Incorporating pushing in exclusion-process models of cell migration

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.

Figure 1

Possible movement and proliferation events in the nonpushing and basic pushing exclusion process models. Occupied sites are black and unoccupied sites are white. Sites of the lattice for which the occupancy is not important for the depicted event are shown in gray. Possible movement or proliferation directions are denoted by green arrows. (a) The selected agent at site $(i,j)$ is free to move (or to place a daughter agent, respectively) into any of its neighboring unoccupied sites, with probability $P^m/4$ $(P^p/4$, respectively). (b) The selected agent at site $(i,j)$ has been chosen to move or proliferate to the right into an occupied site $(i+1,j)$. With pushing probability, $Q^m$ $(Q^p$ for proliferation) such that $0\le Q^m,Q^p\le 1$ this agent, originally at $(i,j)$, pushes the agent at $(i+1,j)$ into unoccupied site $(i+2,j)$ and takes its place (leaving behind a daughter agent at $(i,j)$ in the case of proliferation).

Figure 2

The evolution of the lattice occupancy for [(a)–(c)] the basic exclusion process model $\left(Q^m=0\right)$ and [(d)–(f)] the exclusion process model with agent-agent pushing $\left(Q^m=1\right)$. Agents are less clumped when they are allowed to push each other, although they do not appear to spread much further than in the nonpushing case. For these figures and for other on-lattice individual-level results presented later we have carried out simulations on a lattice with $L_x=L_y=100$ and reflecting boundary conditions on all sides. We simulate on a sufficiently large domain that any boundary effects are negligible. For clarity we only present the $21\times 100$ cross section from the middle of the domain $\left(1\le x\le 100,40\le y\le 60\right)$ at each time point. All lattice sites in the region $41\le x\le 60$ are initially occupied. Simulation parameters are $\tau =1,\mathrm{\Delta }=1,P^m=0.2,P^p=0$.

Figure 3

A comparison of the column-averaged density profiles of the agents in the ILM (red dashed curve) and the corresponding PDE (3) (black continuous curve) for (a) simple diffusion in the absence of pushing $\left(Q^m=0\right)$ and (b) diffusion with basic agent-agent pushing $\left(Q^m=0.5\right)$. The profiles are visualized at times $t=0,t=50$, and $t=200$. The movement probability in these simulations is $P^m=0.8$. All other parameters, domain specifications, boundary conditions, and initial conditions are as in Fig. 2. All individual-level results are averaged over 100 repeats.

Figure 4

Type II adjacent pushing events in which the agent being pushed can move to any unoccupied neighboring lattice site. The occupied sites are in black, whereas the unoccupied neighboring sites are in white. As before, sites which do not affect the movement event are in gray. The selected agent at site $(i,j)$ attempts to push to the right into the occupied site $(i+1,j)$. If the push is successful (with probability $Q^m)$, then the probability with which the pushed agent moves into an unoccupied lattice site depends upon which neighboring sites are unoccupied. (a) The three surrounding sites about the pushed agent are all unoccupied and the pushed agent moves into any of them with probability $1/3$. (b) Only two of the three possible sites are available and the pushed agent moves into either of them with probability $1/2$. (c) There is only one possible site for the pushed agent to move into, which it does with certainty.

Figure 5

A comparison of the averaged column density profiles of the agents in the ILM and the corresponding PDE (6) with coefficients given in Table 1 for (a) type I adjacent pushing, (b) type II adjacent pushing, and (c) multiple-agent pushing $\left(K=2\right)$. The profiles are visualized at times $t=0,t=50$, and $t=200$. The comparison between the averaged individual results and the PDE is good in all three cases, but there is a slight underestimation of the peak density of the ILM by the PDE for the case of multiple-agent pushing. Parameters are (movement probability) $P^m=0.8$ and (pushing probability) $Q^m=0.5$ and for (c) pushing probabilities $Q_1^m=Q_2^m=0.5$. All other parameters, domain specifications, boundary conditions, and initial conditions are as in Fig. 2. All individual-level results are averaged over 100 repeats.

Figure 6

An agent attempting to move into an occupied lattice site can push up to $K$ agents in a line to make room for itself. The selected agent at site $(i,j)$ attempts to push to the right into the occupied site $(i+1,j)$. Sites $(i+2,j)$ and $(i+3,j)$ are also occupied. With probability $Q_3^m$ the focal agent, originally at $(i,j)$, linearly pushes the agents blocking its path and moves into the vacated site $(i+1,j)$. Figure descriptions are as in Fig. 1.

Figure 7

The evolution of the HDE for each of the models presented in Table 1. Model and simulation parameters and descriptions are as in previous figures. In each case the HDE is low for the duration of the simulation.

Figure 8

Schematics of the possible occupancy changes of position $x$ due to spontaneous movement and pushing. In each panel the black agent is the agent which is moving, the red agent (if present) is the agent being pushed, and the gray agent is at the closest position which another agent can be and not affect the movement or pushing of the other agents. Where it appears, $0<\varepsilon <\mathrm{\Delta }$. (a) Occupancy at $x$ decreases as an agent spontaneously moves away. (b) Occupancy decreases as an agent spontaneously moves away and pushes another agent. (c) Occupancy decreases as an agent is pushed away from $x$ by another agent's movement. (d) Occupancy increases as an agent spontaneously moves to $x$. (e) Occupancy increases as an agent spontaneously moves to $x$ and pushes another agent. (f) Occupancy increases as one agent is pushed to position $x$ by the movement of another agent. Note that we have only shown occupancy changes due to rightward agent movement. Equivalent scenarios exist for leftward movements.

Figure 9

A comparison of the averaged and smoothed agent density in the one-dimensional ILM with $N=20$ cells and the corresponding PDE (12) for a range of values of the pushing probability $Q^m$. The profiles are visualized at times $t=50,t=100$, and $t=200$. The comparison between the PDE and the ILM reduces in quality as we increase the pushing probability, $Q^m$. Parameters for these simulations were chosen as $R=0.17,d=0.1,\tau =0.04,P^m=1$. In each ILM simulation agents are initialized quasirandomly in the region [35, 65] so no agents overlap with each other. The initial condition for the PDE is taken to be the average initial condition in the ILM. Movements which would cause an agent to leave the domain are aborted. All individual-level results are averaged over $10000$ repeats.

Figure 10

The evolution of the HDE for the off-lattice model with a range of values of pushing probability, $Q^m$. Model and simulation parameters and descriptions are as in Fig. 9. The scenarios with the best correspondence to the PDE are those with no or very little pushing (red continuous line and blue dotted line, respectively).