The minimal intrinsic stochasticity of constitutively expressed eukaryotic genes is sub-Poissonian

Exploring the central dogma through the lens of gene expression noise

Over the past two decades, cell-to-cell heterogeneity has garnered increasing attention due to its critical role in both developmental and pathological processes. This growing interest has been driven, in part, by the advancements in live-cell and single-molecule imaging techniques. These techniques have provided mechanistic insights into how processes, transcription in particular, contribute to gene expression noise and, ultimately, cell-to-cell heterogeneity. More recently, however, research has expanded to explore how downstream steps in the central dogma influence gene expression noise. In this review, we mostly examine the impact of transcriptional processes on the generation of gene expression noise but also discuss how post-transcriptional mechanisms modulate noise and its propagation to the protein level. This evaluation emphasizes the need for further investigation into how processes beyond transcription shape gene expression noise, highlighting unanswered questions that remain in the field.

Cell-cycle dependence of bursty gene expression: insights from fitting mechanistic models to single-cell RNA-seq data

Bursty gene expression is characterized by two intuitive parameters, burst frequency and burst size, the cell-cycle dependence of which has not been extensively profiled at the transcriptome level. In this study, we estimate the burst parameters per allele in the G1 and G2/M cell-cycle phases for thousands of mouse genes by fitting mechanistic models of gene expression to messenger RNA count data, obtained by sequencing of single cells whose cell-cycle position has been inferred using a deep-learning method. We find that upon DNA replication, the median burst frequency approximately halves, while the burst size remains mostly unchanged. Genome-wide distributions of the burst parameter ratios between the G2/M and G1 phases are broad, indicating substantial heterogeneity in transcriptional regulation. We also observe a significant negative correlation between the burst frequency and size ratios, suggesting that regulatory processes do not independently control the burst parameters. We show that to accurately estimate the burst parameter ratios, mechanistic models must explicitly account for gene copy number variation and extrinsic noise due to the coupling of transcription to cell age across the cell cycle, but corrections for technical noise due to imperfect capture of RNA molecules in sequencing experiments are less critical.

Incorporating spatial diffusion into models of bursty stochastic transcription

The dynamics of gene expression are stochastic and spatial at the molecular scale, with messenger RNA (mRNA) transcribed at specific nuclear locations and then transported to the nuclear boundary for export. Consequently, the spatial distributions of these molecules encode their underlying dynamics. While mechanistic models for molecular counts have revealed numerous insights into gene expression, they have largely neglected now-available subcellular spatial resolution down to individual molecules. Owing to the technical challenges inherent in spatial stochastic processes, tools for studying these subcellular spatial patterns are still limited. Here, we introduce a spatial stochastic model of nuclear mRNA with two-state (telegraph) transcriptional dynamics. Observations of the model can be concisely described as following a spatial Cox process driven by a stochastically switching partial differential equation. We derive analytical solutions for spatial and demographic moments and validate them with simulations. We show that the distribution of mRNA counts can be accurately approximated by a Poisson-beta distribution with tractable parameters, even with complex spatial dynamics. This observation allows for efficient parameter inference demonstrated on synthetic data. Altogether, our work adds progress towards a new frontier of subcellular spatial resolution in inferring the dynamics of gene expression from static snapshot data.

Transient power-law behaviour following induction distinguishes between competing models of stochastic gene expression

What features of transcription can be learnt by fitting mathematical models of gene expression to mRNA count data? Given a suite of models, fitting to data selects an optimal one, thus identifying a probable transcriptional mechanism. Whilst attractive, the utility of this methodology remains unclear. Here, we sample steady-state, single-cell mRNA count distributions from parameters in the physiological range, and show they cannot be used to confidently estimate the number of inactive gene states, i.e. the number of rate-limiting steps in transcriptional initiation. Distributions from over 99% of the parameter space generated using models with 2, 3, or 4 inactive states can be well fit by one with a single inactive state. However, we show that for many minutes following induction, eukaryotic cells show an increase in the mean mRNA count that obeys a power law whose exponent equals the sum of the number of states visited from the initial inactive to the active state and the number of rate-limiting post-transcriptional processing steps. Our study shows that estimation of the exponent from eukaryotic data can be sufficient to determine a lower bound on the total number of regulatory steps in transcription initiation, splicing, and nuclear export.

