Waiting Cycle Times and Generalized Haldane Equality in the Steady-state Cycle Kinetics of Single Enzymes

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.