Combinatorial representation of parameter space for switching networks

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. A computational model based on switching networks is generated from the regulatory network. The phase space dimension of this model equals the number of nodes and the associated parameter space consists of one parameter for each node (a decay rate), and three parameters for each edge (low level of expression, high level of expression, and 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 a compact representation of the global dynamics called an annotated Morse graph that identifies recurrent and nonrecurrent dynamics. 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 dynamical 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 of 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.

Figure 1

(a) Regulatory Network (RN); (b) a set of equations with a particular choice of parameters for RN; (c) phase space; (d) wall graph; (e) domain graph; (f) wall-domain graph; (g) a set of strongly connected components of either of the wall, domain or wall-domain graph; (h) Morse graph representing strongly connected components.

Figure 2

(a) Coarse wall graph where all incoming faces map to all exit faces. (b) Finer wall graph with a disallowed incoming-exit arrow from bottom to top. (c) Finer wall graph with disallowed arrow from left to right.

Figure 3

(a) A self activating one node network. (b) Phase plane for switching network. (c) Annotated Morse graphs: MG(1) has a single node generated by an attracting cell for which the fixed point is less than the threshold and annotated by FP OFF; MG(3) has a single node generated by an attracting cell for which the fixed point is greater than the threshold and annotated by FP ON; and MG(2) has two minimal nodes generated by attracting cells in one of which the fixed point is less than the threshold and in the other the fixed point is greater than the threshold. (d) Parameter graph.

Figure 4

Left: Repressilator. Right: Bistable repressilator.

Figure 5

Left: Repressilator parameter graph with same-colored parameter nodes corresponding to the same Morse graph. Right: Hill function simulation for the repressilator satisfying the inequalities of parameter node 13 with l_i,j = 0.5, θ_i,j = 1.0, u_i,j = 1.5 (see Equation (21)). The Hill exponent is n = 9.

Figure 6

DSGRN Morse graphs for the repressilator and the number of parameter regions at which each Morse graph is realized.

Figure 7

Bistable repressilator parameter graph with colors corresponding to partitioned Morse graph continuation classes. Class A: single stable fixed points; Class B: bistability; Class C: stable cycle; Class D: unstable cycle with a stable fixed point.

Figure 8

DSGRN Morse graphs for the bistable repressilator.

Figure 9

Hill function simulation for the bistable repressilator at parameter node 151. See the text for parameter choices. The Hill exponent is n = 10.

Figure 10

Subnetwork of key species of the p53 signaling network.

Figure 11

Hill function simulation for the p53 model at parameter node 40535. See the text for parameter choices. The Hill exponent for every nonlinearity is n = 8.