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Background: Stochastic modelling plays a crucial role in comprehending the dynamics of intracellular events in various biochemical systems, including gene-expression models. Cell-to-cell variability arises from the stochasticity or noise in the levels of gene products such as messenger RNA (mRNA) and protein. The sources of noise can stem from different factors, including structural elements. Recent studies have revealed that the mRNA structure can be more intricate than previously assumed. Results: Here, we focus on the formation of stem-loops and present a reinterpretation of previous data, offering new insights. Our analysis demonstrates that stem-loops that restrict translation have the potential to reduce noise. Conclusions: In conclusion, we investigate a structured/generalised version of a stochastic gene-expression model, wherein mRNA molecules can be found in one of their finite number of different states and transition between them. By characterising and deriving non-trivial analytical expressions for the steady-state protein distribution, we provide two specific examples which can be readily obtained from the structured/generalised model, showcasing the model's practical applicability. [ABSTRACT FROM AUTHOR]

Abstract Background Stochastic modelling plays a crucial role in comprehending the dynamics of intracellular events in various biochemical systems, including gene-expression models. Cell-to-cell variability arises from the stochasticity or noise in the levels of gene products such as messenger RNA (mRNA) and protein. The sources of noise can stem from different factors, including structural elements. Recent studies have revealed that the mRNA structure can be more intricate than previously assumed. Results Here, we focus on the formation of stem-loops and present a reinterpretation of previous data, offering new insights. Our analysis demonstrates that stem-loops that restrict translation have the potential to reduce noise. Conclusions In conclusion, we investigate a structured/generalised version of a stochastic gene-expression model, wherein mRNA molecules can be found in one of their finite number of different states and transition between them. By characterising and deriving non-trivial analytical expressions for the steady-state protein distribution, we provide two specific examples which can be readily obtained from the structured/generalised model, showcasing the model’s practical applicability.

Regulatory mechanisms set a genes average level of expression, but a genes expression constantly fluctuates around that average. These stochastic fluctuations, or expression noise, play a role in cell-fate transitions, bet hedging in microbes, and the development of chemotherapeutic resistance in cancer. An outstanding question is what regulatory mechanisms contribute to noise. Here, we demonstrate that, for a fixed mean level of expression, strong activation domains (ADs) at low abundance produce high expression noise, while weak ADs at high abundance generate lower expression noise. We conclude that differences in noise can be explained by the interplay between a TFs nuclear concentration and the strength of its ADs effect on mean expression, without invoking differences between classes of ADs. These results raise the possibility of engineering gene expression noise independently of mean levels in synthetic biology contexts and provide a potential mechanism for natural selection to tune the noisiness of gene expression.

Background: Stochastic modelling plays a crucial role in comprehending the dynamics of intracellular events in various biochemical systems, including gene-expression models. Cell-to-cell variability arises from the stochasticity or noise in the levels of gene products such as messenger RNA (mRNA) and protein. The sources of noise can stem from different factors, including structural elements. Recent studies have revealed that the mRNA structure can be more intricate than previously assumed. Results: Here, we focus on the formation of stem-loops and present a reinterpretation of previous data, offering new insights. Our analysis demonstrates that stem-loops that restrict translation have the potential to reduce noise. Conclusions: In conclusion, we investigate a structured/generalised version of a stochastic gene-expression model, wherein mRNA molecules can be found in one of their finite number of different states and transition between them. By characterising and deriving non-trivial analytical expressions for the steady-state protein distribution, we provide two specific examples which can be readily obtained from the structured/generalised model, showcasing the model's practical applicability. [ABSTRACT FROM AUTHOR]

We obtain upper and lower Gaussian-type bounds on the density of each component Y t i of the solution Yt to a multidimensional non-Markovian backward SDE. Our approach is based on the Nourdin-Viens formula and a stochastic version of Wazewski's theorem on the positivity of the components of a solution to an ODE. Furthermore, we apply our results to stochastic gene expression; namely, we estimate the density of the law of the amount of protein generated by a gene in a gene regulatory network. [ABSTRACT FROM AUTHOR]

Stochastic differences among clonal cells can initiate cell fate decisions in development or cause cell-to-cell differences in the responses to drugs or extracellular ligands. One hypothesis is that some of this phenotypic variability is caused by stochastic fluctuations in the activities of transcription factors (TFs). We tested this hypothesis in NIH3T3-CG cells using the response to Hedgehog signaling as a model cellular response. Here, we present evidence for the existence of distinct fast- and slow-responding substates in NIH3T3-CG cells. These two substates have distinct expression profiles, and fluctuations in the Prrx1 TF underlie some of the differences in expression and responsiveness between fast and slow cells. Our results show that fluctuations in TFs can contribute to cell-to-cell differences in Hedgehog signaling. [ABSTRACT FROM AUTHOR]

