Complex kinetics of fluctuating enzymes: phase diagram characterization of a minimal kinetic scheme

Enzyme molecules are dynamic entities with stochastic fluctuation in both protein conformation and enzymatic activity. However, such a notion of fluctuating enzymes, best characterized by recent single-molecule experiments, was not considered in the classic Michaelis-Menten (MM) kinetic scheme. Here we incorporate the fluctuation concept into the reversible MM scheme, and solve analytically all the possible kinetics (i.e., substrate concentration dependent enzymatic velocity) for a minimal model of fluctuating enzymes. Such a minimal model is found to display a variety of distinct kinetic behaviors (phases) in addition to the classic MM kinetics; excess substrate inhibition, sigmoidal kinetics, and concave biphasic kinetics. We find that all these kinetic phases are interrelated and unified under the framework of fluctuating enzymes and can be adequately described by a phase diagram that consists of two master parameters. Functionally, substrate inhibition, sigmoidal kinetics, and convex biphasic phases exhibit positive cooperativity, whereas concave biphasic phases display negative cooperativity. Remarkably, all these complex kinetics are produced by fluctuating enzymes with single substrate binding site, but the two conformations are, therefore, fundamentally different from the classic MWC and KNF models that require multiple subunit or binding sites. This model also suggests that, for a given enzyme/substrate pair, the non-MM behaviors could undergo transitions among different kinetic phases induced by varying product concentrations, owing to the fundamental Haldane symmetry in the reversible MM scheme.