Relaxation to statistical equilibrium in stochastic Michaelis-Menten kinetics

TL;DR

This paper investigates how stochastic Michaelis-Menten enzyme networks reach statistical equilibrium, using detailed asymptotic analysis and the chemical master equation to extend classical deterministic results to molecular and mesoscopic scales.

Contribution

It introduces a novel asymptotic approach based on network decomposition and exact CME solutions to analyze relaxation to equilibrium in stochastic enzyme kinetics.

Findings

Reversible and irreversible sub-networks reach detailed balance and stationarity through relaxation.

New statistical measures characterize the relaxation process.

The work broadens classical enzyme kinetics to include stochastic and mesoscopic effects.

Abstract

The equilibration of enzyme and complex concentrations in deterministic Michaelis-Menten reaction networks underlies the hyperbolic dependence between the input (substrates) and output (products). This relationship was first obtained by Michaelis and Menten and then Briggs and Haldane in two asymptotic limits: `fast equilibrium' and `steady state'. In stochastic Michaelis-Menten networks, relevant to catalysis at single-molecule and mesoscopic concentrations, the classical analysis cannot be directly applied due to molecular discreteness and fluctuations. Instead, as we show here, such networks require a more subtle asymptotic analysis based on the decomposition of the network into reversible and irreversible sub-networks and the exact solution of the chemical master equation (CME). The reversible and irreversible sub-networks reach detailed balance and stationarity, respectively, through a relaxation phase that we characterise in detail through several new statistical measures. Since stochastic enzyme kinetics encompasses the single-molecule, mesoscopic and thermodynamic limits, our work provides a broader molecular viewpoint of the classical results, in much the same manner that statistical mechanics provides a broader understanding of thermodynamics.