Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

Abstract

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

Spatial moment description of birth-death-movement processes incorporating the effects of crowding and obstacles

“…http://dx.doi.org/10.1101/267708 doi: bioRxiv preprint first posted online Feb. 18, 2018; (5). For simplicity, the dispersal kernel µ (p) 1 (ξ) is chosen to be neighbor independent, and specified as a bivariate Gaussian distribution with zero mean and standard deviation σ …”

“…However, these first models that include cell migration are lattice-based, which means that the movement of individuals is restricted to an artificial lattice (Plank and Simpson 2012). Lattice-free moment dynamics models of cell migration and cell proliferation have also been presented (Middleton et al 2014;Binny et al 2015Binny et al , 2016aBinny et al , 2016b. Unlike the latticebased models in which volume exclusion effects are strictly enforced (Dyson and Baker 2015;Bruna and Chapman 2012), lattice-free models use interaction kernels to describe interactions and crowding effects.…”

Lattice-free models of directed cell motility

“…Refs. [33,34] modelled crowding using a neighbour-dependent interaction force, rather than a strict volume exclusion mechanism. These models 60 could include a global bias, as well as local neighbour-dependent bias, but the results presented applied to the case without global bias.…”

“…These models 60 could include a global bias, as well as local neighbour-dependent bias, but the results presented applied to the case without global bias. The different 62 individual-level mechanisms of [28,30,31,33,34] give rise to different nonlinear advection-diffusion equations or integro-differential equations for the average 64 agent density.…”

Identifying the necrotic zone boundary in tumour spheroids with pair-correlation functions

“…The PCF is a second-order summary statistic commonly used for analysing point patterns in cell biology [19][20][21][22][23][24]. Typically, PCFs describe the relative frequency of Euclidean distances between pairs of data points, indicating the extent of deviations from complete spatial randomness (CSR) [25][26][27][28][29].…”