Stochastic gene expression with a multistate promoter: breaking down exact distributions

TL;DR

This paper derives exact stationary distributions for a stochastic gene expression model with multistate promoters, unifying and generalizing previous results through a novel Markov process approach.

Contribution

It introduces a unified framework linking multistate promoter models to multivariate PDMPs, simplifying derivations and extending distributions beyond known cases.

Findings

Exact stationary distributions are derived for multistate promoter models.

In special cases, the distribution simplifies to a Dirichlet distribution.

General distributions extend Beta products with parameters linked to spectral properties.

Abstract

We consider a stochastic model of gene expression in which transcription depends on a multistate promoter, including the famous two-state model and refractory promoters as special cases, and focus on deriving the exact stationary distribution. Building upon several successful approaches, we present a more unified viewpoint that enables us to simplify and generalize existing results. In particular, the original jump process is deeply related to a multivariate piecewise-deterministic Markov process that may also be of interest beyond the biological field. In a very particular case of promoter configuration, this underlying process is shown to have a simple Dirichlet stationary distribution. In the general case, the corresponding marginal distributions extend the well-known class of Beta products, involving complex parameters that directly relate to spectral properties of the promoter transition matrix. Finally, we illustrate these results with biologically plausible examples.