Persistent exclusion processes: Inertia, drift, mixing, and correlation

Abstract

In many biological systems, motile agents exhibit random motion with short-term directional persistence, together with crowding effects arising from spatial exclusion. We formulate and study a class of lattice-based models for multiple walkers with motion persistence and spatial exclusion in one and two dimensions, and use a mean-field approximation to investigate relevant populationlevel partial differential equations in the continuum limit. We show that this model of a persistent exclusion process is in general well described by a nonlinear diffusion equation. With reference to results presented in the current literature, our results reveal that the nonlinearity arises from the combination of motion persistence and volume exclusion, with linearity in terms of the canonical diffusion or heat equation being recovered in either the case of persistence without spatial exclusion, or spatial exclusion without persistence. We generalise our results to include systems of multiple species of interacting, motion-persistent walkers, as well as to incorporate a global drift in addition to persistence. These models are shown to be governed approximately by systems of nonlinear advection-diffusion equations. By comparing the prediction of the mean-field approximation to stochastic simulation results, we assess the performance of our results. Finally, we also address the problem of inferring the presence of persistence from simulation results, with a view to application to experimental cell-imaging data.

From a microscopic solution to a continuum description of active particles with a recoil interaction

“…Perhaps the simplest interaction is hard-core exclusion, which in passive (equilibrium) systems serves only to reduce the volume available to particles to explore: the statistical weights of the accessible microstates remain unchanged. By contrast, a hard-core repulsion between persistent particles induces an effective attraction [36] which can lead to particles clustering [37][38][39][40][41], consistent with the prediction of motility-induced phase separation [42].…”

Jamming of multiple persistent random walkers in arbitrary spatial dimension

“…We consider specifically the case of hard-core exclusion, which we implement by placing the particles on a hypercubic lattice in d dimensions, and disallow any hops that would lead to two particles occupying the same site. This model (and variants with a softer exclusion constraint) has been studied from a variety of viewpoints [7,[14][15][16][17][18][19][20][21] and is sometimes referred to as a persistent exclusion process. Macroscopically, it is expected to exhibit a motility-induced phase separation [22], which arises generically in active particle systems from a feedback between particles accumulating where the propulsion speed is low and this speed decreasing due to crowding from nearby particles.…”

Detection and characterization of chemotaxis without cell tracking

“…Techniques from spatial statistics, such as the pair correlation function, are increasingly being applied to agent-based processes [54,[63][64][65][66][67][68][69][70][71][72][73]. Our approach employs a novel combination of both spatial statistics and functional data analysis.…”

“…Recent results have demonstrated that drift-diffusion PDEs can be produced to approximate the evolution of agent-based processes exhibiting bias and persistence in the setting of modelling bacterial chemotaxis [74]. In addition, drift-diffusion type PDEs can be produced to approximate the evolution of a lattice-based agent-based process that includes bias, persistence, and exclusion, with the drift and diffusion terms being dependent on agent density [72]. Determining whether our approach can royalsocietypublishing.org/journal/rsif J. R. Soc.…”