# De Ronde et al., 2009

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 study, the authors address this problem by solving the chemical master equations for various enzymatic reactions. A perturbation theory analogous to that used in quantum mechanics allows the determination of the first ( ${\displaystyle \langle n\rangle }$) and the second (${\displaystyle \sigma ^{2}}$) 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 ${\displaystyle V=d\langle n\rangle /dt}$ (or ‘dose–response’ curve) and the Fano factor ${\displaystyle F=\sigma ^{2}/\langle n\rangle }$ are both realistically measurable quantities, and whereas 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 the authors explore this possibility in the context of four sample multistep enzymatic reactions. 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.