  In this paper we study an important global regulation mechanism of
transcription of biological cells using specific macro-molecules, 6S RNAs. The
functional property of 6S RNAs is of blocking the transcription of RNAs when
the environment of the cell is not favorable. We investigate the efficiency of
this mechanism with a scaling analysis of a stochastic model. The evolution
equations of our model are driven by the law of mass action and the total
number of polymerases is used as a scaling parameter. Two regimes are analyzed:
exponential phase when the environment of the cell is favorable to its growth,
and the stationary phase when resources are scarce. In both regimes, by
defining properly occupation measures of the model, we prove an averaging
principle for the associated multi-dimensional Markov process on a convenient
timescale, as well as convergence results for fast variables of the system. An
analytical expression of the asymptotic fraction of sequestrated polymerases in
stationary phase is in particular obtained. The consequences of these results
are discussed.
