Analytical Solutions for the Time-Dependent Dynamics of Stochastic Gene Expression with mRNA-sRNA Interactions

The antagonistic interaction between small RNAs (sRNAs) and messenger RNAs (mRNAs) constitutes a fundamental regulatory mechanism of gene expression in both prokaryotic and eukaryotic cells. However, the stochastic nature of transcription renders mean-field approximations inadequate for quantitative analysis of such systems. In the regime of strong sRNA-mRNA antagonism, we generalize the conventional probability-generating-function (PGF) framework and derive a novel approximate solution in the form of a generalized PGF, which can be analytically transformed into the time-dependent joint distribution of sRNA and mRNA via Laurent series expansion. The proposed approximation accurately captures the full stochastic dynamics across diverse systems exhibiting strong antagonism, while incorporating key biological features such as transcriptional burstiness, translation and sRNA recycling over the entire temporal range. Building on this analytical foundation, we further develop a generalized-PGF-based parameter-inference method that enables efficient and precise estimation of kinetic parameters, achieving inference speeds up to three orders of magnitude faster than traditional maximum-likelihood estimation approaches.