A mathematical model of pattern formation

Abstract

This paper presents an explicit mathematical model describing pattern formation in monolayer epithelia. The approach is a generalization of the equations describing soap bubble configurations (Plateau, 1873; Thompson, 1917; Almgren & Taylor, 1976) that allows adjacent cells to adhere with differing intensities (Steinberg, 1962, 1978). The model is a system of simultaneous non-linear equations that considers cell-cell interactions in a two-dimensional sheet. The implementation involves using the equations of the model to predict explicitly the energy-minimizing configuration of a system of cells, based on the adhesivity of their membranes. The model can thus be used to explore the effects of varying adhesions on the dynamics of pattern formation. Following Chichilnisky (1985), such a descriptive system is introduced in this paper, and its predictive properties explored.