Nonlocal dynamics in a gene regulatory system with tempered stable Lévy noise

Abstract

Transcriptional control is an important way of regulating of gene expression and it is often interpreted in terms of protein levels. Taking account of the ubiquity of noises during gene regulation, we construct an one-dimensional two-parameter bistable stochastic dynamical model with both Brownian motion and tempered stable Lévy motion to describe the regulatory mechanism of gene cI in λ-phage. Specifically, the complex nonlocal dynamics, caused by (non-)Gaussian noises, are investigated through three effective dynamical characteristics, mean first exit time (MFET), first escape probability (FEP) and stochastic basin of attraction (SBA). Our results reveal that the gene-expression level can switch from a low-concentration stable state to a high-concentration stable state, and it depends on both different noise intensities, stable and tempering indexes. Interestingly, MFET is shorter under higher noise intensities or larger stable index or smaller tempering index. Indeed, Gaussian noise dominates the switching in the case of an adjacent concentration interval, while non-Gaussian noise is much effective for the non-adjacent case. Additionally, a smaller stable index or larger tempering index results in wider SBA.