A Stochastic Analysis of Autoregulation of Gene Expression

TL;DR

This paper investigates how negative feedback autoregulation in gene expression reduces protein number fluctuations in prokaryotic cells, providing explicit formulas and simulations to evaluate regulation efficiency.

Contribution

It introduces a stochastic model demonstrating that autoregulation via Hill repression effectively limits protein variability, with explicit variance expressions and analysis of convergence.

Findings

Autoregulation reduces protein number fluctuations.

Explicit asymptotic variance formula derived.

Simulations confirm theoretical predictions.

Abstract

This paper analyzes, in the context of a prokaryotic cell, the stochastic variability of the number of proteins when there is a control of gene expression by an autoregulation scheme. The goal of this work is to estimate the efficiency of the regulation to limit the fluctuations of the number of copies of a given protein. The autoregulation considered in this paper relies mainly on a negative feedback: the proteins are repressors of their own gene expression. The efficiency of a production process without feedback control is compared to a production process with an autoregulation of the gene expression assuming that both of them produce the same average number of proteins. The main characteristic used for the comparison is the standard deviation of the number of proteins at equilibrium. With a Markovian representation and a simple model of repression, we prove that, under a scaling regime, the repression mechanism follows a Hill repression scheme with an hyperbolic control. An explicit asymptotic expression of the variance of the number of proteins under this regulation mechanism is obtained. Simulations are used to study other aspects of autoregulation such as the rate of convergence to equilibrium of the production process and the case where the control of the production process of proteins is achieved via the inhibition of mRNAs.