The combined effects of Feller diffusion and transcriptional/translational bursting in simple gene networks

TL;DR

This paper models protein biosynthesis in bacteria considering both stochastic bursting and diffusive fluctuations, analyzing the resulting jump-diffusion process's stability and long-term behavior.

Contribution

It introduces a combined stochastic model incorporating bursting and diffusion effects in gene networks, analyzing its invariant densities and stability.

Findings

Existence of invariant densities established.

Proved asymptotic stability of the density evolution.

Characterized long-term behavior of the jump-diffusion process.

Abstract

We study a stochastic model of biosynthesis of proteins in generic bacterial operons. The stochasticity arises from two different processes, namely from `bursting' production of either mRNA and/or protein (in the transcription/translation process) and from standard diffusive fluctuations. The amount of protein follows the Feller diffusion, while the bursting introduces random jumps between trajectories of the diffusion process. The combined effect leads to a process commonly known as a diffusion process with jumps. We study existence of invariant densities and the long time behavior of distributions of the corresponding Markov process, proving asymptotic stability in the evolution of the density.