  Transcriptional pulsing has been observed in both prokaryotes and eukaryotes
and plays a crucial role in cell to cell variability of protein and mRNA
numbers. The issue is how the time constants associated with episodes of
transcriptional bursting impact cellular mRNA and protein distributions and
reciprocally, to what extent experimentally observed distributions can be
attributed to transcriptional pulsing. We address these questions by
investigating the exact time-dependent solution of the Master equation for a
transcriptional pulsing model of mRNA distributions. We find a plethora of
results: we show that, among others, bimodal and long-tailed (power law)
distributions occur in the steady state as the rate constants are varied over
biologically significant time scales. Since steady state distributions may not
be reached experimentally we present results for the time evolution of the
distributions. Because cellular behavior is essentially determined by proteins,
we investigate the effect of the different mRNA distributions on the
corresponding protein distributions. We delineate the regimes of rate constants
for which the protein distribution mimics the mRNA distribution and those for
which the protein distribution deviates significantly from the mRNA
distribution.
