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The bulk of stochastic gene expression models in the literature do not havean explicit description of the age of a cell within a generation and hence theycannot capture events such as cell division and DNA replication. Instead, manymodels incorporate cell cycle implicitly by assuming that dilution due to celldivision can be described by an effective decay reaction with first-orderkinetics. If it is further assumed that protein production occurs in burststhen the stationary protein distribution is a negative binomial. Here we seekto understand how accurate these implicit models are when compared with moredetailed models of stochastic gene expression. We derive the exact stationarysolution of the chemical master equation describing bursty protein dynamics,binomial partitioning at mitosis, age-dependent transcription dynamicsincluding replication, and random interdivision times sampled from Erlang ormore general distributions; the solution is different for single lineage andpopulation snapshot settings. We show that protein distributions are wellapproximated by the solution of implicit models (a negative binomial) when themean number of mRNAs produced per cycle is low and the cell cycle lengthvariability is large. When these conditions are not met, the distributions areeither almost bimodal or else display very flat regions near the mode andcannot be described by implicit models. We also show that for genes with lowtranscription rates, the size of protein noise has a strong dependence on thereplication time, it is almost independent of cell cycle variability forlineage measurements and increases with cell cycle variability for populationsnapshot measurements. In contrast for large transcription rates, the size ofprotein noise is independent of replication time and increases with cell cyclevariability for both lineage and population measurements.

Exact solution of stochastic gene expression models with bursting, cell cycle and replication dynamics