A model for mesoscale patterns in motile populations

doi:10.1016/j.physa.2009.12.010

Physica A: Statistical Mechanics and its Applications

Volume 389, Issue 7, 1 April 2010, Pages 1412-1424

https://doi.org/10.1016/j.physa.2009.12.010 Get rights and content

Abstract

Experimental observations of cell migration often describe the presence of mesoscale patterns within motile cell populations. These patterns can take the form of cells moving as aggregates or in chain-like formation. Here we present a discrete model capable of producing mesoscale patterns. These patterns are formed by biasing movements to favor a particular configuration of agent–agent attachments using a binding function $f (K)$ , where $K$ is the scaled local coordination number. This discrete model is related to a nonlinear diffusion equation, where we relate the nonlinear diffusivity $D (C)$ to the binding function $f$ . The nonlinear diffusion equation supports a range of solutions which can be either smooth or discontinuous. Aggregation patterns can be produced with the discrete model, and we show that there is a transition between the presence and absence of aggregation depending on the sign of $D (C)$ . A combination of simulation and analysis shows that both the existence of mesoscale patterns and the validity of the continuum model depend on the form of $f$ . Our results suggest that there may be no formal continuum description of a motile system with strong mesoscale patterns.

Introduction

Cell migration is a multiscale phenomenon where experimental observations are made over a range of scales depending on the system of interest. For example, macroscopic observations of cell densities are made to characterize the invasiveness of cell populations [1], [2], whereas microscopic observations of individual cell trajectories are made to characterize the details of the migration mechanism [3], [4]. Observations of cell migration are also made at intermediate scales, which we shall call the mesoscale. These observations correspond to visual patterns that form within a motile population. Two common mesoscale patterns in cell biology include the formation of motile aggregates of cells [5], [6], as well as the formation of chain-like patterns [7], [8], [9].

Traditional modeling approaches were often interested in describing macroscopic properties of a cell population, such as the spatial distribution of cell densities. For such applications continuum models were appropriate, and various motility mechanisms have been considered [1], [2]. Recent advances in microscopy now provide high quality confocal, time-lapse and magnetic resonance imaging data. This new data provides opportunities to develop models at the individual cell level [10].

There is great interest in the correspondence between the discrete and continuum models of cell motility. The starting point can be a spatially discrete, continuous in time, nearest-neighbor master equation for a probability distribution function [11], [12], [13], [14]. In the appropriate limits, this gives rise to a partial differential equation (PDE) description of cell density. An alternative approach starts with a spatially and temporally discrete exclusion process with biologically realistic discrete cell motility rules. The spatially and temporally discrete model can be averaged to provide a continuum description of the system [15], [16], [17]. This is the approach we take here since the spatially and temporally discrete exclusion process is well suited for the interpretation of time-lapse data.

Here we describe and analyze a lattice-based discrete motility model which takes account of contacts or binding between agents. Motility events are governed, in part, by a binding function $f (K)$ , where $K \in [0, 1]$ is the scaled coordination number of the target site and $f$ is a function that will bias movements to favor a particular coordination number at the target site. The discrete model, based on a simple exclusion process [18], is averaged to give a continuum description of the expected behavior of the system in terms of: (i) a PDE description of the population density [15], [18], and (ii) differential equations describing the average trajectory, or pathline, of a tagged agent within the population [16]. Using simulation and visual inspection, we demonstrate that the model can produce a range of mesoscale patterns including chains and aggregates. Different patterns correspond to different choices of $f (K)$ and different initial conditions.

We find that the macroscopic density is governed by a nonlinear diffusion equation. This is an intriguing outcome. Although standard unbiased exclusion processes involve interacting agents, the population-level density obeys a linear diffusion equation, since the interactions are symmetric and do not appear in the continuum model [15], [18]. In the discrete model presented here, the discrete interactions are asymmetric and now appear in the continuum description through the nonlinear diffusivity $D (C)$ , where $C$ is the continuum density.

The PDE model developed here admits a range of solutions with complex behaviors depending on the form of $D (C)$ . In particular, the PDE with appropriate initial conditions admits both smooth and discontinuous solutions, depending on whether $D (C)$ is positive or negative [19], [20]. The discrete model can also generate aggregation patterns when the diffusivity is negative. This behavior is related to the backward heat equation [19], [21]. This contrasts the discrete exclusion process of Deroulers et al. that accounted for maintaining neighbors and gave rise to a positive nonlinear diffusivity [17].

