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This study establishes mathematical bounds on how quickly chemical reaction networks approach equilibrium, measured using Kullback-Leibler divergence. The bounds depend on properties of the stoichiometric matrix, convexity parameters, and reaction activities, and are extended to generalized gradient flow systems. The framework is particularly useful for analyzing biological systems with quasi-steady-state behavior, where reactions exhibit long transient periods and plateau-like dynamics before reaching equilibrium.
Why it matters
The theoretical framework provides quantitative tools for understanding slow relaxation dynamics in chemical and biological networks, which is important for predicting system behavior in enzyme catalysis, metabolic pathways, and other biological processes where transient states have functional significance. The approach offers a systematic way to characterize and bound relaxation times in complex reaction systems.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: We obtain bounds on the Kullback–Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium. The associated trajectory-dependent bounds 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 provide an a posteriori characterization of slow relaxation induced by local convexity and provide a bound-based approach to quantifying relaxation in CRNs.
Source: Convex Analysis of Relaxation Dynamics in Chemical Reaction Networks and Generalized Gradient Flows