Corrected Hill Function in Stochastic Gene Regulatory Networks

TL;DR

This paper derives a stochastic version of the Hill function for gene regulatory networks, accounting for enzymatic reaction fluctuations, and shows how these fluctuations influence the variability of mRNA and protein levels.

Contribution

It introduces a stochastic Hill function derived from the master equation, including corrections for fluctuations, advancing the modeling of stochastic gene regulation.

Findings

Fluctuations in enzymatic reaction rates significantly affect gene expression variability.

The stochastic Hill function provides more accurate reaction rate descriptions under stochastic conditions.

Variability in propensity rates reduces intrinsic fluctuations in mRNA and protein concentrations.

Abstract

Describing reaction rates in stochastic bio-circuits is commonly done by directly introducing the deterministically deduced Hill function into the master equation. However, when fluctuations in enzymatic reaction rates are not neglectable, the Hill function must be derived, considering all the involved stochastic reactions. In this work, we derived the stochastic version of the Hill function from the master equation of the complete set of reactions that, in the macroscopic limit, lead to the Hill function reaction rate. We performed a series expansion around the average values of the concentrations, which allowed us to find corrections for the deterministic Hill function. This process allowed us to quantify the fluctuations of enzymatic reactions. We found that the underlying variability in propensity rates of gene regulatory networks has an important non-linear effect that reduces the intrinsic fluctuations of the mRNA and protein concentrations.