A Stochastic Model for Cell Sorting and Measuring Cell–Cell Adhesion

Spontaneous sorting-out of cells of different types is studied by a lattice-structured model of continuous-time Markovian transition. Cells of two types, called black and white, are arranged on a regular lattice, and the adhesion between cells in contact depends on their types. Cells exchange their location with neighbors at random times, and the rate is affected by the difference in cell-to-cell adhesion caused by the exchange. The model has the unique equilibrium probability distribution of spatial patterns. According to the analysis equivalent to statistical thermodynamics, the spatial pattern should show a clear coarse grained segregation if the ratio of differential adhesion to the intensity of random movement (A/m) exceeds a critical value. However, even for the parameter range in which spatial pattern is predicted to be uniform, the cells of the same type tend to be in contact more often then in a random distribution, forming many small clusters of the same cell types. To quantify this microscopic tendency of aggregation, two statistics are introduced: (1) the fraction of black cells in the neighbor of a randomly chosen black cell (qB B), and (2) the number of isolated black cells (IBC). The equilibrium value of these statistics can be predicted accurately using a pair-approximation method. However, ifA/mis large, the convergence ofqB Bto the equilibrium value takes an extremely long time. IBC is faster in convergence thanqB B. These statistics can be used to measure the strength of differential adhesion relative to random movement (A/m) is based on an observed spatial pattern of cells.