Post-transcriptional regulation of noise in protein distributions during gene expression

First passage time in post-transcriptional regulation by multiple small RNAs

The post-transcriptional regulation of a protein by multiple small RNA molecules has been formulated as a stochastic process. An approximate solution of the master equation shows that the protein statistics can exhibit a generic form applicable for many regulatory scenarios. The first passage time (FPT) statistics has been obtained for regulation by single sRNA, with negative and positive regulations as limiting cases, as well as regulation by multiple sRNAs. The multiple sRNAs are able to independently control protein mean and variance, and we show that this is an advantageous mechanism to control FPT fluctuations in order to improve timing efficiency in post-transcriptional regulation.

Probing Mechanisms of Transcription Elongation Through Cell-to-Cell Variability of RNA Polymerase

The process of transcription initiation and elongation are primary points of control in the regulation of gene expression. Although biochemical studies have uncovered the mechanisms involved in controlling transcription at each step, how these mechanisms manifest in vivo at the level of individual genes is still unclear. Recent experimental advances have enabled single-cell measurements of RNA polymerase (RNAP) molecules engaged in the process of transcribing a gene of interest. In this article, we use Gillespie simulations to show that measurements of cell-to-cell variability of RNAP numbers and interpolymerase distances can reveal the prevailing mode of regulation of a given gene. Mechanisms of regulation at each step, from initiation to elongation dynamics, produce qualitatively distinct signatures, which can further be used to discern between them. Most intriguingly, depending on the initiation kinetics, stochastic elongation can either enhance or suppress cell-to-cell variability at the RNAP level. To demonstrate the value of this framework, we analyze RNAP number distribution data for ribosomal genes in Saccharomyces cerevisiae from three previously published studies and show that this approach provides crucial mechanistic insights into the transcriptional regulation of these genes.

Stochastic Modeling of Gene Regulation by Noncoding Small RNAs in the Strong Interaction Limit

Noncoding small RNAs (sRNAs) are known to play a key role in regulating diverse cellular processes, and their dysregulation is linked to various diseases such as cancer. Such diseases are also marked by phenotypic heterogeneity, which is often driven by the intrinsic stochasticity of gene expression. Correspondingly, there is significant interest in developing quantitative models focusing on the interplay between stochastic gene expression and regulation by sRNAs. We consider the canonical model of regulation of stochastic gene expression by sRNAs, wherein interaction between constitutively expressed sRNAs and mRNAs leads to stoichiometric mutual degradation. The exact solution of this model is analytically intractable given the nonlinear interaction term between sRNAs and mRNAs, and theoretical approaches typically invoke the mean-field approximation. However, mean-field results are inaccurate in the limit of strong interactions and low abundances; thus, alternative theoretical approaches are needed. In this work, we obtain analytical results for the canonical model of regulation of stochastic gene expression by sRNAs in the strong interaction limit. We derive analytical results for the steady-state generating function of the joint distribution of mRNAs and sRNAs in the limit of strong interactions and use the results derived to obtain analytical expressions characterizing the corresponding protein steady-state distribution. The results obtained can serve as building blocks for the analysis of genetic circuits involving sRNAs and provide new insights into the role of sRNAs in regulating stochastic gene expression in the limit of strong interactions.

Distribution of Initiation Times Reveals Mechanisms of Transcriptional Regulation in Single Cells

Transcription is the dominant point of control of gene expression. Biochemical studies have revealed key molecular components of transcription and their interactions, but the dynamics of transcription initiation in cells is still poorly understood. This state of affairs is being remedied with experiments that observe transcriptional dynamics in single cells using fluorescent reporters. Quantitative information about transcription initiation dynamics can also be extracted from experiments that use electron micrographs of RNA polymerases caught in the act of transcribing a gene (Miller spreads). Inspired by these data, we analyze a general stochastic model of transcription initiation and elongation and compute the distribution of transcription initiation times. We show that different mechanisms of initiation leave distinct signatures in the distribution of initiation times that can be compared to experiments. We analyze published data from micrographs of RNA polymerases transcribing ribosomal RNA genes in Escherichia coli and compare the observed distributions of interpolymerase distances with the predictions from previously hypothesized mechanisms for the regulation of these genes. Our analysis demonstrates the potential of measuring the distribution of time intervals between initiation events as a probe for dissecting mechanisms of transcription initiation in live cells.

A stochastic model for post-transcriptional regulation of rare events in gene expression

Post-transcriptional regulation of gene expression is often a critical component of cellular processes involved in cell-fate decisions. Correspondingly, considerable efforts have focused on modeling post-transcriptional regulation of stochastic gene expression and on quantifying its impact on the mean and variance of protein levels. However, the impact of post-transcriptional regulation on rare events corresponding to large deviations in gene expression is less well understood. Here, we study a simple model involving post-transcriptional control of gene expression and characterize the impact of regulation on large deviations in activity fluctuations. We derive analytical results for the large deviation function for protein production rate and for the corresponding driven process which characterizes system fluctuations conditional on the rare event. Our results suggest that fluctuations in burst size rather than frequency play a dominant role in controlling rare events. The results derived also provide insight into how post-transcriptional regulation can be used to fine-tune the probability of rare fluctuations and to thereby control phenotypic variation in a population of cells.

