Spatial moment dynamics for collective cell movement

4 The ability of cells to undergo collective movement plays a fundamental role in tissue repair, 5 development and cancer. Interactions occurring at the level of individual cells may lead to the 6 development of spatial structure which will affect the dynamics of migrating cells at a population 7 level. Models that try to predict population-level behaviour often take a mean-field approach, which 8 assumes that individuals interact with one another in proportion to their average density and ignores 9 the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual10 based model (IBM) that uses random walk theory to model the stochastic interactions occurring 11 at the scale of individual migrating cells. We incorporate a mechanism for local directional bias 12 such that an individual’s direction of movement is dependent on the degree of cell crowding in 13 its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment 14 theory to develop a population-level model which accounts for spatial structure and predicts how 15 these individual-level interactions propagate to the scale of the whole population. The IBM is used 16 to derive an equation for dynamics of the second spatial moment (the average density of pairs of 17 cells) which incorporates the neighbour-dependent directional bias and we solve this numerically for 18 a spatially homogeneous case. 19