ANALYTICAL RESULTS FOR A MULTISTATE GENE MODEL

Gene expression is a complicated multistep process where transcription depends on a chromatin template that can be characterized by stochastic transitions between active and inactive states of promoter. Here, we develop general stochastic models of bursty gene expression, which extend previous models by incorporating slow dynamics of transcription. We derive the analytical expression for the messenger ribonucleic acid (mRNA) probability distribution and discuss it in the limit of some reaction rates. By analyzing experimentally measurable indices such as mean, variance, noise intensity, and Fano factor, we analytically demonstrate that inactive phases of promoter can tune the mRNA or protein noise intensity to the minimum independently from the mean expression, with an inference that previous results based on two-stage models overestimate the noise in mRNA. We also present analytical results for the waiting-time distribution, showing that it has the maximum. These results not only explain well biological phenomena observed in recent experiments, which cannot be explained using previously simplified gene models, but also predict some new phenomena that have not yet been observed in experiments.