Fluctuations in Gene Regulatory Networks as Gaussian Colored Noise

TL;DR

This paper extends the analysis of gene regulatory network fluctuations to include Gaussian colored noise, deriving explicit formulas for variances and covariances, and demonstrating how noise correlation time affects molecular fluctuations.

Contribution

It provides the first explicit formulas for variances and covariances in gene networks under Gaussian colored noise, including detailed examples.

Findings

Finite noise correlation time reduces fluctuations.

Correlation time enhances molecular fluctuation correlation.

Explicit formulas for variances and covariances are derived.

Abstract

The study of fluctuations in gene regulatory networks is extended to the case of Gaussian colored noise. Firstly, the solution of the corresponding Langevin equation with colored noise is expressed in terms of an Ito integral. Then, two important lemmas concerning the variance of an Ito integral and the covariance of two Ito integrals are shown. Based on the lemmas, we give the general formulae for the variances and covariance of molecular concentrations for a regulatory network near a stable equilibrium explicitly. Two examples, the gene auto-regulatory network and the toggle switch, are presented in details. In general, it is found that the finite correlation time of noise reduces the fluctuations and enhances the correlation between the fluctuations of the molecular components.