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Cell migration is frequently modeled using on-lattice agent-based models (ABMs) that employ the excluded volume interaction. However, cells are also capable of exhibiting more complex cell-cell interactions, such as adhesion, repulsion, pulling, pushing, and swapping. Although the first four of these have already been incorporated into mathematical models for cell migration, swapping has not been well studied in this context. In this paper, we develop an ABM for cell movement in which an active agent can "swap" its position with another agent in its neighborhood with a given swapping probability. We consider a two-species system for which we derive the corresponding macroscopic model and compare it with the average behavior of the ABM. We see good agreement between the ABM and the macroscopic density. We also analyze the movement of agents at an individual level in the single-species as well as two-species scenarios to quantify the effects of swapping on an agent's motility.

In this paper, we investigate a generalized model of N particles undergoing second-order nonlocal interactions on a lattice. Our results have applications across many research areas, including the modeling of migration, information dynamics, and Muller's ratchet-the irreversible accumulation of deleterious mutations in an evolving population. Strikingly, numerical simulations of the model are observed to deviate significantly from its mean-field approximation even for large population sizes. We show that the disagreement between deterministic and stochastic solutions stems from finite-size effects that change the propagation speed and cause the position of the wave to fluctuate. These effects are shown to decay anomalously as (lnN)^{-2} and (lnN)^{-3}, respectively-much slower than the usual N^{-1/2} factor. Our results suggest that the accumulation of deleterious mutations in a Muller's ratchet and the loss of awareness in a population may occur much faster than predicted by the corresponding deterministic models. The general applicability of our model suggests that this unexpected scaling could be important in a wide range of real-world applications.

We develop theoretical equivalences between stochastic and deterministic models for populations of individual cells stratified by age. Specifically, we develop a hierarchical system of equations describing the full dynamics of an age-structured multistage Markov process for approximating cell cycle time distributions. We further demonstrate that the resulting mean behavior is equivalent, over large timescales, to the classical McKendrick-von Foerster integropartial differential equation. We conclude by extending this framework to a spatial context, facilitating the modeling of traveling wave phenomena and cell-mediated pattern formation. More generally, this methodology may be extended to myriad reaction-diffusion processes for which the age of individuals is relevant to the dynamics.

Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously well-used method to incorporate domain growth into on-lattice reaction-diffusion models causes a buildup of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here we present an alternative method which addresses this unphysical buildup of particles at the boundaries and demonstrate that it is accurate for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.

There are numerous biological scenarios in which populations of cells migrate in crowded environments. Typical examples include wound healing, cancer growth, and embryo development. In these crowded environments cells are able to interact with each other in a variety of ways. These include excluded-volume interactions, adhesion, repulsion, cell signaling, pushing, and pulling. One popular way to understand the behavior of a group of interacting cells is through an agent-based mathematical model. A typical aim of modellers using such representations is to elucidate how the microscopic interactions at the cell-level impact on the macroscopic behavior of the population. At the very least, such models typically incorporate volume-exclusion. The more complex cell-cell interactions listed above have also been incorporated into such models; all apart from cell-cell pulling. In this paper we consider this under-represented cell-cell interaction, in which an active cell is able to "pull" a nearby neighbor as it moves. We incorporate a variety of potential cell-cell pulling mechanisms into on- and off-lattice agent-based volume exclusion models of cell movement. For each of these agent-based models we derive a continuum partial differential equation which describes the evolution of the cells at a population level. We study the agreement between the agent-based models and the continuum, population-based models and compare and contrast a range of agent-based models (accounting for the different pulling mechanisms) with each other. We find generally good agreement between the agent-based models and the corresponding continuum models that worsens as the agent-based models become more complex. Interestingly, we observe that the partial differential equations that we derive differ significantly, depending on whether they were derived from on- or off-lattice agent-based models of pulling. This hints that it is important to employ the appropriate agent-based model when representing pulling cell-cell interactions.

Identifying and quantifying spatial correlation are important aspects of studying the collective behavior of multiagent systems. Pair correlation functions (PCFs) are powerful statistical tools that can provide qualitative and quantitative information about correlation between pairs of agents. Despite the numerous PCFs defined for off-lattice domains, only a few recent studies have considered a PCF for discrete domains. Our work extends the study of spatial correlation in discrete domains by defining a new set of PCFs using two natural and intuitive definitions of distance for a square lattice: the taxicab and uniform metric. We show how these PCFs improve upon previous attempts and compare between the quantitative data acquired. We also extend our definitions of the PCF to other types of regular tessellation that have not been studied before, including hexagonal, triangular, and cuboidal. Finally, we provide a comprehensive PCF for any tessellation and metric, allowing investigation of spatial correlation in irregular lattices for which recognizing correlation is less intuitive.

Persistence of motion is the tendency of an object to maintain motion in a direction for short time scales without necessarily being biased in any direction in the long term. One of the most appropriate mathematical tools to study this behavior is an agent-based velocity-jump process. In the absence of agent-agent interaction, the mean-field continuum limit of the agent-based model (ABM) gives rise to the well known hyperbolic telegraph equation. When agent-agent interaction is included in the ABM, a strictly advective system of partial differential equations (PDEs) can be derived at the population level. However, no diffusive limit of the ABM has been obtained from such a model. Connecting the microscopic behavior of the ABM to a diffusive macroscopic description is desirable, since it allows the exploration of a wider range of scenarios and establishes a direct connection with commonly used statistical tools of movement analysis. In order to connect the ABM at the population level to a diffusive PDE at the population level, we consider a generalization of the agent-based velocity-jump process on a two-dimensional lattice with three forms of agent interaction. This generalization allows us to take a diffusive limit and obtain a faithful population-level description. We investigate the properties of the model at both the individual and population levels and we elucidate some of the models' key characteristic features. In particular, we show an intrinsic anisotropy inherent to the models and we find evidence of a spontaneous form of aggregation at both the micro- and macroscales.

