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Appropriate regulation of gene expression is essential to ensure that proteinsynthesis occurs in a selective manner. The control of transcription is themost dominant type of regulation mediated by a complex of molecules such astranscription factors. In general, regulatory molecules are of two types:activator and repressor. Activators promote the initiation of transcriptionwhereas repressors inhibit transcription. In many cases, they regulate the genetranscription on binding the promoter mutually exclusively and the observedgene expression response is either graded or binary. In experiments, the geneexpression response is quantified by the amount of proteins produced on varyingthe concentration of an external inducer molecules in the cell. In this paper,we study a gene regulatory network where activators and repressors both bindthe same promoter mutually exclusively. The network is modeled by assuming thatthe gene can be in three possible states: repressed, unregulated and active. Anexact analytical expression for the steady-state probability distribution ofprotein levels is then derived. The exact result helps to explain theexperimental observations that in the presence of activator molecules theresponse is graded at all inducer levels whereas in the presence of bothactivator and repressor molecules, the response is graded at low and highinducer levels and binary at an intermediate inducer level.

Conversion of graded to binary response in an activator-repressor system