Generalized Haldane equation and fluctuation theorem in the steady-state cycle kinetics of single enzymes

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 identical nonexponential distributions: ${\Theta }_{+}\left(t\right)={\Theta }_{-}\left(t\right)$. This equation generalizes the Haldane relation in reversible enzyme kinetics. In terms of the probabilities for the forward $\left(p_{+}\right)$ and backward $\left(p_{-}\right)$ cycles, $k_{B}T\ln (p_{+}∕p_{-})$ is shown to be the chemical driving force of the NESS, $\mathrm{\Delta }\mu$. More interestingly, the moment generating function of the stochastic number of substrate cycle $\nu \left(t\right)$, $⟨e^{-\lambda \nu \left(t\right)}⟩$, follows the fluctuation theorem in the form of Kurchan-Lebowitz-Spohn-type symmetry. When $\lambda =\mathrm{\Delta }\mu ∕k_{B}T$, we obtain the Jarzynski-Hatano-Sasa-type equality $⟨e^{-\nu \left(t\right)\mathrm{\Delta }\mu ∕k_{B}T}⟩$ $\equiv$ 1 for all $t$, where $\nu \mathrm{\Delta }\mu$ is the fluctuating chemical work done for sustaining the NESS. This theory suggests possible methods to experimentally determine the nonequilibrium driving force in situ from turnover data via single-molecule enzymology.