title: A continuum approximation to an off-lattice individual-cell based model of cell migration and adhesion.
abstract: Cell-cell adhesion plays a key role in the collective migration of cells and in determining correlations in the relative cell positions and velocities. Recently, it was demonstrated that off-lattice individual cell based models (IBMs) can accurately capture the correlations observed experimentally in a migrating cell population. However, IBMs are often computationally expensive and difficult to analyse mathematically. Traditional continuum-based models, in contrast, are amenable to mathematical analysis and are computationally less demanding, but typically correspond to a mean-field approximation of cell migration and so ignore cell-cell correlations. In this work, we address this problem by using an off-lattice IBM to derive a continuum approximation which does take into account correlations. We furthermore show that a mean-field approximation of the off-lattice IBM leads to a single partial integro-differential equation of the same form as proposed by Sherratt and co-workers to model cell adhesion. The latter is found to be only effective at approximating the ensemble averaged cell number density when mechanical interactions between cells are weak. In contrast, the predictions of our novel continuum model for the time-evolution of the ensemble cell number density distribution and of the density-density correlation function are in close agreement with those obtained from the IBM for a wide range of mechanical interaction strengths. In particular, we observe 'front-like' propagation of cells in simulations using both our IBM and our continuum model, but not in the continuum model simulations obtained using the mean-field approximation.
authors: University of Heidelberg, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany; Center for Biological Systems Analysis, University of Freiburg, Habsburgerstr. 49, 79104 Freiburg, Germany.
authors: Alistair M
authors: AM
authors: Middleton
authors: Laboratory for Systems and Synthetic Biology, Dreijenplein 10, 6703HB Wageningen, The Netherlands.
authors: Christian
authors: C
authors: Fleck
authors: SynthSys and School of Biological Sciences, University of Edinburgh, EH9 3JR Edinburgh, UK. Electronic address: Ramon.Grima@ed.ac.uk.
authors: Ramon
authors: R
authors: Grima
mesh_terms: D000818:Animals; D002448:Cell Adhesion / Q000502:physiology; D002450:Cell Communication / Q000502:physiology*; D002452:Cell Count; D002465:Cell Movement / Q000502:physiology*; D002478:Cells, Cultured; D004285:Dogs; D006801:Humans; D055595:Mechanical Phenomena; D051379:Mice; D008962:Models, Theoretical
keywords: Cell movement; Correlations; Langevin equations; Partial differential equations; Stochastic simulations