The Michaelis-Menten-Stueckelberg Theorem

TL;DR

This paper generalizes the mass action law for chemical reactions under quasi-steady-state and quasiequilibrium assumptions, deriving entropy production relations without requiring microreversibility, extending classical results by Michaelis, Menten, and Stueckelberg.

Contribution

It introduces a unified framework combining QSS and QE assumptions to derive generalized kinetic relations and entropy production conditions without microreversibility.

Findings

Proves the generalized mass action law under QSS and QE.

Derives relations between kinetic factors ensuring positive entropy production.

Extends Michaelis-Menten-Stueckelberg concepts to broader reaction mechanisms.

Abstract

We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the "rarefied gas" limit or the "molecular chaos" hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density ({\em low density of elementary events}). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann $H$-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.