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We study a stochastic model of gene expression, in which protein productionhas a form of random bursts whose size distribution is arbitrary, whereasprotein decay is a first-order reaction. We find exact analytical expressionsfor the time evolution of the cumulant-generating function for the most generalcase when both the burst size probability distribution and the model parametersdepend on time in an arbitrary (e.g. oscillatory) manner, and for arbitraryinitial conditions. We show that in the case of periodic external activationand constant protein degradation rate, the response of the gene is analogous tothe RC low-pass filter, where slow oscillations of the external driving have agreater effect on gene expression than the fast ones. We also demonstrate thatthe $n$-th cumulant of the protein number distribution depends on the $n$-thmoment of the burst size distribution. We use these results to show thatdifferent measures of noise (coefficient of variation, Fano factor, fractionalchange of variance) may vary in time in a different manner. Therefore, anybiological hypothesis of evolutionary optimization based on the nonmonotonicityof a chosen measure of noise must justify why it assumes that biologicalevolution quantifies noise in that particular way. Finally, we show that notonly for exponentially distributed burst sizes but also for a wider class ofburst size distributions (e.g. Dirac delta and gamma) the control of geneexpression level by burst frequency modulation gives rise to proportionalscaling of variance of the protein number distribution to its mean, whereas thecontrol by amplitude modulation implies proportionality of protein numbervariance to the mean squared.

Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound Poisson process