ATP6V1B1 regulates ovarian cancer progression and cisplatin sensitivity through the mTOR/autophagy pathway

Early detection and effective chemotherapy for ovarian cancer, a serious gynecological malignancy, require further progress. This study aimed to investigate the molecular mechanism of ATPase H+-Transporting V1 Subunit B1 (ATP6V1B1) in ovarian cancer development and chemoresistance. Our data show that ATP6V1B1 is upregulated in ovarian cancer and correlated with decreased progression-free survival. Gain- and loss-of-function experiments demonstrated that ATP6V1B1 promotes the proliferation, migration, and invasion of ovarian cancer cells in vitro, while ATP6V1B1 knockout inhibits tumor growth in vivo. In addition, knocking down ATP6V1B1 increases the sensitivity of ovarian cancer cells to cisplatin. Mechanistic studies showed that ATP6V1B1 regulates the activation of the mTOR/autophagy pathway. Overall, our study confirmed the oncogenic role of ATP6V1B1 in ovarian cancer and revealed that ATP6V1B1 promotes ovarian cancer progression via the mTOR/autophagy axis.

Deterministic patterns in single-cell transcriptomic data

We report the existence of deterministic patterns in statistical plots of single-cell transcriptomic data. We develop a theory showing that the patterns are neither artifacts introduced by the measurement process nor due to underlying biological mechanisms. Rather they naturally emerge from finite sample size effects. The theory precisely predicts the patterns in data from multiplexed error-robust fluorescence in situ hybridization and five different types of single-cell sequencing platforms.

Single-cell RNA sequencing algorithms underestimate changes in transcriptional noise compared to single-molecule RNA imaging

Stochastic fluctuations (noise) in transcription generate substantial cell-to-cell variability. However, how best to quantify genome-wide noise remains unclear. Here, we utilize a small-molecule perturbation (5'-iodo-2'-deoxyuridine [IdU]) to amplify noise and assess noise quantification from numerous single-cell RNA sequencing (scRNA-seq) algorithms on human and mouse datasets and then compare it to noise quantification from single-molecule RNA fluorescence in situ hybridization (smFISH) for a panel of representative genes. We find that various scRNA-seq analyses report amplified noise-without altered mean expression levels-for ∼90% of genes and that smFISH analysis verifies noise amplification for the vast majority of tested genes. Collectively, the analyses suggest that most scRNA-seq algorithms (including a simple normalization approach) are appropriate for quantifying noise, although all algorithms appear to systematically underestimate noise changes compared to smFISH. For practical purposes, this analysis further argues that IdU noise enhancement is globally penetrant-i.e., homeostatically increasing noise without altering mean expression levels-and could enable investigations of the physiological impacts of transcriptional noise.

Exact burst-size distributions for gene-expression models with complex promoter structure

In prokaryotic and eukaryotic cells, most genes are transcribed in a bursty fashion on one hand and complex gene regulations may lead to complex promoter structure on the other hand. This raises an unsolved issue: how does promoter structure shape transcriptional bursting kinetics characterized by burst size and frequency? Here we analyze stochastic models of gene transcription, which consider complex regulatory mechanisms. Notably, we develop an efficient method to derive exact burst-size distributions. The analytical results show that if the promoter of a gene contains only one active state, the burst size indeed follows a geometric distribution, in agreement with the previous result derived under certain limiting conditions. However, if it contains a multitude of active states, the burst size in general obeys a non-geometric distribution, which is a linearly weighted sum of geometric distributions. This superposition principle reveals the essential feature of bursting kinetics in complex cases of transcriptional regulation although it seems that there has been no direct experimental confirmation. The derived burst-size distributions not only highlight the importance of promoter structure in regulating bursting kinetics, but can be also used in the exact inference of this kinetics based on experimental data.

