Hill Function-based Model of Transcriptional Response: Impact of Nonspecific Binding and RNAP Interactions

Hill function is one of the widely used gene transcription regulation models. Its attribute of fitting may result in a lack of an underlying physical picture, yet the fitting parameters can provide information about biochemical reactions, such as the number of transcription factors (TFs) and the binding energy between regulatory elements. However, it remains unclear when and how much biochemical information can Hill function provide in addition to fitting. Here, started from the interactions between TFs and RNA polymerase during transcription regulation and both of their association-dissociation reactions at specific/nonspecific sites on DNA, the regulatory effect of TFs was deduced as fold change. We found that, for weak promoter, fold change can degrade into the regulatory factor (Freg) which is closely correlated with Hill function. By directly comparing and fitting with Hill function, the fitting parameters and corresponding biochemical reaction parameters in Freg were analyzed and discussed, where the single TF and multiple TFs that with cooperativity and basic logic effects were considered. We concluded the strength of promoter and interactions between TFs determine whether Hill function can reflect the corresponding biochemical information. Our findings highlight the role of Hill function in modeling/fitting for transcriptional regulation, which also benefits the preparation of synthetic regulatory elements.