Stochastic activation in a genetic switch model

TL;DR

This paper investigates how stochastic fluctuations influence a genetic switch model with autoregulation, calculating transition rates between stable states using analytical and numerical methods, and exploring effects of protein and mRNA lifetime ratios.

Contribution

It provides analytical expressions for transition rates in a genetic switch model, including the bursting limit and finite lifetime ratios, enhancing understanding of stochastic effects in gene regulation.

Findings

Analytical calculation of the transition barrier $S_0$ matches simulations.

The prefactor $ heta$ varies with protein/mRNA lifetime ratio $ au$, with analytical estimates close to numerical results.

The model captures how fluctuations induce rare transitions in bistable gene regulation systems.

Abstract

We study a biological autoregulation process, involving a protein that enhances its own transcription, in a parameter region where bistability would be present in the absence of fluctuations. We calculate the rate of fluctuation-induced rare transitions between locally-stable states using a path integral formulation and Master and Chapman-Kolmogorov equations. As in simpler models for rare transitions, the rate has the form of the exponential of a quantity $S_0$ (a "barrier") multiplied by a prefactor $\eta$. We calculate $S_0$ and $\eta$ first in the bursting limit (where the ratio $\gamma$ of the protein and mRNA lifetimes is very large). In this limit, the calculation can be done almost entirely analytically, and the results are in good agreement with simulations. For finite $\gamma$ numerical calculations are generally required. However, $S_0$ can be calculated analytically to first order in $1/\gamma$, and the result agrees well with the full numerical calculation for all $\gamma > 1$. Employing a method used previously on other problems, we find we can account qualitatively for the way the prefactor $\eta$ varies with $\gamma$, but its value is 15-20% higher than that inferred from simulations.