Statistical properties of multistep enzyme-mediated reactions

Enzyme-mediated reactions may proceed through multiple intermediate conformational states before creating a final product molecule, and one often wishes to identify such intermediate structures from observations of the product creation. In this paper, we address this problem by solving the chemical master equations for various enzymatic reactions. We devise a perturbation theory analogous to that used in quantum mechanics that allows us to determine the first () and the second (variance) cumulants of the distribution of created product molecules as a function of the substrate concentration and the kinetic rates of the intermediate processes. The mean product flux V=d/dt (or “dose-response” curve) and the Fano factor F=variance/ are both realistically measurable quantities, and while the mean flux can often appear the same for different reaction types, the Fano factor can be quite different. This suggests both qualitative and quantitative ways to discriminate between different reaction schemes, and we explore this possibility in the context of four sample multistep enzymatic reactions. We argue that measuring both the mean flux and the Fano factor can not only discriminate between reaction types, but can also provide some detailed information about the internal, unobserved kinetic rates, and this can be done without measuring single-molecule transition events.

💡 Research Summary

The paper tackles the problem of inferring hidden intermediate steps in enzyme‑mediated reactions from observable product formation data. Rather than relying on the classic Michaelis–Menten picture, which assumes a single catalytic transition, the authors treat the reaction network as a stochastic process described by a chemical master equation (CME). Each intermediate conformation (ES, ES*, EP, etc.) is represented as a distinct state, and the kinetic rates connecting these states (k₁, k₂, …) together with the substrate concentration