Macroscopic limits of individual-based models for motile cell populations with volume exclusion

Partial differential equation models are ubiquitous in studies of motile cell populations, giving a phenomenological description of events which can be analyzed and simulated using a wide range of existing tools. However, these models are seldom derived from individual cell behaviors and so it is difficult to accurately include biological hypotheses on this spatial scale. Moreover, studies which do attempt to link individual- and population-level behavior generally employ lattice-based frameworks in which the artifacts of lattice choice at the population level are unclear. In this work we derive limiting population-level descriptions of a motile cell population from an off-lattice, individual-based model (IBM) and investigate the effects of volume exclusion on the population-level dynamics. While motility with excluded volume in on-lattice IBMs can be accurately described by Fickian diffusion, we demonstrate that this is not the case off lattice. We show that the balance between two key parameters in the IBM (the distance moved in one step and the radius of an individual) determines whether volume exclusion results in enhanced or slowed diffusion. The magnitude of this effect is shown to increase with the number of cells and the rate of their movement. The method we describe is extendable to higher-dimensional and more complex systems and thereby provides a framework for deriving biologically realistic, continuum descriptions of motile populations.

Figure 1

Schematic view of the IBM, showing individuals in one spatial dimension. Each individual is represented by its center point, $x$, and occupies a region around this point, $(x-R,x+R)$. Individuals move with rate $\alpha$, a distance $d$ to the left or right.

Figure 2

Comparison between the IBM and the continuum models with $d>0$ [Eq. (7)] and as $d\to 0$ [Eq. (8)], plotted at $t=0$, 200, and 600 for $x\in [0,65]$. The solution to the diffusion equation is also shown for comparison. Realizations of the IBM were performed using the Gillespie algorithm [25]. Sixty cells were placed at regular intervals in the middle of the domain, with the leftmost cell given an initial position drawn from the normal distribution $\mathcal{N}(25,1)$. For the PDEs we used a finite difference method with initial conditions determined by the average initial distribution from simulations, linearly interpolated onto a mesh with spacing $dx=0.05$. Parameters: $\alpha =2.222$, $R=0.1$, and $d=0.15$, giving $\widehat{\alpha }=0.025$.

Figure 3

Comparison between the IBM and the continuum model with $d>0$ [Eq. (7)], plotted at $t=0,300,600$ for $x\in [0,65]$. The solution to the diffusion equation is also shown for comparison. Realizations of the IBM were performed using the Gillespie algorithm [25]. Two groups of 30 cells were placed at regular intervals giving a density of 80%. The leftmost cells in the first and second groups are given an initial position drawn from $\mathcal{N}(16,1)$ and $\mathcal{N}(38,1)$, respectively. For the PDEs we used a finite difference method with initial conditions determined by the average initial distribution from simulations, linearly interpolated onto a mesh with spacing $dx=0.1$. Parameters: $\alpha =0.8$, $R=0.15$, and $d=0.25$, giving $\widehat{\alpha }=0.025$.

Figure 4

Comparison of the varying importance of the exclusion term in Eq. (7). (a) Plot of ${\int }_{B_L}^{B_R}|C-C_D|dx/(B_R-B_L)$, where $C_D$ is the solution to the diffusion equation, as $R$ and $N$ change, with $d=0.15$. (b) Plot of ${\int }_{B_L}^{B_R}|S-C_D|dx/(B_R-B_L)$, where $C_D$ is the solution to the diffusion equation and $S$ is the averaged simulation results, as $R$ and $d$ change. In each case, $\alpha =2.222$, ten cells were placed on the interval $[-40,40]$, with initial positions drawn from the normal distribution $\mathcal{N}(0,3)$.

Figure 5

Comparison between the IBM and the continuum models with $d\sim \mathcal{N}(0,{\sigma }^2)$ [Eq. (11)] and as $\sigma \to 0$ [Eq. (8)], plotted at $t=0$, 200, and 600 for $x\in [0,65]$. The solution to the diffusion equation is also shown for comparison. Realizations of the IBM were performed using the Gillespie algorithm [25]. Sixty cells were placed at regular intervals in the middle of the domain, with the leftmost cell given an initial condition drawn from the normal distribution $\mathcal{N}(25,1)$. For the PDEs we used a finite difference method with initial positions determined by the average initial distribution from simulations, linearly interpolated onto a mesh with spacing $dx=0.05$. Parameters: $\alpha =2.222$, $R=0.1$, and $\sigma =0.1$.

Figure 6

Comparison of the error between the IBM and the continuum model with (a) fixed jump distance, $d$, [Eq. (8)] and (b) $d\sim \mathcal{N}(0,\sigma )$, [Eq. (11)]. In each case, $\alpha =2.222$, ten cells were placed in the interval $[-40,40]$, with initial positions drawn from the normal distribution $\mathcal{N}(0,3)$.