Pattern Formation by Lateral Inhibition in Large-Scale Networks of Cells

A common pattern formation mechanism in multicellular organisms is lateral inhibition where the cells inhibit a particular function in their immediate neighbors through a contact signaling mechanism. We present a broad dynamical model that captures this mechanism and reveal the key properties that are necessary for patterning. The model consists of subsystems representing the biochemical reactions in individual cells, interconnected according to an undirected graph describing which cells are in contact. By using input-output properties of the subsystems and the spectral properties of the adjacency matrix for the contact graph, we provide verifiable conditions that determine when the spatially homogeneous steady-state loses its stability and what types of patterns emerge. We first study bipartite contact graphs and investigate the existence and stability of a "checkerboard" pattern that demonstrates the ability of neighboring cells to adopt distinct fates under lateral inhibition. Next, we exhibit a monotonicity property of the interconnected model for bipartite graphs and, under mild additional conditions, prove that almost every solution converges to one of the possible steady-states. Finally, we reveal a pattern that emerges in odd-length cycles, which are quintessential examples of non-bipartite graphs. A salient feature of this paper is that, unlike the existing literature, it does not restrict the number of cells and reactants.