Browsing by Author "Harker, Shaun"

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Combinatorial Representation of Parameter Space for Switching Networks
(2016-11) Cummins, Bree; Harker, Shaun; Mischaikow, Konstantin; Mok, Kafung; Gedeon, Tomas
We describe the theoretical and computational framework for the Dynamic Signatures Generated by Regulatory Networks (DSGRN) database. The motivation stems from an 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 have 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 piecewise 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.
DSGRN: Examining the Dynamics of Families of Logical Models
(2018-06) Cummins, Bree; Gedeon, Tomas; Harker, Shaun; Mischaikow, Konstantin
We present a computational tool DSGRN for exploring the dynamics of a network by computing summaries of the dynamics of switching models compatible with the network across all parameters. The network can arise directly from a biological problem, or indirectly as the interaction graph of a Boolean model. This tool computes a finite decomposition of parameter space such that for each region, the state transition graph that describes the coarse dynamical behavior of a network is the same. Each of these parameter regions corresponds to a different logical description of the network dynamics. The comparison of dynamics across parameters with experimental data allows the rejection of parameter regimes or entire networks as viable models for representing the underlying regulatory mechanisms. This in turn allows a search through the space of perturbations of a given network for networks that robustly fit the data. These are the first steps toward discovering a network that optimally matches the observed dynamics by searching through the space of networks.
Global dynamics for steep nonlinearities in two dimensions
(2017-01) Gedeon, Tomas; Harker, Shaun; Kokubu, Hiroshi; Mischaikow, Konstantin; Oka, Hiroe
This paper discusses a novel approach to obtaining mathematically rigorous results on the global dynamics of ordinary differential equations. We study switching models of regulatory networks. To each switching network we associate a Morse graph, a computable object that describes a Morse decomposition of the dynamics. In this paper we show that all smooth perturbations of the switching system share the same Morse graph and we compute explicit bounds on the size of the allowable perturbation. This shows that computationally tractable switching systems can be used to characterize dynamics of smooth systems with steep nonlinearities.