Exact results for stochastic transcription with promoter-proximal pausing and initiation-state-dependent lifetimes

Promoter-proximal pausing is a key regulatory checkpoint in RNA polymerase II transcription, yet current models lack a unified description of how arrival-state-dependent pause heterogeneity shapes nascent transcriptional occupancy, completion flux, and steady-state mature mRNA levels. We introduce a mechanistic framework in which the promoter-proximal pause gate is a continuous-time Markov chain that modulates an M/G/∞ queue. Each initiation event samples a gate-at-initiation state that fixes the entire episode lifetime-pause waiting, early elongation, and termination-yielding exact analytical expressions for transient and steady-state nascent concurrency, and the completion flux. From this structure, we derive an explicit gate-induced pause-initiation limit: a maximal productive initiation rate determined solely by the finite capacity and residence time of the promoter-proximal window. This bound provides a mechanistic explanation for experimentally inferred ceilings on transcriptional activation and clarifies how perturbations affecting pause duration, elongation speed, or termination latency move genes along the same constraint. We further obtain interpretable formulas for the promoter-proximal pausing index and its dependence on pause-state composition. Together, our results establish a compact mathematical theory linking microscopic pausing kinetics to macroscopic transcriptional outputs, offering general principles for how promoter-proximal regulation shapes gene expression dynamics.