Figure 1

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 k_on and k_off 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 k_on and k_off 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 k_on and k_off 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 k_on and k_off 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 [25]. (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.