Statistical physics of a model binary genetic switch with linear feedback

TL;DR

This paper analyzes a stochastic binary genetic switch with linear feedback, providing an exact steady-state solution, exploring flip time distributions, and revealing non-Poissonian statistics and cellular memory effects.

Contribution

It generalizes previous models to include bidirectional feedback, derives analytical solutions for steady-state and flip times, and introduces new measures for correlations and memory in genetic switches.

Findings

Flip time distribution exhibits a peak, indicating non-Poissonian behavior.

Long-lived correlations suggest primitive cellular memory.

Two-switch interactions can be positively or negatively correlated.

Abstract

We study the statistical properties of a simple genetic regulatory network that provides heterogeneity within a population of cells. This network consists of a binary genetic switch in which stochastic flipping between the two switch states is mediated by a "flipping" enzyme. Feedback between the switch state and the flipping rate is provided by a linear feedback mechanism: the flipping enzyme is only produced in the on switch state and the switching rate depends linearly on the copy number of the enzyme. This work generalises the model of [Phys. Rev. Lett., 101, 118104] to a broader class of linear feedback systems. We present a complete analytical solution for the steady-state statistics of the number of enzyme molecules in the on and off states, for the general case where the enzyme can mediate flipping in either direction. For this general case we also solve for the flip time distribution, making a connection to first passage and persistence problems in statistical physics. We show that the statistics of the model are non-Poissonian, leading to a peak in the flip time distribution. The occurrence of such a peak is analysed as a function of the parameter space. We present a new relation between the flip time distributions measured for two relevant choices of initial condition. We also introduce a new correlation measure to show that this model can exhibit long-lived temporal correlations, thus providing a primitive form of cellular memory. Motivated by DNA replication as well as by evolutionary mechanisms involving gene duplication, we study the case of two switches in the same cell. This results in correlations between the two switches; these can either positive or negative depending on the parameter regime.