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

Qian, H. & Xie, X. S. Generalized Haldane equation and fluctuation theorem in the steady-state cycle kinetics of single enzymes. Phys Rev E Stat Nonlin Soft Matter Phys 74, 010902, doi:10.1103/PhysRevE.74.010902 (2006)

Enzyme 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 + (t)=Theta_(t). This equation generalizes the Haldane relation in reversible enzyme kinetics. In terms of the probabilities for the forward (p+) and backward (p-) cycles, kBT 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 v(t), <e-lambda v(t)> , follows the fluctuation theorem in the form of Kurchan-Lebowitz-Spohn-type symmetry. When lambda=delta mu/kBT, we obtain the Jarzynski-Hatano-Sasa-type equality <e-v(t)Delta mu/kBT> identical with 1 for all t, where v 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.