Spectral solutions to stochastic models of gene expression with bursts and regulation

TL;DR

This paper introduces a spectral method for directly calculating probability distributions in stochastic gene expression models, offering improved efficiency and accuracy over traditional simulation approaches, and explores optimal information transmission in gene regulation.

Contribution

The paper develops a spectral approach for solving linear equations in stochastic gene expression models, providing analytic or highly efficient numerical solutions.

Findings

Spectral method outperforms simulations in efficiency and accuracy.

Bimodal responses in slow-switching regimes can be optimal for information transmission.

Analytic solutions are obtained for several gene expression models.

Abstract

Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, which is linear. We apply the resulting spectral method to three general models of stochastic gene expression: a single gene with multiple expression states (often used as a model of bursting in the limit of two states), a gene regulatory cascade, and a combined model of bursting and regulation. In all cases we find either analytic results or numerical prescriptions that greatly outperform simulations in efficiency and accuracy. In the last case, we show that bimodal response in the limit of slow switching is not only possible but optimal in terms of information transmission.