Spatiotemporal dynamics of a gene expression model with positive feedback loop

Abstract

Developing appropriate spatiotemporal models can help us understand at the molecular level how organisms function, when and where certain genes are expressed, and to what extent. In this paper, we establish a reaction-diffusion model with positive feedback loop in the process of gene expression, which describes the expression of genes according to probability and random diffusion. We focus on the long-term behavior of the system, including the determination of invariant regions and the global stability of equilibria. In particular, in the absence of a globally stable equilibrium, we derive some results on the existence and nonexistence of nonconstant steady state solutions, and clarify the dependence of the existence of nonconstant solutions on the diffusion rate of reactants and the size of the reaction environment. These results contribute to understanding the heterogeneity of cell characterization. We also carry out numerical simulations to support our theoretical results.