  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.
