Combinatorial Representation of Parameter Space for Switching Networks

Abstract

We describe the theoretical and computational framework for the Dynamic Signatures for Genetic Regulatory Network ( ) 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 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.

Model Rejection and Parameter Reduction via Time Series

“…By Definition 1.1, the restrictions x| A and x| B onto A = x −1 (ξ) and B = x −1 (ξ ) are again trajectories. By Definition 4.4, x| A and x| B are solutions to Equation (4). Such solutions are monotonic, so it follows that x| A and x| B are increasing.…”

Linear extensions of a partial order subject to algebraic constraints

“…We have introduced a construction and counting problem important to the dynamical classification of switching systems across global parameter space [5]. We have shown that the parameterization of a switching system gives rise to a natural partial order that is subject to algebraic constraints.…”

“…Typically, a regulatory network is represented as an annotated directed graph, where each vertex represents the concentration of a molecular species (e.g. protein, mRNA), and an edge from node i to node j indicates that i directly regulates j, as in [5]. Definition 2.1.…”

“…In particular, these thresholds divide phase space into finitely many domains with the property that flow is mono-directional across each face of each domain. Using this domain structure, Cummins et al [5] have recently developed a method to describe the coarse dynamics of the system (1) over the parameter space. This description of the system's dynamics centers on an object called the parameter graph, a finite undirected graph which discretizes parameter space into a finite set of parameter nodes; each of these is a set of parameters that all give rise to the same direction of flow across each domain's boundary in phase space, and consequentially, the same coarse dynamics.…”

“…This seems to be a difficult problem in algebraic geometry, since it involves enumeration of semi-algebraic sets in the space of parameters that are determined by inequalities involving multilinear functions [5]. The sets W and Υ are not the same in general, since not every linear extension of ≤ satisfies the algebraic constraints from M .…”