Approximate Analytical Protein Distributions for the Three-stage Model of Stochastic Gene Expression
Gene expression is an intrinsically stochastic process that generates phenotypic heterogeneity within genetically identical cell populations. While the exact statistical moments of the protein count can be obtained for a broad range of complex models, the corresponding distributions are significantly harder to obtain and intractable in many cases. The classical three-stage model of gene expression, which predicts fluctuations in protein levels as a function of promoter switching, transcription, translation, and degradation events all occurring with linear propensities, illustrates this perfectly; deriving its exact protein distribution remains elusive. Here, using the partitioning property of time-inhomogeneous Poisson processes, we develop an exact mapping of the three-stage model onto a simplified model. The simplified model allows us to formulate two analytical approximations for the full protein distribution of the three-stage model based on a beta-mixture representation of the exact solution for a simpler model. We show that the two approximations are asymptotically exact in different limiting cases and verify their accuracy against simulations for a broad range of parameters. Although approximate, these are the first analytical expressions for protein distributions for the three-stage model that are highly accurate in intermediate regimes.