Relationship between Decimal Hill Coefficient, Intermediate Processes and Mesoscopic Fluctuations

The Hill function is relevant for describing enzyme binding and other processes in gene regulatory networks. Despite its theoretical foundation, it is often empirically used as a useful fitting function. Theoretical predictions suggest that the Hill coefficient should be an integer. However, it is often assigned a decimal value. The deterministic approximation of binding processes leads to the derivation of the Hill function, which can be expanded around the fluctuation magnitude to derive mesoscopic corrections. This study establishes the relationships between intermediate processes and the decimal Hill coefficient through a direct relationship between the dissociation constants, both with and without fluctuations. This outcome contributes to a deeper understanding of the underlying processes associated with the decimal Hill coefficient while also enabling the prediction of an effective value of the Hill coefficient from the underlying mechanism. This procedure allows us to have a simplified effective description of complex systems.