We analyse a generalisation of the stochastic gene expression model studied recently in Fromion et al. (SIAM J Appl Math 73:195–211, 2013) and Robert (Probab Surv 16:277–332, 2019) that keeps track of the production of both mRNA and protein molecules, using techniques from the theory of point processes, as well as ideas from the theory of matrix-analytic methods. Here, both the activity of a gene and the creation of mRNA are modelled with an arbitrary Markovian Arrival Process governed by finitely many phases, and each mRNA molecule during its lifetime gives rise to protein molecules in accordance with a Poisson process. This modification is important, as Markovian Arrival Processes can be used to approximate many types of point processes on the nonnegative real line, meaning this framework allows us to further relax our assumptions on the overall process of transcription. [ABSTRACT FROM AUTHOR]

Facioscapulohumeral muscular dystrophy (FSHD) is an incurable myopathy linked to the over-expression of the myotoxic transcription factor DUX4. Targeting DUX4 is the leading therapeutic approach, however, it is only detectable in 0.1–3.8% of FSHD myonuclei. How rare DUX4 drives FSHD and the optimal anti-DUX4 strategy are unclear. We combine stochastic gene expression with compartment models of cell states, building a simulation of DUX4 expression and consequences in FSHD muscle fibers. Investigating iDUX4 myoblasts, scRNAseq, and snRNAseq of FSHD muscle we estimate parameters including DUX4 mRNA degradation, transcription and translation rates, and DUX4 target gene activation rates. Our model accurately recreates the distribution of DUX4 and targets gene-positive cells seen in scRNAseq of FSHD myocytes. Importantly, we show DUX4 drives significant cell death despite expression in only 0.8% of live cells. Comparing scRNAseq of unfused FSHD myocytes to snRNAseq of fused FSHD myonuclei, we find evidence of DUX4 protein syncytial diffusion and estimate its rate via genetic algorithms. We package our model into freely available tools, to rapidly investigate the consequences of anti-DUX4 therapy.

mRNA translation is the ubiquitous cellular process of reading messenger-RNA strands into functional proteins. Over the past decade, large strides in microscopy techniques have allowed observation of mRNA translation at a single-molecule resolution for self-consistent time-series measurements in live cells. Dubbed Nascent chain tracking (NCT), these methods have explored many temporal dynamics in mRNA translation uncaptured by other experimental methods such as ribosomal profiling, smFISH, pSILAC, BONCAT, or FUNCAT-PLA. However, NCT is currently restricted to the observation of one or two mRNA species at a time due to limits in the number of resolvable fluorescent tags. In this work, we propose a hybrid computational pipeline, where detailed mechanistic simulations produce realistic NCT videos, and machine learning is used to assess potential experimental designs for their ability to resolve multiple mRNA species using a single fluorescent color for all species. Our simulation results show that with careful application this hybrid design strategy could in principle be used to extend the number of mRNA species that could be watched simultaneously within the same cell. We present a simulated example NCT experiment with seven different mRNA species within the same simulated cell and use our ML labeling to identify these spots with 90% accuracy using only two distinct fluorescent tags. We conclude that the proposed extension to the NCT color palette should allow experimentalists to access a plethora of new experimental design possibilities, especially for cell Signaling applications requiring simultaneous study of multiple mRNAs.

Introduction: Despite continued technological improvements, measurement errors always reduce or distort the information that any real experiment can provide to quantify cellular dynamics. This problem is particularly serious for cell signaling studies to quantify heterogeneity in single-cell gene regulation, where important RNA and protein copy numbers are themselves subject to the inherently random fluctuations of biochemical reactions. Until now, it has not been clear how measurement noise should be managed in addition to other experiment design variables (e.g., sampling size, measurement times, or perturbation levels) to ensure that collected data will provide useful insights on signaling or gene expression mechanisms of interest.Methods: We propose a computational framework that takes explicit consideration of measurement errors to analyze single-cell observations, and we derive Fisher Information Matrix (FIM)-based criteria to quantify the information value of distorted experiments.Results and Discussion: We apply this framework to analyze multiple models in the context of simulated and experimental single-cell data for a reporter gene controlled by an HIV promoter. We show that the proposed approach quantitatively predicts how different types of measurement distortions affect the accuracy and precision of model identification, and we demonstrate that the effects of these distortions can be mitigated through explicit consideration during model inference. We conclude that this reformulation of the FIM could be used effectively to design single-cell experiments to optimally harvest fluctuation information while mitigating the effects of image distortion.

