Structural principles for periodic orbits in glass networks

A piecewise-linear differential equation model framework for gene regulatory interactions (Glass networks) has allowed considerable analysis of qualitative dynamics in such systems, including periodicity, an important class of regulatory behaviors. Here, we present new results relating the structure of the network to its dynamics (structural principles). The structure we refer to is the state space of the network, which is a digraph on an n-cube in the case of a single threshold per gene. In particular, we show that for a wide class of cycles in the state space there exist parameter values, consistent with the graph structure, for which a periodic orbit exists in the network. For some classes, we show in addition that stable periodic orbits exist. These results extend greatly earlier work by Glass and Pasternack (J Math Biol 6:207–223, 1978).