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+) 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(+) and 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 evidence are also included.