Convex Analysis of Relaxation Dynamics in Chemical Reaction Networks and Generalized Gradient Flows

TL;DR

This paper establishes bounds on the relaxation dynamics of chemical reaction networks using convex analysis, linking decay rates to network properties and extending the results within a generalized gradient flow framework relevant to biological systems.

Contribution

It introduces a novel convex analysis approach to bound relaxation times in CRNs and extends these bounds within a generalized gradient flow framework applicable to biological regimes.

Findings

Bounds on Kullback-Leibler divergence decay rates for CRNs.

Extension of bounds within a generalized gradient flow framework.

Application to catalytic CRNs with plateau behavior.

Abstract

We obtain bounds on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium. The associated decay rates are characterized in terms of the singular values of the stoichiometric matrix, convexity parameters, and time-integrated activities via deformed-exponential-type functions. We further extend these bounds within a generalized gradient flow framework. We highlight the biological relevance of this framework: the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important. We illustrate the framework using a catalytic CRN exhibiting plateaus, where the bounds capture slow relaxation induced by local convexity and provide a bound-based approach to quantifying relaxation in CRNs.