Abstract
Mathematical models of collective cell movement often neglect the effects of spatial structure, such as clustering, on the population dynamics. Typically, they assume that individuals interact with one another in proportion to their average density (the mean-field assumption) which means that cell–cell interactions occurring over short spatial ranges are not accounted for. However, in vitro cell culture studies have shown that spatial correlations can play an important role in determining collective behaviour. Here, we take a combined experimental and modelling approach to explore how individual-level interactions give rise to spatial structure in a moving cell population. Using imaging data from in vitro experiments, we quantify the extent of spatial structure in a population of 3T3 fibroblast cells. To understand how this spatial structure arises, we develop a lattice-free individual-based model (IBM) and simulate cell movement in two spatial dimensions. Our model allows an individual’s direction of movement to be affected by interactions with other cells in its neighbourhood, providing insights into how directional bias generates spatial structure. We consider how this behaviour scales up to the population level by using the IBM to derive a continuum description in terms of the dynamics of spatial moments. In particular, we account for spatial correlations between cells by considering dynamics of the second spatial moment (the average density of pairs of cells). Our numerical results suggest that the moment dynamics description can provide a good approximation to averaged simulation results from the underlying IBM. Using our in vitro data, we estimate parameters for the model and show that it can generate similar spatial structure to that observed in a 3T3 fibroblast cell population.
Highlights
Collective cell movement is integral to tissue repair (Martin, 1997; Shaw & Martin, 2009), embryonic development (Kurosaka & Kashina, 2008), the immune response (Rørth, 2009) and cancer (Friedl & Wolf, 2003)
To explore whether our model is capable of generating spatial structure in a simulated cell population we average results from repeated simulations of the individual-based model (IBM) and compute a periodic pair-correlation function (PCF) CIBM(r) as outlined in ‘Pair-correlation function’
We compare this to numerical solutions of our spatial moment model to examine whether it provides a good approximation to the underlying stochastic process
Summary
Collective cell movement is integral to tissue repair (Martin, 1997; Shaw & Martin, 2009), embryonic development (Kurosaka & Kashina, 2008), the immune response (Rørth, 2009) and cancer (Friedl & Wolf, 2003). When detected by neighbouring cells these signals can have a repulsive or attractive effect on an individual’s direction of movement (Painter & Hillen, 2002), or affect the rate at which a cell will move (Cai, Landman & Hughes, 2006) Physical forces, such as cell–cell adhesion (Trepat et al, 2009; Tambe et al, 2011), and crowding effects influence movement (Abercrombie, 1979; Plank & Simpson, 2012). These interactions may generate spatial structure in a cell population which will in turn affect the collective dynamics (Plank & Law, 2015). Repulsive forces such as chemorepellant signals can cause cells to segregate (Kay, Chu & Sanes, 2012; Keeley et al, 2014)
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