Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Keisuke Sugie, Dimitri Loutchko, Tetsuya J. Kobayashi

TL;DR

Borders on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium are obtained and the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important.

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.

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

TL;DR

Borders on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium are obtained and the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important.

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.
Paper Structure (25 sections, 10 theorems, 65 equations, 2 figures)

This paper contains 25 sections, 10 theorems, 65 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Preliminaries
  3. Chemical Reaction Networks: Structure and Dynamics
  4. Structural Properties of CRNs
  5. Dynamics on Thermodynamic CRNs
  6. Assumption on Solutions of the Chemical Rate Equation
  7. Geometric and Thermodynamic Aspects of CRN Dynamics
  8. Information-Geometry of Concentration and Potential Spaces
  9. Dissipative Structure and Generalized Gradient Flow
  10. Thermodynamics of Equilibrium Mass-Action CRNs
  11. Convergence Analysis of Relaxation in Generalized Gradient Flows
  12. Convexity Conditions on Divergences
  13. Global Convexity Conditions
  14. Local Convexity Conditions
  15. Convexity on Solution Orbits
  16. ...and 10 more sections

Key Result

Lemma 3.3

\newlabellemma:global convexity of Bregman divergence on solution orbit0 Suppose that asm:invariance of positive orthant holds for a generalized gradient flow eq:generalized gradient flow. Then there exist positive constants $\underline{\rho_{\bm x_\mathrm{eq}}}(\bm x_0), \overline{\rho_{\bm x_\ma

Figures (2)

  • Figure 1: The diagram of the information-geometric relation between $\bm x$, $\bm x_\mathrm{eq}$, and $\bm x^\mathrm{eff}$. The intersection of the stoichiometric subspace $\mathcal{S}(\bm x_\mathrm{eq})$ and the equilibrium submanifold $\partial\phi^\ast[\mathcal{E}(\partial \phi(\bm x^\mathrm{eff}))]$ contains the unique point $\bm x$. The same relation between the stoichiometric submanifold $\partial\phi[\mathcal{S}(\bm x_0)]$ and the equilibrium subspace $\mathcal{E}(\partial\phi(\bm x^\mathrm{eff}))$ also holds in the potential space $\mathcal{M}$. Both $\partial D_\phi(\bm x\| \bm x_\mathrm{eq})$ (purple vector) and $\partial D_\phi(\bm x^\mathrm{eff}\| \bm x_\mathrm{eq})$ (red vector) are mapped to $f(\bm x)$ by $S^\top$.
  • Figure 1: The time courses of the generalized KL divergence and the corresponding bounds in the CRN1 \ref{['eq:CRN1']}$(0\leq t\leq T = 10^5)$. (a) The time courses of the divergence $D_\mathrm{KL}(\bm x_t \| \bm x_\mathrm{eq})$ and its bounds. We show the local upper bounds $\overline{D}_\mathrm{hyp,loc}$\ref{['eq:hyp_loc_upper']}, $\overline{D}_\mathrm{quad,loc}$\ref{['eq:quad_loc_upper']} for $\rho_{\bm x_\mathrm{eq}}^\mathrm{KL}(\bm x)$, and the global upper bounds $\overline{D}_\mathrm{hyp,glob}, \overline{D}_\mathrm{quad,glob}$ for $\underline{\rho}_{\bm x_\mathrm{eq}}^\mathrm{KL} = \min_{\bm x\in \mathcal{O}_{10^5}(\bm x_0)} \rho_{\bm x_\mathrm{eq}}^\mathrm{KL}(\bm x)$. (b) The time courses of the KL divergence, the local lower bounds $\underline{D}_\mathrm{hyp,loc} \ref{['eq:hyp_loc_lower']}, \underline{D}_\mathrm{quad,loc}$\ref{['eq:hyp_loc_lower']} for $\rho_{\bm x_\mathrm{eq}}^\mathrm{KL}(\bm x)$. (c) The time courses of the force norm $\norm{\bm f (\bm x_t)}_2$, its upper bound \ref{['eq:upper bound of thermodynamic force norm by gradient']} and lower bound \ref{['eq:lower bound of thermodynamic force norm by gradient']}. (d) The time courses of the EPR $\dot\Sigma(\bm x_t, \bm f(\bm x_t))$, the quadratic upper bound $\overline{\dot \Sigma}_\mathrm{quad}$ and lower bound $\underline{\dot \Sigma}_\mathrm{quad}$\ref{['eq:quadratic EPR bounds']}, and the hyperbolic upper bound $\overline{\dot \Sigma}_\mathrm{hyp}$ and lower bound $\underline{\dot \Sigma}_\mathrm{hyp}$\ref{['eq:hyperbolic EPR bounds']}.

Theorems & Definitions (20)

  • Remark 3.1: Local Convexity implies Global Convexity
  • Remark 3.2: Boundedness of orbits
  • Lemma 3.3: Global Convexity of Bregman Divergence on Solution Orbits
  • Proof 1: Proof of \ref{['lemma:global convexity of Bregman divergence on solution orbit']}
  • Lemma 3.4: Bounds of separable EPR by force norm
  • Proof 2: Proof of \ref{['lemma:lower bounds of separable EPR']}
  • Proof 3: Proof of \ref{['lemma:upper bounds of separable EPR']}
  • Lemma 3.5: Sufficient Condition for \ref{['asm: convexity of EPR lower bound', 'asm: convexity of EPR upper bound']} under Separable Dissipation
  • Theorem 3.6: Bounds of Bregman Divergence in Generalized Gradient Flow under Global Convexity Assumption
  • Remark 3.7: Asymptotic Behavior of Bounds
  • ...and 10 more