Exclusion processes: Short-range correlations induced by adhesion and contact interactions

Abstract

We analyze the out-of-equilibrium behavior of exclusion processes where agents interact with their nearest neighbors, and we study the short-range correlations which develop because of the exclusion and other contact interactions. The form of interactions we focus on, including adhesion and contact-preserving interactions, is especially relevant for migration processes of living cells. We show the local agent density and nearest-neighbor two-point correlations resulting from simulations on two dimensional lattices in the transient regime where agents invade an initially empty space from a source and in the stationary regime between a source and a sink. We compare the results of simulations with the corresponding quantities derived from the master equation of the exclusion processes, and in both cases, we show that, during the invasion of space by agents, a wave of correlations travels with velocity v(t) ∼ t −1/2 . The relative placement of this wave to the agent density front and the time dependence of its height may be used to discriminate between different forms of contact interactions or to quantitatively estimate the intensity of interactions. We discuss, in the stationary density profile between a full and an empty reservoir of agents, the presence of a discontinuity close to the empty reservoir. Then, we develop a method for deriving approximate hydrodynamic limits of the processes. From the resulting systems of partial differential equations, we recover the self-similar behavior of the agent density and correlations during space invasion.

Lattice-free descriptions of collective motion with crowding and adhesion

“…Fernando's analysis showed that each of the very different adhesive collective migration mechanisms gave rise to a nonlinear diffusion equation, and they presented an extensive list of the different nonlinear diffusivity functions for each discrete mechanism [14]. Since these initial investigations, others have also considered different aspects of modeling adhesive collective migration processes in a lattice-based framework [15][16][17][18].…”

Cell density and cell size dynamics during in vitro tissue growth experiments: Implications for mathematical models of collective cell behaviour

“…An area of great interest to the mathematical biology community is to understand how the averaged properties of a particular discrete process can be described using continous models such as ordinary differential equations and partial differential equations [30,20,1,[40][41][42]. While some previous work has been completed on understanding how to derive approximate differential equation description of systems with different sized agents [43,44], none of these previous studies considered systems that were composed of several types of different sized agents like we have shown in Fig.…”

“…Most models used to interpret two-dimensional tissue growth experiments do not apply to our experimental data. Standard continuous models neglect to account for individual-level properties, and standard discrete models assume that the size of the agents does not change dynamically [30,36,37,27,38,39,34,31,8,9].…”

“…The increase in average cell size is an interesting feature that is typically neglected by standard continuous [2,5,18] and discrete [30,1,31,3] mathematical models.…”

“…The discrete model consists of a two-dimensional square lattice, with lattice spacing D. Lattice sites are indexed ði; jÞ [1,3,4] such that each lattice site is associated with a position ðx; yÞ ¼ ðiD; jDÞ. The discrete model is an exclusion process [30] which explicitly incorporates crowding effects by allowing each lattice compartment to be occupied by, at most, a single agent or a portion of a single agent. For simplicity we will assume that the system of interest contains two types of agents; however, we note that this assumption can be relaxed and it is possible to generalise our model so that we can simulate any number of different types of agents.…”