title: 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.
authors.affiliation: School of Mathematical Sciences, Queensland University of Technology (QUT) Brisbane, Queensland, Australia.
authors.forename: Oleksii M
authors.initials: OM
authors.lastname: Matsiaka
authors.affiliation: School of Mathematical Sciences, Queensland University of Technology (QUT) Brisbane, Queensland, Australia.
authors.forename: Catherine J
authors.initials: CJ
authors.lastname: Penington
authors.affiliation: Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, United Kingdom.
authors.forename: Ruth E
authors.initials: RE
authors.lastname: Baker
authors.affiliation: School of Mathematical Sciences, Queensland University of Technology (QUT) Brisbane, Queensland, Australia. Electronic address: matthew.simpson@qut.edu.au.
authors.forename: Matthew J
authors.initials: MJ
authors.lastname: Simpson
mesh_terms: D000818:Animals; D002465:Cell Movement / Q000502:physiology*; D006801:Humans; D008954:Models, Biological; D013269:Stochastic Processes
keywords: Cell migration; Kirkwood superposition approximation; Mean field approximation; Moment dynamics approximation