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Recent fluorescence spectroscopy measurements of the turnover timedistribution of single-enzyme turnover kinetics of $\beta$-galactosidaseprovide evidence of Michaelis-Menten kinetics at low substrate concentration.However, at high substrate concentrations, the dimensionless variance of theturnover time distribution shows systematic deviations from theMichaelis-Menten prediction. This difference is attributed to conformationalfluctuations in both the enzyme and the enzyme-substrate complex and to thepossibility of both parallel and off-pathway kinetics. Here, we use thechemical master equation to model the kinetics of a single fluctuating enzymethat can yield a product through either parallel or off-pathway mechanisms. Anexact expression is obtained for the turnover time distribution from which themean turnover time and randomness parameters are calculated. The parallel andoff-pathway mechanisms yield strikingly different dependences of the meanturnover time and the randomness parameter on the substrate concentration. Inthe parallel mechanism, the distinct contributions of enzyme andenzyme-substrate fluctuations are clearly discerned from the variation of therandomness parameter with substrate concentration. From these general resultswe conclude that an off-pathway mechanism, with substantial enzyme-substratefluctuations, is needed to rationalize the experimental findings ofsingle-enzyme turnover kinetics of $\beta$-galactosidase.

Parallel versus off-pathway Michaelis-Menten mechanism for single-enzyme kinetics of a fluctuating enzyme