Revisit of macroscopic dynamics for some non-equilibrium chemical reactions from a Hamiltonian viewpoint

TL;DR

This paper revisits macroscopic dynamics of non-equilibrium chemical reactions using a Hamiltonian framework, decomposing reaction equations, analyzing energy landscapes, and exploring reversibility and transition paths.

Contribution

It introduces a Hamiltonian approach to analyze non-equilibrium chemical reactions, decomposes reaction rate equations, and studies reversibility and transition paths in this context.

Findings

Decomposition of reaction rate equations into conservative and dissipative parts.

Identification of energy landscapes and thermodynamics at non-equilibrium steady states.

Establishment of a Hamiltonian-based reversibility and transition path analysis.

Abstract

Most biochemical reactions in living cells are open systems interacting with environment through chemostats to exchange both energy and materials. At a mesoscopic scale, the number of each species in those biochemical reactions can be modeled by a random time-changed Poisson processes. To characterize macroscopic behaviors in the large volume limit, the law of large numbers in the path space determines a mean-field limit nonlinear reaction rate equation describing the dynamics of the concentration of species, while the WKB expansion for the chemical master equation yields a Hamilton-Jacobi equation (HJE) and the Lagrangian gives the good rate function in the large deviation principle. We decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to HJE. This stationary solution is used to determine the energy landscape and thermodynamics for general chemical reactions, which particularly maintains a positive entropy production rate at a non-equilibrium steady state. The associated energy dissipation law is proved together with the passage from the mesoscopic to macroscopic one. Furthermore, we use a reversible Hamiltonian to study a class of non-equilibrium enzyme reactions, which identifies a new concept of balance within the same reaction vector due to flux grouping degeneracy. This macroscopic reversibility, brought by the reversibility of the chemical reaction jumping process, gives an Onsager-type strong gradient flow. The reversible Hamiltonian also yields a time reversal symmetry for the corresponding Lagrangian. Thus a modified time reversed least action path serves as the transition paths with associated path affinities and energy barriers.