Global analysis of regulatory network dynamics: equilibria and saddle-node bifurcations

In this paper we describe a combined combinatorial/numerical approach to studying equilibria and bifurcations in network models arising in Systems Biology. Often interactions are only coarsely known in terms of a regulatory or signalling network topology. Consequently, ODE models of the dynamics suffer from high dimensional parameters which presents a significant obstruction to studying the global dynamics via numerical methods. Given a network topology describing state variables which regulate one another via monotone and bounded functions, we first use the Dynamic Signatures Generated by Regulatory Networks (DSGRN) software to obtain a combinatorial description which summarizes the dynamics. This summary is coarse but global and we use this information as a first pass to identify "interesting" subsets of parameters in which to focus. We construct an associated ODE model with high parameter dimension using our Network Dynamics Modeling and Analysis (NDMA) Python library. We introduce algorithms for efficiently investigating the dynamics in these ODE models restricted to these parameter subsets. Finally, we perform a statistical validation of the method and several interesting dynamical applications including finding saddle-node bifurcations in a 54 parameter model.