Modelling cell aggregation using a modified swarm model

Cell aggregation and sorting are responsible for the formation, stability, and breakdown of tissue. A key mechanism for cell aggregations and sorting is that of cell-cell adhesion, a process by which cells bind or stick to each other through transmembrane proteins. This process is able to achieve cell sorting via the differential adhesion hypothesis (DAH) (Steinberg (1962b,a,c)). Armstrong et al. (2006) proposed a non-local advection model that was able to simulate the DAH. In their study, cells were modelled using a conservative system acting on cell density. The equations allowed for only two types of movement, random diffusive and directed adhesive movement with the adhesive movement taking into account cells within a finite sensing radius. Using the model with differing cell adhesion values they were able to simulate engulfment, partial engulfment, mixing, and sorting patterns between two cell types in both one and two dimensions. The aggregation of cells can be considered as a type of swarming, in that it is the collective behaviour of a large number of self propelled entities (Loan and Evans (1999)). Examples of macroscopic biological swarms include locust swarms, ungulate herds, fish schools, bird flocks, etc. Non-local swarming models have been used to successfully model these phenomena (see Bernoff and Topaz (2013)). Based on the principle of conservation of mass, a fixed population density moves at a velocity that arises as a result of social interactions (Mogilner and Edelstein-Keshet (1999)), giving rise to an equation of the form t + r(r(Q(x)) = 0; with Q(x) being a social potential function used to describe the social interactions between individuals. In this paper we look at the Armstrong et al. (2006) model of cell-cell adhesion and recreate it by extending the swarm modelling techniques to equations of the form t + r(r(Q(x)f = 0: In doing so we find that by modelling in this way we are able to capture the same qualitative behaviour as the original model with a vastly reduced computational cost. We also derive a numerical scheme to simulate the model in one dimension in such a way that it can be easily adapted to other swarm problems. We find that the convergence rate of the numerical scheme is greater than 1.7 in all of the scenarios presented.