Revisit of Macroscopic Dynamics for Some Non-equilibrium Chemical Reactions from a Hamiltonian Viewpoint

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 number 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 and the Legendre transform of the corresponding Hamiltonian gives the good rate function (action functional) in the large deviation principle. In this paper, we decompose a general macroscopic reaction rate equation into a conservative part and a dissipative part in terms of the stationary solution to the Hamilton–Jacobi equation. 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 at both the mesoscopic and macroscopic levels is proved together with a passage from the mesoscopic to macroscopic one. A non-convex energy landscape emerges from the convex mesoscopic relative entropy functional in the large number limit, which picks up the non-equilibrium features. The existence of this stationary solution is ensured by the optimal control representation at an undetermined time horizon for the weak KAM solution to the stationary Hamilton–Jacobi equation. Furthermore, we use a symmetric Hamiltonian to study a class of non-equilibrium enzyme reactions, which leads to nonconvex energy landscape due to flux grouping degeneracy and reduces the conservative–dissipative decomposition to an Onsager-type strong gradient flow. This symmetric Hamiltonian implies that the transition paths between multiple steady states (rare events in biochemical reactions) is a modified time reversed least action path with associated path affinities and energy barriers. We illustrate this idea through a bistable catalysis reaction and compute the energy barrier for the transition path connecting two steady states via its energy landscape.

Duke Scholars

Author Jian-Guo Liu Physics