Generalized Haldane Equation and Fluctuation Theorem in the Steady State Cycle Kinetics of Single Enzymes

TL;DR

This paper develops a theoretical framework for enzyme cycle kinetics in nonequilibrium steady states, revealing a generalized Haldane relation, fluctuation theorem symmetry, and a Jarzynski-Hatano-Sasa-type equality, with implications for experimental measurement of driving forces.

Contribution

It introduces a Markov renewal process model that generalizes enzyme kinetics relations and establishes fluctuation theorems for enzyme cycle stochasticity in NESS.

Findings

Forward and backward cycle times have identical non-exponential distributions.

The chemical driving force relates to cycle probabilities via $k_BT\,\ln(p_+/p_-)$.

A fluctuation theorem and a Jarzynski-Hatano-Sasa-type equality are derived.

Abstract

Enyzme kinetics are cyclic. We study a Markov renewal process model of single-enzyme turnover in nonequilibrium steady-state (NESS) with sustained concentrations for substrates and products. We show that the forward and backward cycle times have idential non-exponential distributions: $\QQ_+(t)=\QQ_-(t)$. This equation generalizes the Haldane relation in reversible enzyme kinetics. In terms of the probabilities for the forward ($p_+$) and backward ($p_-$) cycles, $k_BT\ln(p_+/p_-)$ is shown to be the chemical driving force of the NESS, $\Delta\mu$. More interestingly, the moment generating function of the stochastic number of substrate cycle $\nu(t)$, $<e^{-\lambda\nu(t)}>$ follows the fluctuation theorem in the form of Kurchan-Lebowitz-Spohn-type symmetry. When $\lambda$ = $\Delta\mu/k_BT$, we obtain the Jarzynski-Hatano-Sasa-type equality: $<e^{-\nu(t)\Delta\mu/k_BT}>$ $\equiv$ 1 for all $t$, where $\nu\Delta\mu$ is the fluctuating chemical work done for sustaining the NESS. This theory suggests possible methods to experimentally determine the nonequilibrium driving force {\it in situ} from turnover data via single-molecule enzymology.