  Many of the existing stochastic models of gene expression contain the
first-order decay reaction term that may describe active protein degradation or
dilution. If the model variable is interpreted as the molecule number, and not
concentration, the decay term may also approximate the loss of protein
molecules due to cell division as a continuous degradation process. The seminal
model of that kind leads to gamma distributions of protein levels, whose
parameters are defined by the mean frequency of protein bursts and mean burst
size. However, such models (whether interpreted in terms of molecule numbers or
concentrations) do not correctly account for the noise due to protein
partitioning between daughter cells.
  We present an exactly solvable stochastic model of gene expression in cells
dividing at random times, where we assume description in terms of molecule
numbers with a constant mean protein burst size. The model is based on a
population balance equation supplemented with protein production in random
bursts.
  If protein molecules are partitioned equally between daughter cells, we
obtain at steady state the analytical expressions for probability distributions
similar in shape to gamma distribution, yet with quite different values of mean
burst size and mean burst frequency than would result from fitting of the
classical continuous-decay model to these distributions. For random
partitioning of protein molecules between daughter cells, we obtain the moment
equations for the protein number distribution and thus the analytical formulas
for the squared coefficient of variation.
