On the Use of the Hill Functions in Mathematical Models of Gene Regulatory Networks

Hill functions follow from the equilibrium state of the reaction in which n ligands simultaneously bind a single receptor. This result if often employed to interpret the Hill coefficient as the number of ligand binding sites in all kinds of reaction schemes. Here, we study the equilibrium states of the reactions in which n ligand bind a receptor sequentially, both non-cooperatively and in a cooperative fashion. The main outcomes of such analysis are that: n is not a good estimate, but only an upper bound, for the Hill coefficient; while the Hill coefficient depends quite strongly on the cooperativity level among ligands. We finally use these results to discuss the feasibility and constrains of using Hill functions to model the regulatory functions in mathematical models of gene regulatory networks.