  We present a theoretical framework to analyze the dynamics of gene expression
with stochastic bursts. Beginning with an individual-based model which fully
accounts for the messenger RNA (mRNA) and protein populations, we propose a
novel expansion of the master equation for the joint process. The resulting
coarse-grained model reduces the dimensionality of the system, describing only
the protein population while fully accounting for the effects of discrete and
fluctuating mRNA population. Closed form expressions for the stationary
distribution of the protein population and mean first-passage times of the
coarse-grained model are derived and large-scale Monte Carlo simulations show
that the analysis accurately describes the individual-based process accounting
for mRNA population, in contrast to the failure of commonly proposed
diffusion-type models.
