Before revision:  A representation of the distribution of the number of proteins and mRNAs at equilibrium is also obtained. The results rely on the representation of the process as a functional of a Poisson process in an extended state space. Classical approximations of the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results.
After revision:  Approximations used in the biological literature for the equilibrium distribution of the number of proteins are discussed and investigated in the light of these results.