This paper analyzes, in the context of a prokaryotic cell, the stochastic
variability of the number of proteins when there is a control of gene
expression by an autoregulation scheme. The goal of this work is to estimate
the efficiency of the regulation to limit the fluctuations of the number of
copies of a given protein. The autoregulation considered in this paper relies
mainly on a negative feedback: the proteins are repressors of their own gene
expression. The efficiency of a production process without feedback control is
compared to a production process with an autoregulation of the gene expression
assuming that both of them produce the same average number of proteins. The
main characteristic used for the comparison is the standard deviation of the
number of proteins at equilibrium. With a Markovian representation and a simple
model of repression, we prove that, under a scaling regime, the repression
mechanism follows a Hill repression scheme with an hyperbolic control. An
explicit asymptotic expression of the variance of the number of proteins under
this regulation mechanism is obtained. Simulations are used to study other
aspects of autoregulation such as the rate of convergence to equilibrium of the
production process and the case where the control of the production process of
proteins is achieved via the inhibition of mRNAs