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 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, $Δμ$. More interestingly, the moment generating function of the stochastic number of substrate cycle $ν(t)$, $<e^{-λν(t)}>$ follows the fluctuation theorem in the form of Kurchan-Lebowitz-Spohn-type symmetry. When $λ$ = $Δμ/k_BT$, we obtain the Jarzynski-Hatano-Sasa-type equality: $<e^{-ν(t)Δμ/k_BT}>$ $\equiv$ 1 for all $t$, where $νΔμ$ 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.