de Mathématiques de Luminy
Positive circuits and two-dimensional spatial differentiation:
Application to the formation of sense organs in Drosophila.
We discuss a rule proposed by the biologist R.Thomas according to which the possibility for a genetic network (represented by a signed directed graph called a regulatory graph) to have several stable states implies the existence of a positive circuit. This result is already known for different models but always with a network of genes contained in a single cell. We consider here the genetics interactions between several cells represented by hexagonals located on a 2-dimensional infinite grid. With this configuration and in the Boolean case, we show that the existence of a positive circuit is a necessary condition for a specific form of multistationnarity, which naturally corresponds to spatial differentiation. We then illustrate this theorem through the example of the formation of sense organs in Drosophila.