Stochastic kinetics of mRNA molecules in a general transcription model

Stochastic modeling of transcription is a classic yet long-standing problem in theoretical biophysics. The lack of unified results and a computationally efficient approach for a general, fine-grained transcription model has confined relevant research to some oversimplified special cases like the telegraph model. This article establishes a general, unified, and computationally efficient framework for studying stochastic transcription kinetics. We consider a chemical reaction model of transcription and construct the time-dependent solution to the corresponding chemical master equation. A well-known matrix-form expression for steady-state binomial moments is recovered by calculating the temporal limit of the time-dependent dynamics. Two novel inequalities for binomial moments and the probability mass function are derived using techniques from functional analysis. It follows that the distribution of mRNA counts is upper-bounded by a constant multiple of Poisson distribution, thus mathematically proving the main statement of the heavy-tailed law. Additionally, the standard binomial moment method is analyzed from a numerical perspective, where truncation error is estimated using our inequalities. Compared with some widely used numerical methods, a key advantage of this result is the significantly lower computational complexity.