Interacting motile agents: Taking a mean-field approach beyond monomers and nearest-neighbor steps

We consider a discrete agent-based model on a one-dimensional lattice, where each agent occupies $L$ sites and attempts movements over a distance of $d$ lattice sites. Agents obey a strict simple exclusion rule. A discrete-time master equation is derived using a mean-field approximation and careful probability arguments. In the continuum limit, nonlinear diffusion equations that describe the average agent occupancy are obtained. Averaged discrete simulation data are generated and shown to compare very well with the solution to the derived nonlinear diffusion equations. This framework allows us to approach a lattice-free result using all the advantages of lattice methods. Since different cell types have different shapes and speeds of movement, this work offers insight into population-level behavior of collective cellular motion.

Figure 1

Monomer agents: If $d=1$, the center [open (green) circle] agent can move in either direction. If $d=2$ the center [open (green) circle] agent can move right but cannot move left, as the left [open (red) circle] agent is occupying the site. If $d⩾3$, the center [open (green) circle] agent cannot move in either direction because there is an agent preventing the move in both directions.

Figure 2

Diffusion coefficient for monomers ($L=1$), $\mathcal{D}\left(C\right)=d^2{(1-C)}^{d-1}$, with movement distance $d=1$, 2, 3, 4, 5 [Eq. (7)]. The arrow indicates increasing $d$.

Figure 3

Monomer agents moving $d$ lattice sites: Solutions to Eq. (7) [medium-gray (red) curve] compared to the average of 50 000 simulations [dark-gray (blue) curve] at times $t=100$, 300, and 500. (a) $d=2$. The initial condition (shown) has agents between $181\le x\le 220$. (b) $d=3$. The initial condition (shown) has agents between $281\le x\le 320$. Note the different scales on the $x$ axes: for larger values of $d$ the outermost agents move farther, and so a larger region is needed to prevent interference with the boundaries for $t\le 500$. Equation (7) was solved using Matlab pdepe with $\delta x=0.1$. Arrows in (a) indicate increasing time.

Figure 4

Fixed movement distance. The diffusion coefficient scaled with fixed distance $d$, and $\mathcal{D}\left(C\right)/d^2$ given by Eq. (24), for two ratios of $L$ and $d$: (a) $d=1,\cdots ,10$ and $L=d$ and (b) $d=2,\cdots ,20$ and $L=\frac{1}{2}d$. Black arrows indicate increasing $d$ and $C_{\max }=1/L$. Note the different scales on the vertical axes.

Figure 5

Dimer agents (i.e., $L=2$) and agents occupying four sites ($L=4$) moving exactly $d$ lattice sites: Solutions to Eq. (21) [medium-gray (red) curves] for $d=2$ and 4 compared to the average of 50 000 simulations [dark-gray (blue) curves] at times $t=100,$ 300, and 500. The initial condition is also shown, with agents between $181\le x\le 220$ for $d=2$ and between $381\le x\le 420$ for $d=4$. Note the different scales on the $x$ axes: for larger values of $d$ the outermost agents move farther, and so a larger region is needed to prevent interference with the boundaries or $t\le 500$. Equations (21) and (22) are solved using Matlab pdepe with $\delta x=0.1$. Arrows in (a) indicate increasing time.

Figure 6

Movement up to a fixed distance $d$, illustrated for a monomeric agent. For any $d⩾3$, the right [open (blue) circle] agent will occupy the green (light gray) site if it attempts to move left, since the left [open (red) circle] agent prevents any further movement.

Figure 7

Movement up to a distance $d$. The diffusion coefficient scaled with distance $\mathcal{D}\left(C\right)/d^2$, given by Eq. (31), for three ratios of $L$ and $d$: (a) $d=1,\cdots ,10$ and $L=2d$, (b) $d=1,\cdots ,10$ and $L=d$, and (c) $d/2=1,\cdots ,10$ and $L=\frac{1}{2}d$. Black arrows indicate increasing $d$, and in all cases $C_{\max }=1/L$. Note the different scales on the $y$ axis: the scaled diffusivity is much larger at maximum density when $L$ is large compared to $d$. Also, note the logarithmic scale in (a).

Figure 8

Monomer agents ($L=1$) and dimer agents ($L=2$) moving up to $d$ lattice sites: Solutions to Eq. (27) [medium-gray (red) curves] compared to the average of 50 000 simulations [dark-gray (blue) curves] at times $t=100,$ 300, and 500. (a) $L=1$ and $d=2$; (b) $L=1$ and $d=3$; (c) $L=2$ and $d=2$; (d) $L=2$ and $d=3$. The initial condition (shown) has agents between $181\le x\le 220$ for $d=2$ and agents between $281\le x\le 320$ for $d=3$. Note the different scales on the $x$ axis: for larger values of $d$ the outermost agents move farther, and so a larger region is needed to prevent interference with the boundaries for $t\le 500$. Equations (27) and (28) were solved using Matlab pdepe with $\delta x=0.1$. Arrows in (a) indicate increasing time.

Figure 9

Comparison of approximations: Eq. (11) [solid (blue) line] compared to simulations [(red) crosses] and the approximation that the occupancies of any two lattice sites at least $L$ lattice sites apart are independent [dashed (blue) line], for (a) $L=2$ and (b) $L=5$. Simulated results are not shown for high densities due to the difficulties of placing long agents randomly. Note the different scales on the $x$ axis: agents of different lengths result in different maximum densities.