This paper investigates the stochastic fluctuations of the number of copies
of a given protein in a cell. This problem has already been addressed in the
past and closed-form expressions of the mean and variance have been obtained
for a simplified stochastic model of the gene expression. These results have
been obtained under the assumption that the duration of all the protein
production steps are exponentially distributed. In such a case, a Markovian
approach (via Fokker-Planck equations) is used to derive analytic formulas of
the mean and the variance of the number of proteins at equilibrium. This
assumption is however not totally satisfactory from a modeling point of view
since the distribution of the duration of some steps is more likely to be
Gaussian, if not almost deterministic. In such a setting, Markovian methods can
no longer be used. A finer characterization of the fluctuations of the number
of proteins is therefore of primary interest to understand the general economy
of the cell. In this paper, we propose a new approach, based on marked Poisson
point processes, which allows to remove the exponential assumption. This is
applied in the framework of the classical three stages models of the
literature: transcription, translation and degradation. The interest of the
method is shown by recovering the classical results under the assumptions that
all the durations are exponentially distributed but also by deriving new
analytic formulas when some of the distributions are not anymore exponential.
Our results show in particular that the exponential assumption may,
surprisingly, underestimate significantly the variance of the number of
proteins when some steps are in fact not exponentially distributed. This
counter-intuitive result stresses the importance of the statistical assumptions
in the protein production process