aimed at describing and analyzing the fascinating phenomenon of locust collective motion. These modeling efforts have a dual role: The first views locusts as a quintessential example of animal collective motion. As such, they aim at abstraction and coarse-graining, often utilizing the tools of statistical physics. The second, which originates from a more biological perspective, views locust swarming as a scientific problem of its own exceptional merit. The main goal should, thus, be the analysis and prediction of natural swarm dynamics. We discuss the properties of swarm dynamics using the tools of statistical physics, as well as the implications for laboratory experiments and natural swarms. Finally, we stress the importance of a combined-interdisciplinary, biological-theoretical effort in successfully confronting the challenges that locusts pose at both the theoretical and practical levels.

Understanding the shape of ant craters: a continuum model

The disposal of soil grains by ants, during excavation of their nest, results in the formation of axisymmetric craters around the nest entrance. We give a simple explanation for the shape of these biological constructs based on basic processes underlying grain transport and grain dropping. We propose that the tendency of an ant to drop a grain, in its next step, keeps increasing as it carries the grain farther away from the nest. Based on this hypothesis, a continuum mathematical model is developed to describe the soil dumping activity of ants, averaged over space and time. Consisting of a single, first-order differential equation, the model resembles that used to describe simultaneous convection and reaction of a chemical species, thus establishing a connection between ant craters and reacting flows. The model is shown to accurately describe the soil disposal data for two species of ants—M. barbarus and P. ambigua—using only two adjustable parameters- one less than previous empirical distributions. The characteristic single-hump shape of the crater is explained as follows: While the tendency to drop grains is greater at distances further away from the nest, the density of grain-bearing ants is highest close to the nest, thus most of the grains are dropped at an intermediate location and form a peak. The model predicts that steep craters with a sharp peak are always located closer to the nest entrance than craters which are more spread out; this new prediction is verified by data for M. barbarus and P. ambiguaants.

On the interplay between mathematics and biology: hallmarks toward a new systems biology

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.

Quantification of cell co-migration occurrences during cell aggregation on fibroin substrates

A quantitative analytical method was proposed for measuring cell co-migration, which was defined as two or more cells migrating together. To accurately identify and quantify this behavior, cell migration on fibroin substrates was analyzed with respect to intercellular distance. Specifically, cell size was characterized by major diameter, and then, based on these measurements and cell center data, a specific threshold distance for defining co-migration was determined after analyzing cell motion using the Voronoi diagram method. The results confirmed that co-migration occurrences of rounded cells were significantly more stable on fibroin than on ProNectin substrates under the present experimental conditions. The cell co-migration analysis method in this article was shown to be successful in evaluating the stability of cell co-migration and also suggested the presence of "critical distance" where two cells interact on fibroin substrates. With further research, the cell co-migration analysis method and "critical distance" may prove to be capable of identifying the aggregation behavior of other cells on different materials, making it a valuable tool that can be used in tissue engineering design.

Simultaneous use of different communication mechanisms leads to spatial sorting and unexpected collective behaviours in animal groups

Communication among individuals forms the basis of social interactions in every animal population. In general, communication is influenced by the physiological and psychological constraints of each individual, and in large aggregations this means differences in the reception and emission of communication signals. However, studies on the formation and movement of animal aggregations usually assume that all individuals communicate with neighbours in the same manner. Here, we take a new approach on animal aggregations and use a nonlocal mathematical model to investigate theoretically the simultaneous use of two communication mechanisms by different members of a population. We show that the use of multiple communication mechanisms can lead to behaviours that are not necessarily predicted by the behaviour of subpopulations that use only one communication mechanism. In particular, we show that while the use of one communication mechanism by the entire population leads to deterministic movement, the use of multiple communication mechanisms can lead in some cases to chaotic movement. Finally, we show that the use of multiple communication mechanisms leads to the sorting of individuals inside aggregations: individuals that are aware of the location and the movement direction of all their neighbours usually position themselves at the centre of the groups, while individuals that are aware of the location and the movement direction of only some neighbours position themselves at the edges of the groups.

Controllability in hybrid kinetic equations modeling nonequilibrium multicellular systems

This paper is concerned with the derivation of hybrid kinetic partial integrodifferential equations that can be proposed for the mathematical modeling of multicellular systems subjected to external force fields and characterized by nonconservative interactions. In order to prevent an uncontrolled time evolution of the moments of the solution, a control operator is introduced which is based on the Gaussian thermostat. Specifically, the analysis shows that the moments are solution of a Riccati-type differential equation.