Stochastic gene expression plays a leading developmental role through its contribution to cell differentiation. It is also proposed to promote phenotypic diversification in malignant cells. However, it remains unclear if these two forms of cellular bet‐hedging are identical or rather display distinct features. Here we argue that bet‐hedging phenomena in cancer cells are more similar to those occurring in unicellular organisms than to those of normal metazoan cells. We further propose that the atavistic bet‐hedging strategies in cancer originate from a hijacking of the normal developmental bet‐hedging of metazoans. Finally, we discuss the constraints that may shape the atavistic bet‐hedging strategies of cancer cells. [ABSTRACT FROM AUTHOR]

A common feature of bacterial, fungal and cancer cell populations upon treatment is the presence of tolerant and persistent cells able to survive, and sometimes grow, even in the presence of usually inhibitory or lethal drug concentrations, driven by non-genetic differences among individual cells in a population. Here we review and compare data obtained on drug survival in bacteria, fungi and cancer cells to unravel common characteristics and cellular pathways, and to point their singularities. This comparative work also allows to cross-fertilize ideas across fields. We particularly focus on the role of gene expression variability in the emergence of cell-cell non-genetic heterogeneity because it represents a possible common basic molecular process at the origin of most persistence phenomena and could be monitored and tuned to help improve therapeutic interventions., Competing Interests: Declaration of competing interest The authors declare no competing interests., (Copyright © 2024. Published by Elsevier Inc.)

RNA and protein concentrations within cells constantly fluctuate. Some molecular species typically have very low copy numbers, so stochastic changes in their abundances can dramatically alter cellular concentration levels. Such noise can be harmful through constrained functionality or reduced efficiency. Gene regulatory networks have evolved to be robust in the face of noise. We obtain exact analytical expressions for noise dissipation in an idealised stochastic model of a gene regulatory network. We show that noise decays exponentially fast. The decay rate for RNA molecular counts is given by the integral of the tail of the cumulative distribution function of the degradation time. For proteins, it is given by the slowest rate-limiting step of RNA degradation or proteolytic breakdown. This is intuitive because memory of the chemical composition of the system is manifested through molecular persistence. The results are obtained by analysing a non-standard tandem of infinite server queues, in which the number of customers present in one queue modulates the arrival rate into the next. [ABSTRACT FROM AUTHOR]

Stochastic gene expression in regulatory networks is conventionally modelled via the chemical master equation (CME). As explicit solutions to the CME, in the form of so-called propagators, are oftentimes not readily available, various approximations have been proposed. A recently developed analytical method is based on a separation of time scales that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficient approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to simulated data from a 'telegraph' model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be applied successfully for parameter inference in the non-regulated model; moreover, we show that, in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. These results indicate that our approach may allow for successful parameter inference and model identification from longitudinal single cell data. [ABSTRACT FROM AUTHOR]

Abstract Genetically identical individuals in bacterial populations can display significant phenotypic variability. This variability can be functional, for example by allowing a fraction of stress prepared cells to survive an otherwise lethal stress. The optimal fraction of stress prepared cells depends on environmental conditions. However, how bacterial populations modulate their level of phenotypic variability remains unclear. Here we show that the alternative sigma factor σV circuit in Bacillus subtilis generates functional phenotypic variability that can be tuned by stress level, environmental history and genetic perturbations. Using single‐cell time‐lapse microscopy and microfluidics, we find the fraction of cells that immediately activate σV under lysozyme stress depends on stress level and on a transcriptional memory of previous stress. Iteration between model and experiment reveals that this tunability can be explained by the autoregulatory feedback structure of the sigV operon. As predicted by the model, genetic perturbations to the operon also modulate the response variability. The conserved sigma‐anti‐sigma autoregulation motif is thus a simple mechanism for bacterial populations to modulate their heterogeneity based on their environment.

Abstract Genetic variability, epigenetic variability, and gene expression variability (noise) are generally considered independently in their relationship with phenotypic variation. However, they appear to be intrinsically interconnected and influence it in combination. The study of the interplay between genetic and epigenetic variability has the longest history. This article rather considers the introduction of gene expression variability in its relationships with the two others and reviews for the first time experimental evidences over the four relationships connected to gene expression noise. They show how introducing this third source of variability complicates the way of thinking evolvability and the emergence of biological novelty. Finally, cancer cells are proposed to be an ideal model to decipher the dynamic interplay between genetic, epigenetic, and gene expression variability when one of them is either experimentally increased or therapeutically targeted. This interplay is also discussed in an evolutionary perspective in the context of cancer cell drug resistance.