Stochastic gene expression conditioned on large deviations

The intrinsic stochasticity of gene expression can give rise to large fluctuations and rare events that drive phenotypic variation in a population of genetically identical cells. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression. Recent developments in non-equilibrium statistical mechanics have led to a framework for analyzing Markovian processes conditioned on rare events and for representing such processes by conditioning-free driven Markovian processes. We use this framework, in combination with approaches based on queueing theory, to analyze a general class of stochastic models of gene expression. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. We find that the conditioning-free driven process can also be represented by a BMAP that has the same form as the original process, but with renormalized parameters. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.

Frequency modulation of stochastic gene expression bursts by strongly interacting small RNAs

The sporadic nature of gene expression at the single-cell level-long periods of inactivity punctuated by bursts of mRNA or protein production-plays a critical role in diverse cellular processes. To elucidate the cellular role of bursting in gene expression, synthetic biology approaches have been used to design simple genetic circuits with bursty mRNA or protein production. Understanding how such genetic circuits can be designed with the ability to control burst-related parameters requires the development of quantitative stochastic models of gene expression. In this work, we analyze stochastic models for the regulation of gene expression bursts by strongly interacting small RNAs. For the parameter range considered, results based on mean-field approaches are significantly inaccurate and alternative analytical approaches are needed. Using simplifying approximations, we obtain analytical results for the corresponding steady-state distributions that are in agreement with results from stochastic simulations. These results indicate that regulation by small RNAs, in the strong interaction limit, can be used to effectively modulate the frequency of bursting. We explore the consequences of such regulation for simple genetic circuits involving feedback effects and switching between promoter states.

Mathematical modeling of combinatorial regulation suggests that apparent positive regulation of targets by miRNA could be an artifact resulting from competition for mRNA

MicroRNAs bind to and regulate the abundance and activity of target messenger RNA through sequestration, enhanced degradation, and suppression of translation. Although miRNA have a predominantly negative effect on the target protein concentration, several reports have demonstrated a positive effect of miRNA, i.e., increase in target protein concentration on miRNA overexpression and decrease in target concentration on miRNA repression. miRNA-target pair-specific effects such as protection of mRNA degradation owing to miRNA binding can explain some of these effects. However, considering such pairs in isolation might be an oversimplification of the RNA biology, as it is known that one miRNA interacts with several targets, and conversely target mRNA are subject to regulation by several miRNAs. We formulate a mathematical model of this combinatorial regulation of targets by multiple miRNA. Through mathematical analysis and numerical simulations of this model, we show that miRNA that individually have a negative effect on their targets may exhibit an apparently positive net effect when the concentration of one miRNA is experimentally perturbed by repression/overexpression in such a multi-miRNA multitarget situation. We show that this apparent unexpected effect is due to competition and will not be observed when miRNA interact noncompetitively with the target mRNA. This result suggests that some of the observed unusual positive effects of miRNA may be due to the combinatorial complexity of the system rather than due to any inherently unusual positive effect of the miRNA on its target.

Quantification of noise in bifunctionality-induced post-translational modification

We present a generic analytical scheme for the quantification of fluctuations due to bifunctionality-induced signal transduction within the members of a bacterial two-component system. The proposed model takes into account post-translational modifications in terms of elementary phosphotransfer kinetics. Sources of fluctuations due to autophosphorylation, kinase, and phosphatase activity of the sensor kinase have been considered in the model via Langevin equations, which are then solved within the framework of linear noise approximation. The resultant analytical expression of phosphorylated response regulators are then used to quantify the noise profile of biologically motivated single and branched pathways. Enhancement and reduction of noise in terms of extra phosphate outflux and influx, respectively, have been analyzed for the branched system. Furthermore, the role of fluctuations of the network output in the regulation of a promoter with random activation-deactivation dynamics has been analyzed.

Stabilizing gene regulatory networks through feedforward loops

The global dynamics of gene regulatory networks are known to show robustness to perturbations in the form of intrinsic and extrinsic noise, as well as mutations of individual genes. One molecular mechanism underlying this robustness has been identified as the action of so-called microRNAs that operate via feedforward loops. We present results of a computational study, using the modeling framework of stochastic Boolean networks, which explores the role that such network motifs play in stabilizing global dynamics. The paper introduces a new measure for the stability of stochastic networks. The results show that certain types of feedforward loops do indeed buffer the network against stochastic effects.

