Theory of transcription bursting: stochasticity in the transcription rates

Transcription bursting creates variation among the individuals of a given population. Bursting emerges as the consequence of turning on and off the transcription process randomly. There are at least three sub-processes involved in the bursting phenomenon with different timescale regimes viz. flipping across the on–off state channels, microscopic transcription elongation events and the mesoscopic transcription dynamics along with the mRNA recycling. We demonstrate that when the flipping dynamics is coupled with the microscopic elongation events, then the distribution of the resultant transcription rates will be over-dispersed. This in turn reflects as the transcription bursting with over-dispersed non-Poisson type distribution of mRNA numbers. We further show that there exist optimum flipping rates (αC, βC) at which the stationary state Fano factor and variance associated with the mRNA numbers attain maxima. These optimum points are connected via αC=βCβC+γr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \alpha_{C} = \sqrt {\beta_{C} \left( {\beta_{C} + \gamma_{r} } \right)} $$\end{document}. Here α is the rate of flipping from the on-state to the off-state, β is the rate of flipping from the off-state to the on-state and γr is the decay rate of mRNA. When α = β = χ with zero rate in the off-state channel, then there exist optimum flipping rates at which the non-stationary Fano factor and variance attain maxima. Here χC,v≃3kr+/21+kr+t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi_{C,v} \simeq {{3k_{r}^{ + } } \mathord{\left/ {\vphantom {{3k_{r}^{ + } } {2\left( {1 + k_{r}^{ + } t} \right)}}} \right. \kern-0pt} {2\left( {1 + k_{r}^{ + } t} \right)}} $$\end{document} (here kr+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k_{r}^{ + } $$\end{document} is the rate of transcription purely through the on-state elongation channel) is the optimum flipping rate at which the variance of mRNA attains a maximum and χC,κ≃1.72/t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \chi_{C,\kappa } \simeq {{1.72} \mathord{\left/ {\vphantom {{1.72} t}} \right. \kern-0pt} t} $$\end{document} is the optimum flipping rate at which the Fano factor attains a maximum. Close observation of the transcription mechanism reveals that the RNA polymerase performs several rounds of stall-continue type dynamics before generating a complete mRNA. Based on this observation, we model the transcription event as a stochastic trajectory of the transcription machinery across these on–off state elongation channels. Each mRNA transcript follows different trajectory. The total time taken by a given trajectory is the first passage time (FPT). Inverse of this FPT is the resultant transcription rate associated with the particular mRNA. Therefore, the time required to generate a given mRNA transcript will be a random variable. For a stall-continue type dynamics of RNA polymerase, we show that the overall average transcription rate can be expressed as kr≃h∞+kr+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k_{r} \simeq h_{\infty }^{ + } k_{r}^{ + } $$\end{document} where kr+≃λr+/L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ k_{r}^{ + } \simeq {{\lambda_{r}^{ + } } \mathord{\left/ {\vphantom {{\lambda_{r}^{ + } } L}} \right. \kern-0pt} L} $$\end{document}, λr+ is the microscopic transcription elongation rate in the on-state channel and L is the length of a complete mRNA transcript and h∞+ = [β/(α + β)] is the stationary state probability of finding the transcription machinery in the on-state channel.