Embryonic lateral inhibition as optical modes: An analytical framework for mesoscopic pattern formation

Cellular checkerboard patterns are observed at many stages of embryonic development. We study an analytically tractable model for lateral inhibition and show that the steady states are analogous to optical phonons at the Gamma point, which have the wave number k = 0. We study the cases of cells arranged in linear and hexagonal lattices. To determine how the final pattern is selected it is necessary to take into account the granularity of the pattern and, analogously to solid-state physics, to redefine the basis and lattice sites in terms of a periodic crystal. The sites and basis are determined by looking at the symmetries of inhibitory interactions between cells. The redefined basis for cells placed in a linear lattice is composed by two cells which are embedded in another linear lattice, while for cells placed in a hexagonal lattice the redefined basis consists of three cells embedded in another hexagonal lattice. The pattern in hexagonal lattices can be driven into three different states: two of those states are periodic checkerboards and a third in which both periodic states coexist. These observations provides new predictions for experiments.