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, $k_{B}T\ln (p_{+}/p_{-})$ 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 $\lambda$ = $\Delta \mu /k_{B}T$, we obtain the Jarzynski-Hatano-Sasa-type equality: $<e^{-\nu (t)\Delta \mu /k_{B}T}>$ $\equiv$ 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.

4 pages, 3 figures

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