Poisson-beta model. (A) Schematic of a two-state kinetic model for stochastic gene expression. (B) Heat map of the maximum P values of two goodness-of-fit tests for Poisson and negative binomial distributions. One thousand combinations of kon and koff were uniformly sampled from the log space by fixing s to 100. For each combination of the sampled parameters, 1,000 independent samples were generated from the Poisson-beta distribution to evaluate the fit of the data to the Poisson and negative binomial distributions using a bootstrap-based goodness-of-fit test. The colors represent minus log10-transformed P values and the heat map is interpolated from the scattered data by using a Delaunay triangulation method. (C) Heat map of the Fano factor as a function of kon and koff with a fixed rate of transcription (s = 100). Along the black dashed line fixing the average number of mRNA molecules to 20, the four combinations of kon and koff give the varied level of the Fano factor and show different patterns of the variability of the number of mRNA molecules between cells. At point 1 with the highest Fano factor, the transitions between the two promoter states are slow, and the standardized expression level of a gene exhibits a U-shaped distribution, resulting in a bimodal distribution. At point 2, the transition to the inactive state is faster than the transition to the active state, and therefore the mRNA distribution has a long right tail resulting from occasional transcriptional bursts. As kon and koff increase at points 3 and 4, transitions between promoter states become fast, resulting in a Poisson-like distribution of the number of mRNA molecules with the Fano factor approaching 1. Note that this plot is similar to a recent figure generated by . (D) Representative Poisson-beta distributions from four points in (C), which were computed with the auxiliary variable approach. (E) The corresponding beta distributions of p.