  The bulk of stochastic gene expression models in the literature do not have
an explicit description of the age of a cell within a generation and hence they
cannot capture events such as cell division and DNA replication. Instead, many
models incorporate cell cycle implicitly by assuming that dilution due to cell
division can be described by an effective decay reaction with first-order
kinetics. If it is further assumed that protein production occurs in bursts
then the stationary protein distribution is a negative binomial. Here we seek
to understand how accurate these implicit models are when compared with more
detailed models of stochastic gene expression. We derive the exact stationary
solution of the chemical master equation describing bursty protein dynamics,
binomial partitioning at mitosis, age-dependent transcription dynamics
including replication, and random interdivision times sampled from Erlang or
more general distributions; the solution is different for single lineage and
population snapshot settings. We show that protein distributions are well
approximated by the solution of implicit models (a negative binomial) when the
mean number of mRNAs produced per cycle is low and the cell cycle length
variability is large. When these conditions are not met, the distributions are
either almost bimodal or else display very flat regions near the mode and
cannot be described by implicit models. We also show that for genes with low
transcription rates, the size of protein noise has a strong dependence on the
replication time, it is almost independent of cell cycle variability for
lineage measurements and increases with cell cycle variability for population
snapshot measurements. In contrast for large transcription rates, the size of
protein noise is independent of replication time and increases with cell cycle
variability for both lineage and population measurements.
