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The detailed balance property and chemical systems out of equilibrium

E. Franco, J. J. L. Velázquez

TL;DR

This work investigates how detailed balance (DB) constrains chemical reaction networks and how open, non-equilibrium behavior emerges from environmental exchange and model reductions. It develops a rigorous framework linking closed DB systems to their open counterparts via completion and reduction, showing that bidirectional networks can be completed to DB systems and that reducing DB systems by freezing certain concentrations preserves DB under specific cycle-topology conditions. The authors introduce reduced chemical networks, reduced kinetics, and the notion of cycles, establishing when DB is inherited robustly and when it can fail under perturbations. They further quantify energy landscapes through a DB-compatible energy vector $E$ and associated free-energy functionals, providing Lyapunov-based guarantees of global stability and long-time behavior, even in the presence of fluxes, by decomposing dissipation and external energy contributions. Collectively, the results offer a principled way to measure non-equilibrium Todoity via cycle structure, enabling systematic design or analysis of biochemical networks with controlled departures from DB in open settings.

Abstract

The detailed balance property is a fundamental property that must be satisfied in all the macroscopic systems with a well defined temperature at each point. On the other hand, many biochemical networks work in non-equilibrium conditions and they can be effectively modelled using sets of equations in which the detailed balance condition fails. In this paper we study a class of "out of equilibrium" chemical networks that can be obtained freezing the concentration of some substances in chemical networks for which the detailed balance property holds. In particular, we prove that any chemical system with bidirectional chemical reactions can be extended to a system having additional substances and for which the detailed balance property holds.

The detailed balance property and chemical systems out of equilibrium

TL;DR

This work investigates how detailed balance (DB) constrains chemical reaction networks and how open, non-equilibrium behavior emerges from environmental exchange and model reductions. It develops a rigorous framework linking closed DB systems to their open counterparts via completion and reduction, showing that bidirectional networks can be completed to DB systems and that reducing DB systems by freezing certain concentrations preserves DB under specific cycle-topology conditions. The authors introduce reduced chemical networks, reduced kinetics, and the notion of cycles, establishing when DB is inherited robustly and when it can fail under perturbations. They further quantify energy landscapes through a DB-compatible energy vector and associated free-energy functionals, providing Lyapunov-based guarantees of global stability and long-time behavior, even in the presence of fluxes, by decomposing dissipation and external energy contributions. Collectively, the results offer a principled way to measure non-equilibrium Todoity via cycle structure, enabling systematic design or analysis of biochemical networks with controlled departures from DB in open settings.

Abstract

The detailed balance property is a fundamental property that must be satisfied in all the macroscopic systems with a well defined temperature at each point. On the other hand, many biochemical networks work in non-equilibrium conditions and they can be effectively modelled using sets of equations in which the detailed balance condition fails. In this paper we study a class of "out of equilibrium" chemical networks that can be obtained freezing the concentration of some substances in chemical networks for which the detailed balance property holds. In particular, we prove that any chemical system with bidirectional chemical reactions can be extended to a system having additional substances and for which the detailed balance property holds.

Paper Structure

This paper contains 23 sections, 30 theorems, 164 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Kinetic systems
  4. Chemical reaction networks
  5. Conservative networks
  6. Kinetic systems: chemical networks endowed with a rate function
  7. Detailed balance property of kinetic systems
  8. Detailed balance property
  9. Energy associated to kinetic systems with detailed balance
  10. Global stability of the steady states for kinetic systems with the detailed balance property
  11. Reduction of a kinetic system
  12. Reduction of a chemical network
  13. Reduction of the cycles
  14. Reduction of a kinetic system
  15. The detailed balance property of reduced kinetic systems
  16. ...and 8 more sections

Key Result

Theorem 1.1

Consider a kinetic system that satisfies the detailed balance property. Assume that the concentrations of some substances are kept at constant values by influxes and outfluxes of chemicals. Moreover, assume that if the substances whose concentrations are frozen appear in the cycles, then their conce

Figures (1)

  • Figure 1: On the left we have a cycle in the reduced chemical network $(U, \mathcal{R}_V)$ that is not a cycle in the non-reduced chemical network $(\Omega , \mathcal{R})$. In the right we have a cycle in the reduced system that is also a cycle in the non reduced system.

Theorems & Definitions (82)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Chemical network
  • Definition 2.2: Cycles of a chemical network
  • Example 2.3
  • Definition 2.4: Stoichiometric subspace
  • Definition 2.5: Stoichiometric compatibility classes
  • Definition 2.6: Set of conservation laws
  • Definition 2.7: Conservative chemical network
  • ...and 72 more