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