Before revision:  These distributions are shown to be well described in the continuous approximation, which is then used to discuss the qualitative features of the protein equilibrium distributions as a function of the biological parameters in the fast mRNA regime  .
After revision:  These distributions are shown to be well described in the continuous approximation, which is then used to discuss the qualitative features of the protein equilibrium distributions as a function of the biological parameters in the fast mRNA regime . The performance of the timescale approximation is assessed by comparison with simulations of the full stochastic system, and a good quantitative agreement is found for a wide range of parameter values. We show that either unimodal or bimodal equilibrium protein distributions can arise, and we discuss the auto-regulation mechanisms associated with bimodality .