On detailed balance and reversibility of semi-Markov processes and single-molecule enzyme kinetics

Semi-Markov processes have found increasing applications in modeling the kinetics of single enzyme molecules. Detailed balance is a widely accepted condition for Markov models of closed chemical systems and well known to be equivalent to the reversibility of a stationary Markov process. We show that for a semi-Markov process detailed balance is only a necessary condition, but not sufficient, for its time reversibility. The statistical independence between the transition direction and the sojourn time is also necessary. We show that the direction-time independence naturally arises from the exit problem of Markov models for enzyme kinetics with detailed balance. Detailed balance and the direction-time independence together are equivalent to the time reversibility of a stationary semi-Markov process. Applications of the present theory to single-molecule enzymology are also presented.