  We introduce a reaction-path statistical mechanics formalism based on the
principle of large deviations to quantify the kinetics of single-molecule
enzymatic reaction processes under the Michaelis-Menten mechanism, which
exemplifies an out-of-equilibrium process in the living system. Our theoretical
approach begins with the principle of equal a priori probabilities and defines
the reaction path entropy to construct a new nonequilibrium ensemble as a
collection of possible chemical reaction paths. As a result, we evaluate a
variety of path-based partition functions and free energies using the formalism
of statistical mechanics. They allow us to calculate the timescales of a given
enzymatic reaction, even in the absence of an explicit boundary condition that
is necessary for the equilibrium ensemble. We also consider the large deviation
theory under a closed-boundary condition of the fixed observation time to
quantify the enzyme-substrate unbinding rates. The result demonstrates the
presence of a phase-separation-like, bimodal behavior in unbinding events at a
finite timescale, and the behavior vanishes as its rate function converges to a
single phase in the long-time limit.
