Relationship between Decimal Hill Coefficient, Intermediate Processes, and Mesoscopic Fluctuations in Gene Expression

The Hill function is relevant for describing enzyme binding and other processes in gene regulatory networks. Despite its theoretical foundation, based on the mechanism of ligand-receptor binding, it is often used as a proper fitting function with a noninteger Hill coefficient in the description of gene expression. In this study, we explicitly considered intermediate processes in the transcription factor binding sites and mesoscopic concentration fluctuations, which, in contrast to the case of a single binding site without conformal states or all-or-none binding, lead to a noninteger Hill coefficient for the transcription rate. The relationships between the intermediate processes and the decimal Hill coefficient were established 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 of gene expression rates while also enabling the prediction of an effective value of the Hill coefficient from the underlying mechanism. This procedure provides a simplified and effective description of the complex mechanisms that underlie gene expression.