  In this paper we analyze the equilibrium properties of a large class of
stochastic processes describing the fundamental biological process within
bacterial cells, {\em the production process of proteins}. Stochastic models
classically used in this context to describe the time evolution of the numbers
of mRNAs and proteins are presented and discussed. An extension of these
models, which includes elongation phases of mRNAs and proteins, is introduced.
A convergence result to equilibrium for the process associated to the number of
proteins and mRNAs is proved and a representation of this equilibrium as a
functional of a Poisson process in an extended state space is obtained.
Explicit expressions for the first two moments of the number of mRNAs and
proteins at equilibrium are derived, generalizing some classical formulas.
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. Several convergence results for the distribution of the
number of proteins at equilibrium are in particular obtained under different
scaling assumptions.
