Fluctuation-induced corrections to the Hill function: implications for gene regulatory network dynamics

Abstract Gene regulatory networks are essential for understanding the mechanisms underlying genetic disorders. However, traditional models based on deterministic Hill functions (without fluctuations) often fail to capture critical stochastic effects, particularly in small systems in which molecular fluctuations play a significant role. In this study, we introduced a modified Hill function derived from the stochastic dynamics of ligand-receptor interactions. By performing a second-order expansion of the chemical master equation, we derived a set of ordinary differential equations governing the mean concentrations and covariances. This framework allows the identification of effective reaction rates using Hill functions that include stochastic correction terms that account for fluctuations in ligand-receptor binding. We applied this approach to canonical models, such as the toggle switch, repressilator, and activator-repressor clock, and showed that the Hill function with stochastic corrections provides a more accurate description of system dynamics with intrinsic fluctuations. Moreover, the proposed method offers substantial computational advantages over direct stochastic simulations using the Gillespie algorithm. Our results demonstrate that this approximation captures the essential features of stochastic biochemical systems more accurately than models that use deterministic Hill functions, offering a powerful tool for studying the role of fluctuations in complex reaction networks and uncovering potential emergent behaviors driven by intrinsic fluctuations in the system.