Transcriptional bursting: from fundamentals to novel insights

Transcription occurs as irregular bursts in a very wide range of systems, including numerous different species and many genes within these. In this review, we examine the underlying theories, discuss how these relate to experimental measurements, and explore some of the discrepancies that have emerged among various studies. Finally, we consider more recent works that integrate novel concepts, such as the involvement of biomolecular condensates in enhancer-promoter interactions and their effects on the dynamics of transcriptional bursting.

Quantitative comparison of single-cell RNA sequencing versus single-molecule RNA imaging for quantifying transcriptional noise

Stochastic fluctuations (noise) in transcription generate substantial cell-to-cell variability. However, how best to quantify genome-wide noise, remains unclear. Here we utilize a small-molecule perturbation (IdU) to amplify noise and assess noise quantification from numerous scRNA-seq algorithms on human and mouse datasets, and then compare to noise quantification from single-molecule RNA FISH (smFISH) for a panel of representative genes. We find that various scRNA-seq analyses report amplified noise, without altered mean-expression levels, for ~90% of genes and that smFISH analysis verifies noise amplification for the vast majority of genes tested. Collectively, the analyses suggest that most scRNA-seq algorithms are appropriate for quantifying noise including a simple normalization approach, although all of these systematically underestimate noise compared to smFISH. From a practical standpoint, this analysis argues that IdU is a globally penetrant noise-enhancer molecule-amplifying noise without altering mean-expression levels-which could enable investigations of the physiological impacts of transcriptional noise.

Population Diversity at the Single-Cell Level

Population-scale single-cell genomics is a transformative approach for unraveling the intricate links between genetic and cellular variation. This approach is facilitated by cutting-edge experimental methodologies, including the development of high-throughput single-cell multiomics and advances in multiplexed environmental and genetic perturbations. Examining the effects of natural or synthetic genetic variants across cellular contexts provides insights into the mutual influence of genetics and the environment in shaping cellular heterogeneity. The development of computational methodologies further enables detailed quantitative analysis of molecular variation, offering an opportunity to examine the respective roles of stochastic, intercellular, and interindividual variation. Future opportunities lie in leveraging long-read sequencing, refining disease-relevant cellular models, and embracing predictive and generative machine learning models. These advancements hold the potential for a deeper understanding of the genetic architecture of human molecular traits, which in turn has important implications for understanding the genetic causes of human disease.

Single gene analysis in yeast suggests nonequilibrium regulatory dynamics for transcription

Fluctuations in the initiation rate of transcription, the first step in gene expression, ensue from the stochastic behavior of the molecular process that controls transcription. In steady state, the regulatory process is often assumed to operate reversibly, i.e., in equilibrium. However, reversibility imposes fundamental limits to information processing. For instance, the assumption of equilibrium is difficult to square with the precision with which the regulatory process executes its task in eukaryotes. Here we provide evidence - from microscopic analyses of the transcription dynamics at a single gene copy of yeast - that the regulatory process for transcription is cyclic and irreversible (out of equilibrium). The necessary coupling to reservoirs of free energy occurs via sequence-specific transcriptional activators and the recruitment, in part, of ATP-dependent chromatin remodelers. Our findings may help explain how eukaryotic cells reconcile the dual but opposing requirements for fast regulatory kinetics and high regulatory specificity.

Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction

We introduce a biologically detailed, stochastic model of gene expression describing the multiple rate-limiting steps of transcription, nuclear pre-mRNA processing, nuclear mRNA export, cytoplasmic mRNA degradation and translation of mRNA into protein. The processes in sub-cellular compartments are described by an arbitrary number of processing stages, thus accounting for a significantly finer molecular description of gene expression than conventional models such as the telegraph, two-stage and three-stage models of gene expression. We use two distinct tools, queueing theory and model reduction using the slow-scale linear-noise approximation, to derive exact or approximate analytic expressions for the moments or distributions of nuclear mRNA, cytoplasmic mRNA and protein fluctuations, as well as lower bounds for their Fano factors in steady-state conditions. We use these to study the phase diagram of the stochastic model; in particular we derive parametric conditions determining three types of transitions in the properties of mRNA fluctuations: from sub-Poissonian to super-Poissonian noise, from high noise in the nucleus to high noise in the cytoplasm, and from a monotonic increase to a monotonic decrease of the Fano factor with the number of processing stages. In contrast, protein fluctuations are always super-Poissonian and show weak dependence on the number of mRNA processing stages. Our results delineate the region of parameter space where conventional models give qualitatively incorrect results and provide insight into how the number of processing stages, e.g. the number of rate-limiting steps in initiation, splicing and mRNA degradation, shape stochastic gene expression by modulation of molecular memory.

