Variance-corrected Michaelis-Menten equation predicts transient rates of single-enzyme reactions and response times in bacterial gene-regulation

TL;DR

This paper introduces a universal variance-corrected Michaelis-Menten equation that accurately predicts reaction rates and response times in low-concentration biological reactions, accounting for inherent concentration fluctuations.

Contribution

It presents a novel correction to the Michaelis-Menten equation that incorporates substrate concentration variance, applicable to enzymatic and gene-regulation reactions.

Findings

The correction accurately predicts reaction rates under large fluctuations.

The approach relies only on mean and variance, making it experimentally accessible.

Validation through theory and simulations confirms high predictive accuracy.

Abstract

Many chemical reactions in biological cells occur at very low concentrations of constituent molecules. Thus, transcriptional gene-regulation is often controlled by poorly expressed transcription-factors, such as E.coli lac repressor with few tens of copies. Here we study the effects of inherent concentration fluctuations of substrate-molecules on the seminal Michaelis-Menten scheme of biochemical reactions. We present a universal correction to the Michaelis-Menten equation for the reaction-rates. The relevance and validity of this correction for enzymatic reactions and intracellular gene-regulation is demonstrated. Our analytical theory and simulation results confirm that the proposed variance-corrected Michaelis-Menten equation predicts the rate of reactions with remarkable accuracy even in the presence of large non-equilibrium concentration fluctuations. The major advantage of our approach is that it involves only the mean and variance of the substrate-molecule concentration. Our theory is therefore accessible to experiments and not specific to the exact source of the concentration fluctuations.