Effect of reaction step-size noise on the switching dynamics of stochastic populations

TL;DR

This paper investigates how bursty, stochastic influx of particles influences the switching dynamics of populations, revealing that burstiness significantly reduces mean escape times and providing analytical expressions near bifurcation points.

Contribution

It introduces a novel analytical framework for understanding how step-size noise from bursty influx affects population switching, extending previous models to include arbitrary burst-size distributions.

Findings

Burstiness exponentially decreases mean escape time.

Analytical expression for escape time near bifurcation depends on burst mean and variance.

Model matches well with numerical simulations.

Abstract

In genetic circuits, when the mRNA lifetime is short compared to the cell cycle, proteins are produced in geometrically-distributed bursts, which greatly affects the cellular switching dynamics between different metastable phenotypic states. Motivated by this scenario, we study a general problem of switching or escape in stochastic populations, where influx of particles occurs in groups or bursts, sampled from an arbitrary distribution. The fact that the step size of the influx reaction is a-priori unknown, and in general, may fluctuate in time with a given correlation time and statistics, introduces an additional non-demographic step-size noise into the system. Employing the probability generating function technique in conjunction with Hamiltonian formulation, we are able to map the problem in the leading order onto solving a stationary Hamilton-Jacobi equation. We show that bursty influx exponentially decreases the mean escape time compared to the "usual case" of single-step influx. In particular, close to bifurcation we find a simple analytical expression for the mean escape time, which solely depends on the mean and variance of the burst-size distribution. Our results are demonstrated on several realistic distributions and compare well with numerical Monte-Carlo simulations.