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The equilibration of enzyme and complex concentrations in deterministicMichaelis-Menten reaction networks underlies the hyperbolic dependence betweenthe input (substrates) and output (products). This relationship was firstobtained by Michaelis and Menten and then Briggs and Haldane in two asymptoticlimits: `fast equilibrium' and `steady state'. In stochastic Michaelis-Mentennetworks, relevant to catalysis at single-molecule and mesoscopicconcentrations, the classical analysis cannot be directly applied due tomolecular discreteness and fluctuations. Instead, as we show here, suchnetworks require a more subtle asymptotic analysis based on the decompositionof the network into reversible and irreversible sub-networks and the exactsolution of the chemical master equation (CME). The reversible and irreversiblesub-networks reach detailed balance and stationarity, respectively, through arelaxation phase that we characterise in detail through several new statisticalmeasures. Since stochastic enzyme kinetics encompasses the single-molecule,mesoscopic and thermodynamic limits, our work provides a broader molecularviewpoint of the classical results, in much the same manner that statisticalmechanics provides a broader understanding of thermodynamics.

Relaxation to statistical equilibrium in stochastic Michaelis-Menten kinetics