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

Gene transcription 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. Our results demonstrate that distinct promoter dynamics can produce identical mean expression levels and variances while exhibiting markedly different skewness and kurtosis. The explicit analytical expressions derived here reveal how higher-order moments encode mechanistically informative signatures of transcriptional bursting through distributional asymmetry and heavy-tailed behavior. These results demonstrate that higher-order moments encode mechanistic information beyond mean–variance statistics and provide a powerful framework for distinguishing between different promoter-switching mechanisms in stochastic gene transcription.