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.

Solving stochastic gene-expression models using queueing theory: A tutorial review

Stochastic models of gene expression are typically formulated using the chemical master equation, which can be solved exactly or approximately using a repertoire of analytical methods. Here, we provide a tutorial review of an alternative approach based on queueing theory that has rarely been used in the literature of gene expression. We discuss the interpretation of six types of infinite-server queues from the angle of stochastic single-cell biology and provide analytical expressions for the stationary and nonstationary distributions and/or moments of mRNA/protein numbers and bounds on the Fano factor. This approach may enable the solution of complex models that have hitherto evaded analytical solution.

RASP: Optimal Single Puncta Detection in Complex Cellular Backgrounds

Super-resolution and single-molecule microscopies have been increasingly applied to complex biological systems. A major challenge of these approaches is that fluorescent puncta must be detected in the low signal, high noise, heterogeneous background environments of cells and tissue. We present RASP, Radiality Analysis of Single Puncta, a bioimaging-segmentation method that solves this problem. RASP removes false-positive puncta that other analysis methods detect and detects features over a broad range of spatial scales: from single proteins to complex cell phenotypes. RASP outperforms the state-of-the-art methods in precision and speed using image gradients to separate Gaussian-shaped objects from the background. We demonstrate RASP's power by showing that it can extract spatial correlations between microglia, neurons, and α-synuclein oligomers in the human brain. This sensitive, computationally efficient approach enables fluorescent puncta and cellular features to be distinguished in cellular and tissue environments, with sensitivity down to the level of the single protein. Python and MATLAB codes, enabling users to perform this RASP analysis on their own data, are provided as Supporting Information and links to third-party repositories.

Power-law behavior of transcriptional bursting regulated by enhancer-promoter communication

Revealing how transcriptional bursting kinetics are genomically encoded is challenging because genome structures are stochastic at the organization level and are suggestively linked to gene transcription. To address this challenge, we develop a generic theoretical framework that integrates chromatin dynamics, enhancer-promoter (E-P) communication, and gene-state switching to study transcriptional bursting. The theory predicts that power law can be a general rule to quantitatively describe bursting modulations by E-P spatial communication. Specifically, burst frequency and burst size are up-regulated by E-P communication strength, following power laws with positive exponents. Analysis of the scaling exponents further reveals that burst frequency is preferentially regulated. Bursting kinetics are down-regulated by E-P genomic distance with negative power-law exponents, and this negative modulation desensitizes at large distances. The mutual information between burst frequency (or burst size) and E-P spatial distance further reveals essential characteristics of the information transfer from E-P communication to transcriptional bursting kinetics. These findings, which are in agreement with experimental observations, not only reveal fundamental principles of E-P communication in transcriptional bursting but also are essential for understanding cellular decision-making.

Regulatory Mechanisms for Transcriptional Bursting Revealed by an Event-Based Model

Gene transcription often occurs in discrete bursts, and it can be difficult to deduce the underlying regulatory mechanisms for transcriptional bursting with limited experimental data. Here, we categorize numerous states of single eukaryotic genes and identify 6 essential transcriptional events, each comprising a series of state transitions; transcriptional bursting is characterized as a sequence of 4 events, capable of being organized in various configurations, in addition to the beginning and ending events. By associating transcriptional kinetics with mean durations and recurrence probabilities of the events, we unravel how transcriptional bursting is modulated by various regulators including transcription factors. Through analytical derivation and numerical simulation, this study reveals key state transitions contributing to transcriptional sensitivity and specificity, typical characteristics of burst profiles, global constraints on intrinsic transcriptional noise, major regulatory modes in individual genes and across the genome, and requirements for fast gene induction upon stimulation. It is illustrated how biochemical reactions on different time scales are modulated to separately shape the durations and ordering of the events. Our results suggest that transcriptional patterns are essentially controlled by a shared set of transcriptional events occurring under specific promoter architectures and regulatory modes, the number of which is actually limited.