In this work we study the effect of domain growth on spatial correlations in agent populations containing multiple species. This is important as heterogenous cell populations are ubiquitous during the embryonic development of many species. We have previously shown that the long-term behavior of an agent population depends on the way in which domain growth is implemented. We extend this work to show that, depending on the way in which domain growth is implemented, different species dominate in multispecies simulations. Continuum approximations of the lattice-based model that ignore spatial correlations cannot capture this behavior, while those that explicitly account for spatial correlations can. The results presented here show that the precise mechanism of domain growth can determine the long-term behavior of multispecies populations and, in certain circumstances, establish spatially varying species densities.

Domain growth plays an important role in many biological systems, and so the inclusion of domain growth in models of these biological systems is important to understanding how these systems function. In this work we present methods to include the effects of domain growth on the evolution of spatial correlations in a continuum approximation of a lattice-based model of cell motility and proliferation. We show that, depending on the way in which domain growth is implemented, different steady-state densities are predicted for an agent population. Furthermore, we demonstrate that the way in which domain growth is implemented can result in the evolution of the agent density depending on the size of the domain. Continuum approximations that ignore spatial correlations cannot capture these behaviors, while those that account for spatial correlations do. These results will be of interest to researchers in developmental biology, as they suggest that the nature of domain growth can determine the characteristics of cell populations.

Diffusive transport is a universal phenomenon, throughout both biological and physical sciences, and models of diffusion are routinely used to interrogate diffusion-driven processes. However, most models neglect to take into account the role of volume exclusion, which can significantly alter diffusive transport, particularly within biological systems where the diffusing particles might occupy a significant fraction of the available space. In this work we use a random walk approach to provide a means to reconcile models that incorporate crowding effects on different spatial scales. Our work demonstrates that coarse-grained models incorporating simplified descriptions of excluded volume can be used in many circumstances, but that care must be taken in pushing the coarse-graining process too far.

The macroscale movement behavior of a wide range of isolated migrating cells has been well characterized experimentally. Recently, attention has turned to understanding the behavior of cells in crowded environments. In such scenarios it is possible for cells to interact, inducing neighboring cells to move in order to make room for their own movements or progeny. Although the behavior of interacting cells has been modeled extensively through volume-exclusion processes, few models, thus far, have explicitly accounted for the ability of cells to actively displace each other in order to create space for themselves. In this work we consider both on- and off-lattice volume-exclusion position-jump processes in which cells are explicitly allowed to induce movements in their near neighbors in order to create space for themselves to move or proliferate into. We refer to this behavior as pushing. From these simple individual-level representations we derive continuum partial differential equations for the average occupancy of the domain. We find that, for limited amounts of pushing, comparison between the averaged individual-level simulations and the population-level model is nearly as good as in the scenario without pushing. Interestingly, we find that, in the on-lattice case, the diffusion coefficient of the population-level model is increased by pushing, whereas, for the particular off-lattice model that we investigate, the diffusion coefficient is reduced. We conclude, therefore, that it is important to consider carefully the appropriate individual-level model to use when representing complex cell-cell interactions such as pushing.

We present a minimal model to describe the onset of collective motion seen when a population of locusts are placed in an annular arena. At low densities motion is disordered, while at high densities locusts march in a common direction, which may reverse during the experiment. The data are well captured by an individual-based model, in which demographic noise leads to the observed density-dependent effects. By fitting the model parameters to equation-free coefficients, we give a quantitative comparison, showing time series, stationary distributions, and the mean switching times between states.

Position-jump processes are used for the mathematical modeling of spatially extended chemical and biological systems with increasing frequency. A large subset of the literature concerning such processes is concerned with modeling the effect of stochasticity on reaction-diffusion systems. Traditionally, computational domains have been divided into regular voxels. Molecules are assumed well mixed within each of these voxels and are allowed to react with other molecules within the same voxel or to jump to neighboring voxels with predefined transition rates. For a variety of reasons implementing position-jump processes on irregular grids is becoming increasingly important. However, it is not immediately clear what form an appropriate irregular partition of the domain should take if it is to allow the derivation of mean molecular concentrations that agree with a given partial differential equation for molecular concentrations. It has been demonstrated, in one dimension, that the Voronoi domain partition is the appropriate method with which to divide the computational domain. In this Brief Report, we investigate theoretically the propriety of the Voronoi domain partition as an appropriate method to partition domains for position-jump models in higher dimensions. We also provide simulations of diffusion processes in two dimensions in order to corroborate our results.

There exists a plethora of mathematical models for cluster growth and/or aggregation on regular lattices. Almost all suffer from inherent anisotropy caused by the regular lattice upon which they are grown. We analyze the little-known model for stochastic cluster growth on a regular lattice first introduced by Ferreira Jr. and Alves [J. Stat. Mech. Theo. & Exp. (2006) P11007], which produces circular clusters with no discernible anisotropy. We demonstrate that even in the noise-reduced limit the clusters remain circular. We adapt the model by introducing a specific rearrangement algorithm so that, rather than adding elements to the cluster from the outside (corresponding to apical growth), our model uses mitosis-like cell splitting events to increase the cluster size. We analyze the surface scaling properties of our model and compare it to the behavior of more traditional models. In "1+1" dimensions we discover and explore a new, nonmonotonic surface thickness scaling relationship which differs significantly from the Family-Vicsek scaling relationship. This suggests that, for models whose clusters do not grow through particle additions which are solely dependent on surface considerations, the traditional classification into "universality classes" may not be appropriate.

Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [Bull. Math. Biol. 72, 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations.

We obtain a Fokker-Planck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.