Effects of post-transcriptional regulation on phenotypic noise in Escherichia coli

Cell-to-cell variations in protein abundance, called noise, give rise to phenotypic variability between isogenic cells. Studies of noise have focused on stochasticity introduced at transcription, yet the effects of post-transcriptional regulatory processes on noise remain unknown. We study the effects of RyhB, a small-RNA of Escherichia coli produced on iron stress, on the phenotypic variability of two of its downregulated target proteins, using dual chromosomal fusions to fluorescent reporters and measurements in live individual cells. The total noise of each of the target proteins is remarkably constant over a wide range of RyhB production rates despite cells being in stress. In fact, coordinate downregulation of the two target proteins by RyhB reduces the correlation between their levels. Hence, an increase in phenotypic variability under stress is achieved by decoupling the expression of different target proteins in the same cell, rather than by an increase in the total noise of each. Extrinsic noise provides the dominant contribution to the total protein noise over the total range of RyhB production rates. Stochastic simulations reproduce qualitatively key features of our observations and show that a feed-forward loop formed by transcriptional extrinsic noise, an sRNA and its target genes exhibits strong noise filtration capabilities.

Stochastic modeling of regulation of gene expression by multiple small RNAs

A wealth of new research has highlighted the critical roles of small noncoding RNAs (sRNAs) in diverse processes, such as quorum sensing and cellular responses to stress. The pathways controlling these processes often have a central motif composed of a master regulator protein whose expression is controlled by multiple sRNAs. However, the stochastic gene expression of a single target gene regulated by multiple sRNAs is currently not well understood. To address this issue, we analyze a stochastic model of regulation of gene expression by multiple sRNAs. For this model, we derive exact analytic results for the regulated protein distribution, including compact expressions for its mean and variance. The derived results provide insights into the roles of multiple sRNAs in fine-tuning the noise in gene expression. In particular, we show that, in contrast to regulation by a single sRNA, multiple sRNAs provide a mechanism for independently controlling the mean and variance of the regulated protein distribution.

Stochastic single-gene autoregulation

A detailed stochastic model of single-gene autoregulation is established and its solutions are explored when mRNA dynamics is fast compared with protein dynamics and in the opposite regime. The model includes all the sources of randomness that are intrinsic to the autoregulation process and it considers both transcriptional and post-transcriptional regulation. The time-scale separation allows the derivation of analytic expressions for the equilibrium distributions of protein and mRNA. These distributions are generally well described in the continuous approximation, which is used to discuss the qualitative features of the protein equilibrium distributions as a function of the biological parameters in the fast mRNA regime. The performance of the time-scale approximation is assessed by comparison with simulations of the full stochastic system, and a good quantitative agreement is found for a wide range of parameter values. We show that either unimodal or bimodal equilibrium protein distributions can arise, and we discuss the autoregulation mechanisms associated with bimodality.

Gene expression noise facilitates adaptation and drug resistance independently of mutation

We show that the effect of stress on the reproductive fitness of noisy cell populations can be modeled as a first-passage time problem, and demonstrate that even relatively short-lived fluctuations in gene expression can ensure the long-term survival of a drug-resistant population. We examine how this effect contributes to the development of drug-resistant cancer cells, and demonstrate that permanent immunity can arise independently of mutations.

Connecting protein and mRNA burst distributions for stochastic models of gene expression

The intrinsic stochasticity of gene expression can lead to large variability in protein levels for genetically identical cells. Such variability in protein levels can arise from infrequent synthesis of mRNAs which in turn give rise to bursts of protein expression. Protein expression occurring in bursts has indeed been observed experimentally and recent studies have also found evidence for transcriptional bursting, i.e. production of mRNAs in bursts. Given that there are distinct experimental techniques for quantifying the noise at different stages of gene expression, it is of interest to derive analytical results connecting experimental observations at different levels. In this work, we consider stochastic models of gene expression for which mRNA and protein production occurs in independent bursts. For such models, we derive analytical expressions connecting protein and mRNA burst distributions which show how the functional form of the mRNA burst distribution can be inferred from the protein burst distribution. Additionally, if gene expression is repressed such that observed protein bursts arise only from single mRNAs, we show how observations of protein burst distributions (repressed and unrepressed) can be used to completely determine the mRNA burst distribution. Assuming independent contributions from individual bursts, we derive analytical expressions connecting means and variances for burst and steady-state protein distributions. Finally, we validate our general analytical results by considering a specific reaction scheme involving regulation of protein bursts by small RNAs. For a range of parameters, we derive analytical expressions for regulated protein distributions that are validated using stochastic simulations. The analytical results obtained in this work can thus serve as useful inputs for a broad range of studies focusing on stochasticity in gene expression.

Intrinsic noise in stochastic models of gene expression with molecular memory and bursting

Regulation of intrinsic noise in gene expression is essential for many cellular functions. Correspondingly, there is considerable interest in understanding how different molecular mechanisms of gene expression impact variations in protein levels across a population of cells. In this work, we analyze a stochastic model of bursty gene expression which considers general waiting-time distributions governing arrival and decay of proteins. By mapping the system to models analyzed in queueing theory, we derive analytical expressions for the noise in steady-state protein distributions. The derived results extend previous work by including the effects of arbitrary probability distributions representing the effects of molecular memory and bursting. The analytical expressions obtained provide insight into the role of transcriptional, post-transcriptional, and post-translational mechanisms in controlling the noise in gene expression.