Our analysis suggests that the formation of strong mesoscale patterns (chains, aggregates) and the validity of the continuum models described here depend on the form of $f (K)$ . Furthermore our results suggest that strong mesoscale patterns may not have any continuum description. This means that the application of a continuum model to capture a mesoscale chain-like structure may be inappropriate. Although this study is motivated with examples from cell biology, this work has implications for related applications involving collective migration that include ecological applications [22], malignant invasion [23] and pedestrian motility [24].

Section snippets

The discrete model

Simulation data presented in this work will use the two-dimensional square lattice as a specific illustration of the model. However, most of the analysis applies to arbitrary $d$ -dimensional periodic lattices with spacing $Δ$ . Agents can be viewed as occupying sites. Alternatively, agents can be regarded as residing in regions, since each site $s$ is associated with a spatial region consisting of all points closer to site $s$ than to any other. In various contexts these regions are known as Voronoi

Discrete simulation data: Influence of $f (K)$

To demonstrate the influence of different binding functions, we present a suite of simulations on a two-dimensional square lattice of size 100×100. Periodic boundary conditions are imposed along all boundaries. Initially, each site $(x, y) = (i Δ, j Δ)$ with $35 \leq i, j \leq 65$ is occupied with probability 0.5. Snapshots of four simulations, after 200 time steps, are shown in Fig. 2. Results in Fig. 2(a), with $f (K) = 1$ , show a symmetric distribution of agents where there is no apparent preference for the local

Deriving a continuous description

To connect the discrete mechanism with a continuum model we define the average occupancy of site $s$ , averaged over many statistically identical realizations, as $〈 C_{s} 〉$ . After averaging, we form a discrete conservation statement describing $δ 〈 C_{s} 〉$ , which is the change in average occupancy of site $s$ during the time interval from $t$ to $t + τ$ [15], [18]: $δ 〈 C_{s} 〉 = P (1 - 〈 C_{s} 〉) \sum_{s^{'} \in N {s}} 〈 C_{s^{'}} 〉 \frac{f (K_{s})}{\sum_{s^{″} \in N {s^{'}}} f (K_{s^{″}})} - P 〈 C_{s} 〉 \sum_{s^{'} \in N {s}} (1 - 〈 C_{s^{'}} 〉) \frac{f (K_{s^{'}})}{\sum_{s^{″} \in N {s}} f (K_{s^{″}})} .$ The positive terms on the right of Eq. (1) represent the

Results

Simulation data will be generated and compared to the solution of the continuum models for two types of problems. First, we consider the evolution of an initially close-packed group of agents. Second, we consider the evolution of a uniformly seeded lattice for which, on average, there are no spatial gradients present initially in the system. These two problems demonstrate a wide range of behaviors supported by the discrete model and demonstrate the relationship between the discrete mechanism

Discussion and conclusion

We have presented and analyzed a discrete motility model capable of producing a range of mesoscale patterns. Our approach enables us to visualize discrete simulations, as well as relating our discrete model to continuum equations describing agent density and agent pathline data. Previous approaches to modeling mesoscale patterns have either taken a discrete or continuum approach separately [13], [37], [38], without any explicit connection between the two. The development and application of

Acknowledgements

This work is supported by the Australian Research Council (ARC). Mat Simpson is an ARC Postdoctoral Fellow and Kerry Landman is an ARC Professorial Fellow. We thank Dr. Heather Young and Dr. Don Newgreen whose experimental work was the motivation for this paper.

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A model for mesoscale patterns in motile populations

Abstract

Introduction

Section snippets

The discrete model

Discrete simulation data: Influence of f(K)

Deriving a continuous description

Results

Discussion and conclusion

Acknowledgements

Appl. Math. Lett.

Dev. Biol.

Chem. Eng. Sci.

Dev. Biol.

J. Theoret. Biol.

Physica A

Appl. Math. Lett.

Physica A

Phys. Life Rev.

Physica A

J. Theoret. Biol.

J. Theoret. Biol.

J. Neurosci.

Neoplasia

Biotechnol. Bioeng.

Development

Cancer Res.

Phys. Rev. E

SIAM. J. Appl. Math.

Bull. Math. Biol.

J. Math. Biol.

Discrete simulation data: Influence of $f (K)$