  We study a stochastic model of gene expression, in which protein production
has a form of random bursts whose size distribution is arbitrary, whereas
protein decay is a first-order reaction. We find exact analytical expressions
for the time evolution of the cumulant-generating function for the most general
case when both the burst size probability distribution and the model parameters
depend on time in an arbitrary (e.g. oscillatory) manner, and for arbitrary
initial conditions. We show that in the case of periodic external activation
and constant protein degradation rate, the response of the gene is analogous to
the RC low-pass filter, where slow oscillations of the external driving have a
greater effect on gene expression than the fast ones. We also demonstrate that
the $n$-th cumulant of the protein number distribution depends on the $n$-th
moment of the burst size distribution. We use these results to show that
different measures of noise (coefficient of variation, Fano factor, fractional
change of variance) may vary in time in a different manner. Therefore, any
biological hypothesis of evolutionary optimization based on the nonmonotonicity
of a chosen measure of noise must justify why it assumes that biological
evolution quantifies noise in that particular way. Finally, we show that not
only for exponentially distributed burst sizes but also for a wider class of
burst size distributions (e.g. Dirac delta and gamma) the control of gene
expression level by burst frequency modulation gives rise to proportional
scaling of variance of the protein number distribution to its mean, whereas the
control by amplitude modulation implies proportionality of protein number
variance to the mean squared.
