Poisson indicator and Fano factor for probing dynamic disorder in single-molecule enzyme inhibition kinetics

We consider a generic stochastic model to describe the kinetics of single-molecule enzyme inhibition reactions in which the turnover events correspond to conversion of substrate into a product by a single enzyme molecule in the presence of an inhibitor. We observe that slow fluctuations between the active and inhibited state of the enzyme or the enzyme substrate complex can induce dynamic disorder, which is manifested in the measurement of the Poisson indicator and the Fano factor as functions of substrate concentrations for different inhibition reactions. For a single enzyme molecule inhibited by the product, we derive a single-molecule Michaelis-Menten equation for the reaction rate, which shows a dependence on the substrate concentration similar to the ensemble enzymatic catalysis rate as obtained from bulk experimental results. The measurement of Fano factor is shown to be able to discriminate reactions following different inhibition mechanisms and also extract kinetic rates.