Continuum approximations for lattice-free multi-species models of collective cell migration

Abstract

Cell migration within tissues involves the interaction of many cells from distinct subpopulations. In this work, we present a discrete model of collective cell migration where the motion of individual cells is driven by random forces, short range repulsion forces to mimic crowding, and longer range attraction forces to mimic adhesion. This discrete model can be used to simulate a population of cells that is composed of K ≥ 1 distinct subpopulations. To analyse the discrete model we formulate a hierarchy of moment equations that describe the spatial evolution of the density of agents, pairs of agents, triplets of agents, and so forth. To solve the hierarchy of moment equations we introduce two forms of closure: (i) the mean field approximation, which effectively assumes that the distributions of individual agents are independent; and (ii) a moment dynamics description that is based on the Kirkwood superposition approximation. The moment dynamics description provides an approximate way of incorporating spatial patterns, such as agent clustering, into the continuum description. Comparing the performance of the two continuum descriptions confirms that both perform well when adhesive forces are sufficiently weak.In contrast, the moment dynamics description outperforms the mean field model when adhesive forces are sufficiently large. This is a first attempt to provide an accurate continuum description of a lattice-free, multi-species model of collective cell migration.

Discrete and continuum approximations for collective cell migration in a scratch assay with cell size dynamics

“…This choice of notation allows us to distinguish between the positions of agents in the discrete simulations and the coordinates of fixed points in the continuum description. Furthermore, it is consistent with our previous work which does not involve dynamical changes in agent size (Matsiaka et al 2017).…”

“…Using the framework presented it would be possible to consider different subpopulations with different initial sizes, and to consider different subpopulations that grow at different rates. This kind of model could be described using a more complicated multi-species framework (Matsiaka et al 2017). However, since this is the first time that a model of collective cell migration in a scratch assay that incorporates crowding effects and cell size dynamics has been explored, we leave this extension to the multi-species case for future consideration.…”

“…Middleton et al 2014;Matsiaka et al 2017). The focus of the current study is to consider the continuum approximations for the case where the agent diameter increases with time.…”

Asymptotic expansion approximation for spatial structure arising from directionally biased movement

“…Similar neighbour-dependent mechanisms have been modelled by Middleton et al [11], Matsiaka et al [12], Surendran et al [13] and Johnston and Painter [14]. All of these studies resort to computationally expensive solutions of partial integrodifferential equations for the second spatial moment, pair density or comparable correlation function.…”

Living in groups: Spatial-moment dynamics with neighbour-biased movements

“…A zone of repulsion is also supported by data (Krause et al, 2002). Zones of repulsion and attraction have also been modelled in the cell behaviour literature, for example using the Lennard-Jones kernel (Jeon et al, 2010) and the Morse potential (Middleton et al, 2014;Matsiaka et al, 2017). Our model incorporates and builds on these ideas, including the possibility for multiple zones of attraction and repulsion with different spatial scales.…”