Transcriptional bursting in gene expression: analytical results for general stochastic models

Transcriptional pulsing has been observed in both prokaryotes and eukaryotes and plays a crucial role in cell to cell variability of protein and mRNA numbers. The issue is how the time constants associated with episodes of transcriptional bursting impact cellular mRNA and protein distributions and reciprocally, to what extent experimentally observed distributions can be attributed to transcriptional pulsing. We address these questions by investigating the exact time-dependent...

This paper considers adiabatic reduction in both discrete and continuous models of stochastic gene expression. In gene expression models, the concept of bursting is a production of several molecules simultaneously and is generally represented as a compound Poisson process of random size. In a general two-dimensional birth and death discrete model, we prove that under specific assumptions and scaling (that are characteristics of the mRNA-protein system) an adiabatic reduction ...

Gene expression is a fundamental process in a living system. The small RNAs (sRNAs) is widely observed as a global regulator in gene expression. The inherent nonlinearity in this regulatory process together with the bursty production of messenger RNA (mRNA), sRNA and protein make the exact solution for this stochastic process intractable. This is particularly the case when quantifying the protein noise level, which has great impact on multiple cellular processes. Here we prop...

The bulk of stochastic gene expression models in the literature do not have an explicit description of the age of a cell within a generation and hence they cannot capture events such as cell division and DNA replication. Instead, many models incorporate cell cycle implicitly by assuming that dilution due to cell division can be described by an effective decay reaction with first-order kinetics. If it is further assumed that protein production occurs in bursts then the station...

Signal-processing molecules inside cells are often present at low copy number, which necessitates probabilistic models to account for intrinsic noise. Probability distributions have traditionally been found using simulation-based approaches which then require estimating the distributions from many samples. Here we present in detail an alternative method for directly calculating a probability distribution by expanding in the natural eigenfunctions of the governing equation, wh...

The canonical model of mRNA expression is the telegraph model, describing a gene that switches on and off, subject to transcription and decay. It describes steady-state mRNA distributions that subscribe to transcription in bursts with first-order decay, referred to as super-Poissonian expression. Using a telegraph-like model, I propose an answer to the question of why gene expression is bursty in the first place, and what benefits it confers. Using analytics for the entropy p...

Here we develop an effective approach to simplify two-time-scale Markov chains with infinite state spaces by removal of states with fast leaving rates, which improves the simplification method of finite Markov chains. We introduce the concept of fast transition paths and show that the effective transitions of the reduced chain can be represented as the superposition of the direct transitions and the indirect transitions via all the fast transition paths. Furthermore, we apply...

We explore a Markov model used in the analysis of gene expression, involving the bursty production of pre-mRNA, its conversion to mature mRNA, and its consequent degradation. We demonstrate that the integration used to compute the solution of the stochastic system can be approximated by the evaluation of special functions. Furthermore, the form of the special function solution generalizes to a broader class of burst distributions. In light of the broader goal of biophysical p...

Due to the stochastic nature of biochemical processes, the copy number of any given type of molecule inside a living cell often exhibits large temporal fluctuations. Here, we develop analytic methods to investigate how the noise arising from a bursting input is reshaped by a transport reaction which is either linear or of the Michaelis-Menten type. A slow transport rate smoothes out fluctuations at the output end and minimizes the impact of bursting on the downstream cellular...

The processes, resulting in the transcription of RNA, are intrinsically noisy. It was observed experimentally that the synthesis of mRNA molecules is driven by short, burst-like, events. An accurate prediction of the protein level often requires one to take these fluctuations into account. Here, we consider the stochastic model of gene expression regulated by small RNAs. Small RNA post-transcriptional regulation is achieved by base-pairing with mRNA. We show that in a strong ...