Enzyme kinetics are cyclic. A more realistic reversible three-step mechanism of the Michaelis-Menten kinetics is investigated in detail, and three kinds of waiting cycle times $T$, $T_{ +}$, $T_{ -}$ are defined. It is shown that the mean waiting cycle times $<T>$, $<T_{ +}>$, and $<T_{ -}>$ are the reciprocal of the steady-state cycle flux $J^{ ss}$, the forward steady-state cycle flux $J^{ ss}_{ +}$ and the backward steady-state cycle flux $J^{ ss}_{ -}$ respectively. We also show that the distribution of $T_{ +}$ conditioned on $T_{ +}<T_{ -}$ is identical to the distribution of $T_{ -}$ conditioned on $T_{ -}<T_{ +}$, which is referred as generalized Haldane equality. Consequently, the mean waiting cycle time of $T_{ +}$ conditioned on $T_{ +}<T_{ -}$ ($<T_{ +}| T_{ +}<T_{ -}>$) and the one of $T_{ -}$ conditioned on $T_{ -}<T_{ +}$ ($<T_{ -}| T_{ -}<T_{ +} >$) are both just the same as $<T>$. In addition, the forward and backward stepping probabilities $p^{ +},p^{ -}$ are also defined and discussed, especially their relationship with the cycle fluxes and waiting cycle times. Furthermore, we extend the same results to the $n$-step cycle, and finally, experimental and theoretically based evidences are also included.
Categories: cond-mat.stat-mech physics.chem-ph