The Limiting Dynamics of a Bistable Molecular Switch With and Without Noise

TL;DR

This paper analyzes the dynamics of a bistable gene regulatory network, characterizing how noise influences the steady state distribution, and finds that different noise sources produce similar stationary behaviors.

Contribution

It provides a complete analytical characterization of the local and global dynamics of a bistable switch with and without noise, including explicit stationary density forms.

Findings

Steady state distribution can be unimodal or bimodal.

Stationary density behavior is similar for bursting and Gaussian noise.

The model with bursting noise and two slow genes has a stable stationary density.

Abstract

We consider the dynamics of a population of organisms containing two mutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in the absence of any noise, and then go on to consider the effects of either noise coming from bursting (transcription or translation), or Gaussian noise in molecular degradation rates when there is a dominant slow variable in the system. We show analytically how the steady state distribution in the population can range from a single unimodal distribution through a bimodal distribution and give the explicit analytic form for the invariant stationary density which is globally asymptotically stable. Rather remarkably, the behaviour of the stationary density with respect to the parameters characterizing the molecular behaviour of the bistable switch is qualitatively identical in the presence of noise coming from bursting as well as in the presence of Gaussian noise in the degradation rate. This implies that one cannot distinguish between either the dominant source or nature of noise based on the stationary molecular distribution in a population of cells. We finally show that the switch model with bursting but two dominant slow genes has an asymptotically stable stationary density.