Abstract
A gene regulatory network can be described at a high level by a directed graph with signed edges, and at a more detailed level by a system of ordinary differential equations (ODEs). The former qualitatively models the causal regulatory interactions between ordered pairs of genes, while the latter quantitatively models the time-varying concentrations of mRNA and proteins. This paper clarifies the connection between the two types of models. We propose a property, called the constant sign property, for a general class of ODE models. The constant sign property characterizes the set of conditions (system parameters, external signals, or internal states) under which an ODE model is consistent with a signed, directed graph. If the constant sign property for an ODE model holds globally for all conditions, then the ODE model has a single signed, directed graph. If the constant sign property for an ODE model only holds locally, which may be more typical, then the ODE model corresponds to different graphs under different sets of conditions. In addition, two versions of constant sign property are given and a relationship between them is proved. As an example, the ODE models that capture the effect of cis-regulatory elements involving protein complex binding, based on the model in the GeneNetWeaver source code, are described in detail and shown to satisfy the global constant sign property with a unique consistent gene regulatory graph. Even a single gene regulatory graph is shown to have many ODE models of GeneNetWeaver type consistent with it due to combinatorial complexity and continuous parameters. Finally the question of how closely data generated by one ODE model can be fit by another ODE model is explored. It is observed that the fit is better if the two models come from the same graph.
Highlights
A gene regulatory network is a collection of molecular classes such that each molecular class interacts with a small number of other molecular classes, creating a sparse graph structure [1]
We propose a property of the ordinary differential equations (ODEs) models, called the constant sign property (CSP), such that an ODE model corresponds to a single graph model under a set of conditions if and only if the ODE model satisfies CSP under that set of conditions
Graph models with signed directed edges provide circuit-like characterization of gene regulation, while ODE models quantify detailed dynamics for various molecular classes
Summary
A gene regulatory network is a collection of molecular classes such that each molecular class interacts with a small number of other molecular classes, creating a sparse graph structure [1]. A directed graph with vertices representing genes and signed edges representing geneto-gene interactions, known as a circuit model [4] or a logical model [5], is a model with a high level of abstraction (see S1 Appendix). The vertices of such graph models often only. B.H. and Y.H., and by the Communication and Information Foundations program from the National Science Foundation (NSF-CCF-CIF1900636) to B.H
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