Uncovering the effect of RNA polymerase steric interactions on gene expression noise: Analytical distributions of nascent and mature RNA numbers

The telegraph model is the standard model of stochastic gene expression, which can be solved exactly to obtain the distribution of mature RNA numbers per cell. A modification of this model also leads to an analytical distribution of nascent RNA numbers. These solutions are routinely used for the analysis of single-cell data, including the inference of transcriptional parameters. However, these models neglect important mechanistic features of transcription elongation, such as the stochastic movement of RNA polymerases and their steric (excluded-volume) interactions. Here we construct a model of gene expression describing promoter switching between inactive and active states, binding of RNA polymerases in the active state, their stochastic movement including steric interactions along the gene, and their unbinding leading to a mature transcript that subsequently decays. We derive the steady-state distributions of the nascent and mature RNA numbers in two important limiting cases: constitutive expression and slow promoter switching. We show that RNA fluctuations are suppressed by steric interactions between RNA polymerases, and that this suppression can in some instances even lead to sub-Poissonian fluctuations; these effects are most pronounced for nascent RNA and less prominent for mature RNA, since the latter is not a direct sensor of transcription. We find a relationship between the parameters of our microscopic mechanistic model and those of the standard models that ensures excellent consistency in their prediction of the first and second RNA number moments over vast regions of parameter space, encompassing slow, intermediate, and rapid promoter switching, provided the RNA number distributions are Poissonian or super-Poissonian. Furthermore, we identify the limitations of inference from mature RNA data, specifically showing that it cannot differentiate between highly distinct RNA polymerase traffic patterns on a gene.

Evaluating single-cell variability in proteasomal decay

Gene expression is a stochastic process that leads to variability in mRNA and protein abundances even within an isogenic population of cells grown in the same environment. This variation, often called gene-expression noise, has typically been attributed to transcriptional and translational processes while ignoring the contributions of protein decay variability across cells. Here we estimate the single-cell protein decay rates of two degron GFPs in Saccharomyces cerevisiae using time-lapse microscopy. We find substantial cell-to-cell variability in the decay rates of the degron GFPs. We evaluate cellular features that explain the variability in the proteasomal decay and find that the amount of 20s catalytic beta subunit of the proteasome marginally explains the observed variability in the degron GFP half-lives. We propose alternate hypotheses that might explain the observed variability in the decay of the two degron GFPs. Overall, our study highlights the importance of studying the kinetics of the decay process at single-cell resolution and that decay rates vary at the single-cell level, and that the decay process is stochastic. A complex model of decay dynamics must be included when modeling stochastic gene expression to estimate gene expression noise.

The impossible challenge of estimating non-existent moments of the Chemical Master Equation

MOTIVATION

The Chemical Master Equation (CME) is a set of linear differential equations that describes the evolution of the probability distribution on all possible configurations of a (bio-)chemical reaction system. Since the number of configurations and therefore the dimension of the CME rapidly increases with the number of molecules, its applicability is restricted to small systems. A widely applied remedy for this challenge is moment-based approaches which consider the evolution of the first few moments of the distribution as summary statistics for the complete distribution. Here, we investigate the performance of two moment-estimation methods for reaction systems whose equilibrium distributions encounter fat-tailedness and do not possess statistical moments.

RESULTS

We show that estimation via stochastic simulation algorithm (SSA) trajectories lose consistency over time and estimated moment values span a wide range of values even for large sample sizes. In comparison, the method of moments returns smooth moment estimates but is not able to indicate the non-existence of the allegedly predicted moments. We furthermore analyze the negative effect of a CME solution's fat-tailedness on SSA run times and explain inherent difficulties. While moment-estimation techniques are a commonly applied tool in the simulation of (bio-)chemical reaction networks, we conclude that they should be used with care, as neither the system definition nor the moment-estimation techniques themselves reliably indicate the potential fat-tailedness of the CME's solution.