Discrete model of periodic pattern formation through a combined autocrine-juxtacrine cell signaling.

We model the formation of periodic patterns of gene expression in epithelial cell sheets driven by autocrine signaling coupled to juxtacrine lateral inhibition. The mathematical model is based on a continuous description of the extracellular matrix and a discrete cell-level description of the layer of cells, coupling the dynamics of diffusible ligands to the threshold-controlled cell-autonomous regulation with randomly fluctuating production rates. The results of numerical simulations indicate that propagating signaling waves emerge in a certain parametric domain, leading to the formation of a variety of either periodic or irregular patterns. For some selections of parameters, a propagating stripe of uniform expression leaves in its wake stationary periodic arrays. Coupling of autocrine and juxtacrine cell communication is essential for the pattern regularity and for the selection of expression patterns. Moreover, weak but non-vanishing noise levels are essential for the formation of regular patterns. Additional autocrine and cell-autonomous regulatory interactions can be introduced to increase the spacing of a periodic pattern.