Quantitative ergodicity for gene regulatory networks with transcriptional bursting

TL;DR

This paper analyzes the long-term behavior of stochastic gene regulatory networks with bursting, proving existence, uniqueness, and convergence rates of their stationary distributions.

Contribution

It establishes the existence and uniqueness of stationary distributions for complex gene networks and provides explicit convergence bounds using coupling methods.

Findings

Existence and uniqueness of stationary distribution proven for arbitrary gene networks.

Explicit upper bounds for convergence to equilibrium derived.

Applicable to models with arbitrary interaction strength and number of genes.

Abstract

We study the long-term behavior of two piecewise-deterministic Markov processes used to model stochastic gene regulatory networks with bursting dynamics. Under regularity assumptions on the jump rate, we prove the existence and uniqueness of the stationary distribution for an arbitrary number of interacting genes and an arbitrary strength of interaction. Using coupling methods, we also provide explicit upper bounds for the convergence to equilibrium in terms of Wasserstein distances.