Quantitative dynamics of irreversible enzyme reaction-diffusion systems

TL;DR

This paper develops a novel entropy-based method to analyze the convergence to equilibrium in irreversible enzyme reaction-diffusion systems, including cases with non-diffusing large molecules, providing explicit exponential convergence results.

Contribution

Introduces a new cut-off partial entropy approach to prove exponential convergence in irreversible enzyme reaction-diffusion models, applicable even when some molecules do not diffuse.

Findings

Established explicit exponential convergence to equilibrium.

Extended method to systems with non-diffusing enzyme and complex molecules.

Provided a framework for analyzing irreversible reaction-diffusion systems.

Abstract

In this work we investigate the convergence to equilibrium for mass action reaction-diffusion systems which model irreversible enzyme reactions. Using the standard entropy method in this situation is not feasible as the irreversibility of the system implies that the concentrations of the substrate and the complex decay to zero. The key idea we utilise in this work to circumvent this issue is to introduce a family of cut-off partial entropy functions which, when combined with the dissipation of a mass like term of the substrate and the complex, yield an explicit exponential convergence to equilibrium. This method is also applicable in the case where the enzyme and complex molecules do not diffuse, corresponding to chemically relevant situation where these molecules are large in size.