  The production processes of proteins in prokaryotic cells are investigated.
Most of the mathematical models in the literature study the production of {\em
one} fixed type of proteins. When several classes of proteins are considered,
an important additional aspect has to be taken into account, the limited common
resources of the cell (polymerases and ribosomes) used by the production
process. Understanding the impact of this limitation is a key issue in this
domain. In this paper we focus on the allocation of ribosomes in the case of
the production of multiple proteins. The cytoplasm of the cell being a
disorganized medium subject to thermal noise, the protein production process
has an important stochastic component. For this reason, a Markovian model of
this process is introduced. Asymptotic results of the equilibrium are obtained
under a scaling procedure and a realistic biological assumption of saturation
of the ribosomes available in the cell. It is shown in particular that, in the
limit, the number of non-allocated ribosomes at equilibrium converges in
distribution to a Poisson distribution whose parameter satisfies a fixed point
equation. It is also shown that the production process of different types of
proteins can be seen as independent production processes but with modified
parameters.
