  Describing reaction rates in stochastic bio-circuits is commonly done by
directly introducing the deterministically deduced Hill function into the
master equation. However, when fluctuations in enzymatic reaction rates are not
neglectable, the Hill function must be derived, considering all the involved
stochastic reactions. In this work, we derived the stochastic version of the
Hill function from the master equation of the complete set of reactions that,
in the macroscopic limit, lead to the Hill function reaction rate. We performed
a series expansion around the average values of the concentrations, which
allowed us to find corrections for the deterministic Hill function. This
process allowed us to quantify the fluctuations of enzymatic reactions. We
found that the underlying variability in propensity rates of gene regulatory
networks has an important non-linear effect that reduces the intrinsic
fluctuations of the mRNA and protein concentrations.
