Single-Molecule Kinetic Theory of Heterogeneous and Enzyme Catalysis

Abstract

Recent experimental advances in single-molecule enzymology stimulated many efforts to develop single-molecule kinetic theories of enzyme catalysis, especially for the classic Michaelis−Menten mechanism. Our group recently studied redox catalysis by single metal nanoparticles at single-turnover resolution. Compared with enzymes, which are homogeneous catalysts and have well-defined active sites, nanoparticles are heterogeneous catalysts and have many different surface sites for catalysis. To provide a theoretical framework to understand nanoparticle catalysis at the single-molecule level, here we formulate in detail the single-molecule kinetic theory of a Langmuir−Hinshelwood mechanism for heterogeneous catalysis, which includes the multitude of surface sites on one nanoparticle. We consider two parallel product dissociation pathways that give complex single-molecule kinetics of the product dissociation reaction. We derive the probability density functions of the stochastic waiting times for both the product formation and the product dissociation reactions and describe their complex behaviors at different kinetic limiting conditions. We also obtain a single-molecule Langmuir−Hinshelwood equation that describes the saturation kinetics of the product formation rate over substrate concentrations and evaluate the randomness parameter of single-turnover waiting times. We further compare the single-molecule kinetics between the Langmuir−Hinshelwood mechanism for heterogeneous catalysis and the Michaelis−Menten mechanism for enzyme catalysis and formulate the modified single-molecule Michaelis−Menten kinetics with multiple product dissociation pathways. In the end, we suggest that the Langmuir−Hinshelwood mechanism is also applicable to describe the single-molecule kinetics of oligomeric enzymes that contain multiple catalytic sites. We expect that these theories will enable quantitative analysis of single-turnover kinetics of heterogeneous and enzyme catalysis and provide a theoretical foundation to understand the catalytic dynamics of nanoparticles and enzymes at the single-molecule level.