Mathematical Modeling of Gene Regulatory Networks

Genes operate in complex networks which control fundamental cellular processes. To better understand the highly integrated regulatory mechanisms used in such processes, mathematicians and biologists have developed a variety of mathematical and computational models. The accuracy of such models is highly important in predicting the behavior of the system and evaluating potential experimental alterations of its function. A vast literature exists describing the wide variety of approaches used to model these networks.

After a brief general introduction to the mathematical modeling of gene regulatory networks (GRNs), we investigate the dynamical behavior of simple networks with feedback loops. Standard methods for analyzing such networks for gene activity use continuous models (systems of differential equations tracking the concentrations of mRNA and protein) or Boolean models (where the genes are considered either on or off). These two types of models give rise to qualitatively different predictions. We establish the existence of Hopf bifurcations in the continuous models and use these bifurcations to compare the models more closely. This analysis enables us to identify the regions in the parameter space where the dynamical behavior of the models agree and where they disagree.

In GRNs, the expression of genes is subject to not only the input from the other genes but also possible internal and/or external noise. These factors may be modeled by introducing stochasticity at gene levels. Here we build upon an established stochastic discrete dynamical system model; we study the effects of choosing propensity parameters from a set of a priori distributions. We examine the effects of different probability distributions on the dynamics of GRN. We create state space diagrams using different probabilistic models.