Exact burst-size distributions for gene-expression models with complex promoter structure

In prokaryotic and eukaryotic cells, most genes are transcribed in a bursty fashion on one hand and complex gene regulations may lead to complex promoter structure on the other hand. This raises an unsolved issue: how does promoter structure shape transcriptional bursting kinetics characterized by burst size and frequency? Here we analyze stochastic models of gene transcription, which consider complex regulatory mechanisms. Notably, we develop an efficient method to derive exact burst-size distributions. The analytical results show that if the promoter of a gene contains only one active state, the burst size indeed follows a geometric distribution, in agreement with the previous result derived under certain limiting conditions. However, if it contains a multitude of active states, the burst size in general obeys a non-geometric distribution, which is a linearly weighted sum of geometric distributions. This superposition principle reveals the essential feature of bursting kinetics in complex cases of transcriptional regulation although it seems that there has been no direct experimental confirmation. The derived burst-size distributions not only highlight the importance of promoter structure in regulating bursting kinetics, but can be also used in the exact inference of this kinetics based on experimental data.