Combinatorial Representation of Parameter Space for Switching Systems

Abstract

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network (DSGRN) database. The motivation stems from urgent need to understand the global dynamics of biologically relevant signal transduction/gene regulatory networks that have at least 5 to 10 nodes, involve multiple interactions, and decades of parameters. The input to the database computations is a regulatory network, i.e.\ a directed graph with edges indicating up or down regulation, from which a computational model based on switching networks is generated. The phase space dimension equals the number of nodes. The associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low and high levels of expression, and a threshold at which expression levels change). Since the nonlinearities of switching systems are piece-wise constant, there is a natural decomposition of phase space into cells from which the dynamics can be described combinatorially in terms of a state transition graph. This in turn leads to compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent. The focus of this paper is on the construction of a natural computable finite decomposition of parameter space into domains where the annotated Morse graph description of dynamics is constant. We use this decomposition to construct an SQL database that can be effectively searched for dynamic signatures such as bistability, stable or unstable oscillations, and stable equilibria. We include two simple 3-node networks to provide small explicit examples of the type information stored in the DSGRN database. To demonstrate the computational capabilities of this system we consider a simple network associated with p53 that involves 5-nodes and a 29-dimensional parameter space.