Skewness and Kurtosis of mRNA Distributions in Stochastic Gene Transcription with Promoter Switching

Gene expression is inherently stochastic, and promoter switching–induced transcriptional bursting generates substantial cell-to-cell variability in mRNA abundance. Such variability is commonly characterized by the mean and variance; however, these low-order statistics fail to capture the geometric features of mRNA copy number distributions and may obscure mechanistic differences in promoter dynamics. In this work, we analyze a two-state stochastic gene transcription model and derive explicit analytical expressions for higher-order moments of mRNA abundance. We show that skewness and kurtosis provide mechanistically informative signatures of transcriptional bursting, explicitly depending on promoter switching kinetics and burst size. In particular, positive skewness increases with slower promoter switching and larger burst sizes, even when the mean expression level is fixed, while elevated kurtosis distinguishes burst-dominated, low-expression regimes from Gaussian-like high-expression regimes. Our results demonstrate that distinct promoter dynamics can produce identical mean expression levels and variances while exhibiting markedly different skewness and kurtosis. Incorporating higher-order statistics, therefore, extends conventional mean–variance analyses and enables improved discrimination between competing stochastic gene expression mechanisms in single-cell data.