Comparison of Combinatorial Signatures of Global Network Dynamics Generated by Two Classes of ODE Models

Abstract

Modeling the dynamics of biological networks introduces many challenges, among them the lack of first principle models, the size of the networks, and difficulties with parameterization. Discrete time Boolean networks and related continuous time switching systems provide a computationally accessible way to translate the structure of the network to predictions about the dynamics. Recent work has shown that the parameterized dynamics of switching systems can be captured by a combinatorial object, called a Dynamic Signatures Generated by Regulatory Networks (DSGRN) database, that consists of a parameter graph characterizing a finite parameter space decomposition, whose nodes are assigned a Morse graph that captures global dynamics for all corresponding parameters. We show that for a given network there is a way to associate the same type of object by considering a continuous time ODE system with a continuous right-hand side, which we call an L-system. The main goal of this paper is to compare the two DSGRN databases for the same network. Since the L-systems can be thought of as perturbations (not necessarily small) of the switching systems, our results address the correspondence between global parameterized dynamics of switching systems and their perturbations. We show that, at corresponding parameters, there is an order preserving map from the Morse graph of the switching system to that of the L-system that is surjective on the set of attractors and bijective on the set of fixed-point attractors. We provide important examples showing why this correspondence cannot be strengthened.context of cell biology, the problems of nonlinear dynamics are compounded by the lack of first principles that would determine appropriate nonlinearities, the difficulty in obtaining precise experimental data needed to determine parameters, and the need to analyze the dynamics of 5--10 dimensional systems over 30--50 parameters.Recently, we introduced in [16, 15] a new approach to this problem that assigns two finite objects to any network with positive and negative edges. First is a parameter graph whose nodes are in 1-1 correspondence with regions in the parameter space, where these regions form a decomposition of the parameter space. To each region of the parameter space, i.e., a node of the parameter graph, there is an associated state transition graph that characterizes allowable transitions between well-defined states that partition the phase space. Since state transition graphs can be large, a useful description of the recurrent trajectories is a Morse graph, which is the graph of strongly connected path components of the state transition graph. This information is compiled into a database of Dynamic Signatures Generated by Regulatory Networks (DSGRN database), where to each node of the parameter graph there is an associated Morse graph that captures the recurrent dynamics valid for all parameters in the corresponding parameter region.The advantages of such a description of global dynamics is its finiteness and the resulting computa...

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“… 2017 ; Crawford-Kahrl et al. 2019 ) turns into a smooth sigmoid function by the application of ( 30 ); see Fig. 1 , top panels.…”

Mixture distributions in a stochastic gene expression model with delayed feedback

“…Nevertheless, the main contribution tof (x) comes from the values of f s whose argument s is close to x. Due to the contributions of neighbouring terms, any sharp features of f s are "mollified" in the functionf (x); for example, a step function (Gedeon et al, 2017;Crawford-Kahrl et al, 2019)…”