Stochastic Kinetics of mRNA Molecules in a General Transcription Model

TL;DR

This paper develops a comprehensive, efficient stochastic model for mRNA transcription, providing new mathematical bounds and insights into the distribution of mRNA counts, applicable to complex biological systems.

Contribution

It introduces a unified framework for stochastic transcription modeling, deriving new inequalities and proving the Heavy-Tailed Law for mRNA count distributions.

Findings

Distribution of mRNA counts is upper-bounded by a scaled Poisson distribution.

New inequalities for binomial moments and probability mass functions are established.

The proposed method reduces computational complexity compared to existing numerical approaches.

Abstract

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 over-simplified 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.