Analytical distribution and tunability of noise in a model of promoter progress

Chromatin template (CT), which accumulates over time until the promoter becomes active, determines upstream dynamics of transcription, but how upstream sequential steps impact downstream dynamics qualitatively and quantitatively is unclear. Here, we analyze a stochastic gene model with a simple yet typical CT that contains one active state and several inactive states of the promoter. We derive the analytical expressions for the noise in mRNA probability distributions governed by master equations. The derived results extend previous work by including the effects of promoter progress on variability and bimodality. Specifically, given a CT for transcription, we analytically demonstrate that inactive phases of the promoter can modulate the noise intensity to the minimum independently of the mean expression of mRNA. If one new inactive state is added to the CT, then the resulting noise will be reduced, implying that the multi-off mechanism plays a role of attenuating the noise. In contrast to the simple on-off mechanism, the multi-off mechanism can also narrow bimodal regions in a certain parameter plane and obscure two peaks, explaining why bimodal distributions are rarely observed in experiments. Our results provide insight into the role of promoter progress in determining the level of cell-to-cell variability in gene expression.