Error Bound for Hill-Function Approximations in a Class of Stochastic Transcriptional Network Models

Dylan Hirsch, Theodore W. Grunberg, Domitilla Del Vecchio

Abstract: Hill functions are often used in stochastic models of gene regulation to approximate the dependence of gene activity on the concentration of the transcription factor (TF) that regulates the gene. However, it is generally unknown how much error one may incur from this approximation. We investigate this question in the context of transcriptional networks (TNs). Under the assumption of rapid binding and unbinding of TFs with their gene targets, we bound the approximation error (in terms of the total variation distance) between a mass-action stochastic model and a corresponding model with Hill function propensities. To do so, we use a combination of singular perturbation theory and moment analysis for stochastic chemical reaction networks. We assume throughout that TFs regulate genes in a one-to-one fashion, each regulated gene produces a single TF, TFs do not multimerize, and each gene only has a single TF binding site. These results are pertinent for the modeling of TNs and may also carry relevance for more general biological processes.