Homogeneous cell populations can exhibit considerable cell-to-cell variability in protein levels arising from the stochastic nature of the gene-expression process. In particular, transcriptional bursting of mRNAs from the promoter has been implicated as a major source of stochasticity in the expression of many genes. In eukaryotes, transcribed pre-mRNAs have to be exported outside the nucleus and in many cases, export rates can be slow and comparable to mRNA turnover rates. We investigate whether such export processes can be effective mechanisms in buffering protein levels from transcriptional bursting of pre-mRNAs in the nucleus. For a stochastic gene-expression model with both transcriptional bursting and export, we derive an exact solution of the steady-state probability-generating function for both the nuclear and the cytoplasmic mRNA levels. These formulas reveal that decreasing export rates can dramatically reduce variability in cytoplasmic mRNA levels. However, our results also show that decreasing export rates enhance mRNA autocorrelation times, which function to increase heterogeneity in protein levels. Our overall analysis concludes that under physiologically relevant parameter regimes, a pre-mRNA export step can decrease steady-state variability at the mRNA level but not at the protein level. Finally, we reinforce previous observations that saturation in the pre-mRNA transport machinery can be an important mechanism in suppressing protein variability from underlying transcriptional bursts.

Gene expression at the single-cell level incorporates reaction mechanisms which are intrinsically stochastic as they involve molecular species present at low copy numbers. The dynamics of these mechanisms can be described quantitatively using stochastic master-equation modelling; in this paper we study a generic gene-expression model of this kind which explicitly includes the representations of the processes of transcription and translation. For this model we determine the generating function of the steady-state distribution of mRNA and protein counts and characterise the underlying probability law using a combination of analytic, asymptotic and numerical approaches, finding that the distribution may assume a number of qualitatively distinct forms. The results of the analysis are suitable for comparison with single-molecule resolution gene-expression data emerging from recent experimental studies.

Stochastic phenomena in gene regulatory networks can be modelled by the chemical master equation for gene products such as mRNA and proteins. If some of these elements are present in significantly higher amounts than the rest, or if some of the reactions between these elements are substantially faster than others, it is often possible to reduce the master equation to a simpler problem using asymptotic methods. We present examples of such a procedure and analyse the relationship between the reduced models and the original.

Transcription factors (TFs) interact with a multitude of binding sites on DNA and partner proteins inside cells. We investigate how nonspecific binding/unbinding to such decoy binding sites affects the magnitude and time-scale of random fluctuations in TF copy numbers arising from stochastic gene expression. A stochastic model of TF gene expression, together with decoy site interactions is formulated. Distributions for the total (bound and unbound) and free (unbound) TF levels are derived by analytically solving the chemical master equation under physiologically relevant assumptions. Our results show that increasing the number of decoy binding sides considerably reduces stochasticity in free TF copy numbers. The TF autocorrelation function reveals that decoy sites can either enhance or shorten the time-scale of TF fluctuations depending on model parameters. To understand how noise in TF abundances propagates downstream, a TF target gene is included in the model. Intriguingly, we find that noise in the expression of the target gene decreases with increasing decoy sites for linear TF-target protein dose-responses, even in regimes where decoy sites enhance TF autocorrelation times. Moreover, counterintuitive noise transmissions arise for nonlinear dose-responses. In summary, our study highlights the critical role of molecular sequestration by decoy binding sites in regulating the stochastic dynamics of TFs and target proteins at the single-cell level.

Recent experimental results indicate that, in single Escherichia coli cells, the fluctuations in mRNA level are uncorrelated with those of protein. However, a basic two-stage model for prokaryotic gene expression suggests that there ought to be a degree of correlation between the two. Therefore, it is important to investigate realistic modifications of the basic model that have the potential to reduce the theoretical level of the correlation. In this work, we focus on translational and reporter maturation delay, reporting that its introduction into the two-stage model reduces the cross correlation between instantaneous mRNA and protein levels. Our results indicate that the experimentally observed sample correlation coefficient between mRNA and protein levels may increase if the protein measurements are shifted back in time by the value of the delay.

Inside individual cells, expression of genes is stochastic across organisms ranging from bacterial to human cells. A ubiquitous feature of stochastic expression is burst-like synthesis of gene products, which drives considerable intercellular variability in protein levels across an isogenic cell population. One common mechanism by which cells control such stochasticity is negative feedback regulation, where a protein inhibits its own synthesis. For a single gene that is expressed in bursts, negative feedback can affect the burst frequency or the burst size. In order to compare these feedback types, we study a piecewise deterministic model for gene expression of a self-regulating gene. Mathematically tractable steady-state protein distributions are derived and used to compare the noise suppression abilities of the two feedbacks. Results show that in the low noise regime, both feedbacks are similar in term of their noise buffering abilities. Intriguingly, feedback in burst size outperforms the feedback in burst frequency in the high noise regime. Finally, we discuss various regulatory strategies by which cells implement feedback to control burst sizes of expressed proteins at the level of single cells.

Hybrid models for gene expression combine stochastic and deterministic representations of the underlying biophysical mechanisms. According to one of the simplest hybrid formalisms, protein molecules are produced in randomly occurring bursts of a randomly distributed size while they are degraded deterministically. Here, we use this particular formalism to study two key regulatory motifs-the autoregulation loop and the toggle switch. The distribution of burst times is determined and used as a basis for the development of exact simulation algorithms for gene expression dynamics. For the autoregulation loop, the simulations are compared to an analytic solution of a master equation. Simulations of the toggle switch reveal a number of qualitatively distinct scenarios with implications for the modelling of cell-fate selection.

A single transcription factor may interact with a multitude of targets on the genome, some of which are at gene promoters, others being part of DNA repeat elements. Being sequestered at binding sites, protein molecules can be prevented from partaking in other pathways, specifically, from regulating the expression of the very gene that encodes them. Acting as decoys at the expense of the autoregulatory loop, the binding sites can have a profound impact on protein abundance—on its mean as well as on its cell-to-cell variability. In order to quantify this impact, we study in this paper a mathematical model for pulsatile expression of a transcription factor that autoregulates its expression and interacts with decoys. We determine the exact stationary distribution for protein abundance at the single-cell level, showing that in the case of non-cooperative positive autoregulation, the distribution can be bimodal, possessing a basal expression mode and a distinct, up-regulated, mode. Bimodal protein distributions are more feasible if the rate of degradation is the same irrespective of whether protein is bound or not. Contrastingly, the presence of decoy binding sites which protect the protein from degradation reduces the availability of the bimodal scenario.