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

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.