  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.
