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Many of the existing stochastic models of gene expression contain thefirst-order decay reaction term that may describe active protein degradation ordilution. If the model variable is interpreted as the molecule number, and notconcentration, the decay term may also approximate the loss of proteinmolecules due to cell division as a continuous degradation process. The seminalmodel of that kind leads to gamma distributions of protein levels, whoseparameters are defined by the mean frequency of protein bursts and mean burstsize. However, such models (whether interpreted in terms of molecule numbers orconcentrations) do not correctly account for the noise due to proteinpartitioning between daughter cells.  We present an exactly solvable stochastic model of gene expression in cellsdividing at random times, where we assume description in terms of moleculenumbers with a constant mean protein burst size. The model is based on apopulation balance equation supplemented with protein production in randombursts.  If protein molecules are partitioned equally between daughter cells, weobtain at steady state the analytical expressions for probability distributionssimilar in shape to gamma distribution, yet with quite different values of meanburst size and mean burst frequency than would result from fitting of theclassical continuous-decay model to these distributions. For randompartitioning of protein molecules between daughter cells, we obtain the momentequations for the protein number distribution and thus the analytical formulasfor the squared coefficient of variation.

Exactly solvable model of gene expression in proliferating bacterial cell population with stochastic protein bursts and protein partitioning