Stochastic Gene Expression in Proliferating Cells: Differing Noise Intensity in Single-Cell and Population Perspectives.

Random fluctuations (noise) in gene expression can be studied from two complementary perspectives: following expression in a single cell over time or comparing expression between cells in a proliferating population at a given time. Here, we systematically investigated scenarios where both perspectives lead to different levels of noise in a given gene product. We first consider a stable protein, whose concentration is diluted by cellular growth, and the protein inhibits growth at high concentrations, establishing a positive feedback loop. For a stochastic model with molecular bursting of gene products, we analytically predict and contrast the steady-state distributions of protein concentration in both frameworks. Although positive feedback amplifies the noise in expression, this amplification is much higher in the population framework compared to following a single cell over time. We also study other processes that lead to different noise levels even in the absence of such dilution-based feedback. When considering randomness in the partitioning of molecules between daughters during mitosis, we find that in the single-cell perspective, the noise in protein concentration is independent of noise in the cell cycle duration. In contrast, partitioning noise is amplified in the population perspective by increasing randomness in cell-cycle time. Overall, our results show that the commonly used single-cell framework that does not account for proliferating cells can, in some cases, underestimate the noise in gene product levels. These results have important implications for studying the inter-cellular variation of different stress-related expression programs across cell types that are known to inhibit cellular growth.

MicroRNA Based Feedforward Control of Intrinsic Gene Expression Noise.

Intrinsic noise, which arises in gene expression at low copy numbers, can be controlled by diverse regulatory motifs, including feedforward loops. Here, we study an example of a feedforward control system based on the interaction between an mRNA molecule and an antagonistic microRNA molecule encoded by the same gene, aiming to quantify the variability (or noise) in molecular copy numbers. Using linear noise approximation, we show that the mRNA noise is sub-Poissonian in case of non-bursty transcription, and exhibits a nonmonotonic response both to the species natural lifetime ratio and to the strength of the antagonistic interaction. Additionally, we use the Chemical Reaction Network Theory to prove that the mRNA copy number distribution is Poissonian in the absence of spontaneous mRNA decay channel. In case of transcriptional bursts, we show that feedforward control can attenuate the super-Poissonian gene-expression noise that is due to bursting, and that the effect is more considerable at the protein than at the mRNA level. Our results indicate that the strong coupling between mRNA and microRNA in the sense of burst stoichiometry and also of timing of production events renders the microRNA based feedforward motif an effective mechanism for the control of gene expression noise.

Mixture distributions in a stochastic gene expression model with delayed feedback: a WKB approximation approach.

Noise in gene expression can be substantively affected by the presence of production delay. Here we consider a mathematical model with bursty production of protein, a one-step production delay (the passage of which activates the protein), and feedback in the frequency of bursts. We specifically focus on examining the steady-state behaviour of the model in the slow-activation (i.e. large-delay) regime. Using a formal asymptotic approach, we derive an autonomous ordinary differential equation for the inactive protein that applies in the slow-activation regime. If the differential equation is monostable, the steady-state distribution of the inactive (active) protein is approximated by a single Gaussian (Poisson) mode located at the globally stable fixed point of the differential equation. If the differential equation is bistable (due to cooperative positive feedback), the steady-state distribution of the inactive (active) protein is approximated by a mixture of Gaussian (Poisson) modes located at the stable fixed points; the weights of the modes are determined from a WKB approximation to the stationary distribution. The asymptotic results are compared to numerical solutions of the chemical master equation.

Limit-cycle oscillatory coexpression of cross-inhibitory transcription factors: a model mechanism for lineage promiscuity.

Lineage switches are genetic regulatory motifs that govern and maintain the commitment of a developing cell to a particular cell fate. A canonical example of a lineage switch is the pair of transcription factors PU.1 and GATA-1, of which the former is affiliated with the myeloid and the latter with the erythroid lineage within the haematopoietic system. On a molecular level, PU.1 and GATA-1 positively regulate themselves and antagonize each other via direct protein-protein interactions. Here we use mathematical modelling to identify a novel type of dynamic behaviour that can be supported by such a regulatory architecture. Guided by the specifics of the PU.1-GATA-1 interaction, we formulate, using the law of mass action, a system of differential equations for the key molecular concentrations. After a series of systematic approximations, the system is reduced to a simpler one, which is tractable to phase-plane and linearization methods. The reduced system formally resembles, and generalizes, a well-known model for competitive species from mathematical ecology. However, in addition to the qualitative regimes exhibited by a pair of competitive species (exclusivity, bistable exclusivity, stable-node coexpression) it also allows for oscillatory limit-cycle coexpression. A key outcome of the model is that, in the context of cell-fate choice, such oscillations could be harnessed by a differentiating cell to prime alternately for opposite outcomes; a bifurcation-theory approach is adopted to characterize this possibility.

High Cooperativity in Negative Feedback can Amplify Noisy Gene Expression.

Burst-like synthesis of protein is a significant source of cell-to-cell variability in protein levels. Negative feedback is a common example of a regulatory mechanism by which such stochasticity can be controlled. Here we consider a specific kind of negative feedback, which makes bursts smaller in the excess of protein. Increasing the strength of the feedback may lead to dramatically different outcomes depending on a key parameter, the noise load, which is defined as the squared coefficient of variation the protein exhibits in the absence of feedback. Combining stochastic simulation with asymptotic analysis, we identify a critical value of noise load: for noise loads smaller than critical, the coefficient of variation remains bounded with increasing feedback strength; contrastingly, if the noise load is larger than critical, the coefficient of variation diverges to infinity in the limit of ever greater feedback strengths. Interestingly, feedbacks with lower cooperativities have higher critical noise loads, suggesting that they can be preferable for controlling noisy proteins.

Gene expression noise is affected differentially by feedback in burst frequency and burst size.

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.

Protein copy number distributions for a self-regulating gene in the presence of decoy binding sites.

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.

Transcriptional bursting diversifies the behaviour of a toggle switch: hybrid simulation of stochastic gene expression.

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.

Delayed protein synthesis reduces the correlation between mRNA and protein fluctuations.

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.

Consequences of mRNA transport on stochastic variability in protein levels.

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.

Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression.

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.

Multiscale stochastic modelling of gene expression.

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.

A bistable genetic switch which does not require high co-operativity at the promoter: a two-timescale model for the PU.1-GATA-1 interaction.

The transcription factors PU.1 and GATA-1 antagonize each other in common myeloid progenitors and their relative abundance is thought to decide whether the cell follows the erythrocyte/megakaryocyte lineage or the granulocyte/macrophage lineage. We propose a kinetic model for the PU.1-GATA-1 interaction, analyse its phase space and interpret the results of our analysis. The conclusions have broader implications for the modelling of cell-fate selection.