Periodic solutions of gene networks with steep sigmoidal regulatory functions

•Existence and qualitative stability of periodic solutions in gene network models.•Determination of periodicity for a rigorous automated analyzer of gene networks.•Algorithm for establishing whether a periodic solution exists for a given cycle.•Quantitative stability of given periodic orbits may be assessed in some cases.

We address the question of existence and stability of periodic solutions in gene regulatory networks. The threshold-dependent network dynamics divides the phase space into domains and a qualitative description can be derived, specifying which transitions between domains can occur. Any periodic solution must follow a cyclic sequence of domains, but the problem of determining when such a cyclic sequence of domains contains a periodic solution, and when it is stable, has not been completely resolved, though results have been obtained before for restricted classes of networks. Here, we develop a method by which existence or non-existence of such solutions can be demonstrated analytically in any given example of a general class of gene networks with steep sigmoidal interactions, under the assumption that any gene product that regulates multiple other genes does so at distinct thresholds. Our method determines qualitative stability, but we also give a procedure that, where applicable, allows determination of quantitative stability of a periodic solution. This complements the previous development of a local analysis method for this class of systems, which allows computation of trajectories through any sequence of domains. Together the previous and current work form the basis for rigorous computer-aided assessment of qualitative dynamics of a very general class of gene network models. The ability to handle periodic solutions will also increase the applicability of such a computational tool to the design of synthetic networks.