Continuum models describing probabilistic motion of tagged agents in exclusion processes

TL;DR

This paper develops a continuum framework for probabilistic motion in crowding-limited exclusion processes by deriving a PDE for the PDF of tagged agents, $P^{(s)}(x,y,t)$, and coupling it to macroscopic densities $C^{(s)}(x,y,t)$ under total density $T= extstyleigl( extstyle\sum_s C^{(s)}igr)$. The authors start from a stochastic lattice-based ABM with crowding, motility bias, and proliferation, and obtain continuum limits that yield nonlinear advection-diffusion equations for densities and a corresponding PDF equation for tagged trajectories, with flux terms that explicitly include crowding through factors like $(1-C)$ or $(1-T)$. They demonstrate through extensive numerical comparisons that the PDE for $P^{(s)}$ captures both the mean trajectory and distributional variability of tagged agents across unbiased/bias cases and with/without proliferation, and they discuss the accuracy relative to existing moment-based approaches, highlighting the added value of full PDFs for likelihood-based inference and trajectory-data analysis. The framework generalizes to multiple subpopulations, enabling analysis of heterogeneous migration and proliferation, and lays the groundwork for parameter inference and model selection from combined density and trajectory data, with potential extensions to lattice-free formulations and lineage tracking. Overall, the work provides a physically interpretable, computationally efficient method to quantify and predict trajectory distributions in crowded cellular systems.

Abstract

Lattice-based random walk models are widely used to study populations of migrating cells with motility bias and proliferation. Crowding is typically represented by volume exclusion, where each lattice site can be occupied by at most one agent and conflicting moves are aborted. This framework enables simulations that yield both population-level spatiotemporal agent density profiles and individual agent trajectories, comparable to experimental cell-tracking data. Previous continuum models for tagged-agent trajectories captured trajectory information only, and overlooked any measure of variability. This is an important limitation since trajectory data is inherently variable. To address this limitation, here we derive partial differential equations for the probability density function of tagged-agent trajectories. This continuum description has a clear physical interpretation, agrees well with distributional data from stochastic simulations, reveals the role of stochasticity in different contexts, and generalises to multiple subpopulations of distinct agents.

Figures (4)

• Figure 1: Schematic and experimental motivation. (a) Schematic scaled density profile showing the spatial expansion of a population of cells, undergoing migration and proliferation, leading to the macroscopic propagation of a density front in the positive $x$-direction. (b) In vivo tagged cell trajectories reported by Druckenbrod and Epstein Druckenbrod2006. The direction of the population density front motion is shown with the blue arrow and individual cell trajectories within the population are given by the red and green traces. (c) In vitro tagged cell trajectories in a wound healing experiment reported by Cai et al. Cai2007. The direction of the population density front motion is shown with the blue arrow and individual trajectories show that cells at the edge of population front are biased to move in the same direction as the population front. All images are reproduced with permission.
• Figure 2: Comparison of discrete and continuum models for a motile population: (a)-(c) without bias or proliferation; (d)-(f) with bias and without proliferation; (g)-(i) without bias and with proliferation; (j)-(l) with bias and proliferation. Left column of plots show agent density $C(x,t)$ at $t=300$; middle column shows distribution of the location at $t=300$ of tagged agents initially located near the left-hand leading edge ($x_0=-18$, blue), in the centre of the population ($x_0=0$, red) and near the right-hand leading edge ($x_0=18$, yellow); right column shows the median and 90% PrI of tagged agent locations over time time according to the ABM (thick solid curve = median, shaded band = 90% PrI) and the PDE (thin solid curve = median, dashed curves = 90% PrI). Vertical dashed lines in middle column show the initial location of tagged agents. Discrete parameter values $M = 1$ and $\rho_x=Q=0$ corresponding to $D=0.25$ and $v=\lambda=0$ for a simulation with $\Delta = \tau = 1$.
• Figure S1: Graphs of the mean $\langle x(t)\rangle$ and standard deviation $\sigma_x(t)$ of tagged agent locations over time in the ABM (solid curves), PDE model (dashed curves) and the SLH approximation [27] (dot-dash curves), for agents initially located at $x_0=-18$ (blue), $x_0=0$ (red) and $x_0=18$ (yellow). The four rows of plots show the four cases investigated: (a-b) unbiased, no proliferation; (c-d) biased, no proliferation; (e-f) unbiased, with proliferation; (g-h) biased, with proliferation. Note in (b) and (f) the blue and yellow curves for standard deviation coincide almost exactly for all three models due to the symmetry in the model.
• Figure S2: Comparison of ABM and PDE results for a test case in which all lattice sites initially occupied with probability $C_0=0.5$ and with no bias or proliferation. (a) Agent density $C(x,t)$ at $t=300$. (b) Distribution of the location at $t=300$ of tagged agents initially located at $x_0=-18$ (blue), $x_0=0$ (red) and $x_0=18$ (yellow). Vertical dashed lines show the initial location of tagged agents. (c) Median and 90% PrI of tagged agent locations as a function of time: ABM results are shown as thick solid curve (median) and shaded band (90% PrI); PDE results are shown as thin solid curve (median) and dashed curves (90% PrI). Discrete parameter values $M = 1$ and $\rho_x=Q=0$ corresponding to $D=0.25$ and $v=\lambda=0$ for a simulation with $\Delta = \tau = 1$. Notice that the PDE solution for $P(x,t)$ slightly overestimates the variance in the distribution of tagged agent locations in the ABM.