Abstract Background Single-cell RNA sequencing (scRNA-seq) is the most widely used technique to obtain gene expression profiles from complex tissues. Cell subsets and developmental states are often identified via differential gene expression patterns. Most of the single-cell tools utilized highly variable genes to annotate cell subsets and states. However, we have discovered that a group of genes, which sensitively respond to environmental stimuli with high coefficients of variation (CV), might impose overwhelming influences on the cell type annotation. Result In this research, we developed a method, based on the CV-rank and Shannon entropy, to identify these noise genes, and termed them as “sensitive genes”. To validate the reliability of our methods, we applied our tools in 11 single-cell data sets from different human tissues. The results showed that most of the sensitive genes were enriched pathways related to cellular stress response. Furthermore, we noticed that the unsupervised result was closer to the ground-truth cell labels, after removing the sensitive genes detected by our tools. Conclusion Our study revealed the prevalence of stochastic gene expression patterns in most types of cells, compared the differences among cell marker genes, housekeeping genes (HK genes), and sensitive genes, demonstrated the similarities of functions of sensitive genes in various scRNA-seq data sets, and improved the results of unsupervised clustering towards the ground-truth labels. We hope our method would provide new insights into the reduction of data noise in scRNA-seq data analysis and contribute to the development of better scRNA-seq unsupervised clustering algorithms in the future.

Abstract The circadian clock is an endogenous and self‐sustained oscillator that anticipates daily environmental cycles. While rhythmic gene expression of circadian genes is well‐described in populations of cells, the single‐cell mRNA dynamics of multiple core clock genes remain largely unknown. Here we use single‐molecule fluorescence in situ hybridisation (smFISH) at multiple time points to measure pairs of core clock transcripts, Rev‐erbα (Nr1d1), Cry1 and Bmal1, in mouse fibroblasts. The mean mRNA level oscillates over 24 h for all three genes, but mRNA numbers show considerable spread between cells. We develop a probabilistic model for multivariate mRNA counts using mixtures of negative binomials, which accounts for transcriptional bursting, circadian time and cell‐to‐cell heterogeneity, notably in cell size. Decomposing the mRNA variability into distinct noise sources shows that clock time contributes a small fraction of the total variability in mRNA number between cells. Thus, our results highlight the intrinsic biological challenges in estimating circadian phase from single‐cell mRNA counts and suggest that circadian phase in single cells is encoded post‐transcriptionally.

A natural phenomenon occurring in a living system is an outcome of the dynamics of the specific biological network underlying the phenomenon. The collective dynamics have both deterministic and stochastic components. The stochastic nature of the key processes like gene expression and cell differentiation give rise to fluctuations (noise) at the levels of the biomolecules, and this combined with nonlinear interactions gives rise to a number of emergent phenomena. In this review, we describe and discuss some of these phenomena which have the character of phase transitions in physical systems. We specifically focus on noise-induced transitions in a stochastic model of gene expression and in a population genetics model which have no analogs when the dynamics are solely deterministic in nature. Some of these transitions exhibit critical-point phenomena belonging to the mean-field Ising universality class of equilibrium phase transitions. A number of other examples, ranging from biofilms to homeostasis in adult tissues, are also discussed, which exhibit behaviour similar to critical phenomena in equilibrium and nonequilbrium phase transitions. The examples illustrate how the subject of statistical mechanics provides a bridge between theoretical models and experimental observations. [ABSTRACT FROM AUTHOR]

Biochemical reaction systems, in particular those that govern the expression of genes in single cells, involve discrete numbers of molecules and are inherently stochastic. This study concerns reaction systems with two molecular species and an arbitrary number of reactions, the rates of which conform to a scaling of the classical type in the first species (but not the second). For limiting values of the scaling parameter, the first species is highly abundant, evolves slowly, and buffers the relatively fast fluctuation of the second, scarce, species. The scale separation facilitates the construction of asymptotic approximations to the molecular distributions: the large-time behavior is described by a WKB approximation, which is further resolved into a tractable mixture of metastable modes; the earlier-time dynamics are described by quasi-stationary and linear noise approximations. The paper presents the theory and the algebraic recipes that are required to implement it. The framework is applied on two reaction systems with positive feedback: a delayed feedback circuit and a gene autoregulation model with cooperative binding of the protein to the promoter. For the former, the approximation scheme results into a mixture of Gaussian/Poisson modes for the inactive/active protein distribution. For the latter, the analysis elucidates the effects of bursty gene expression, promoter responsivity, and protein sequestration on bimodality of protein distributions. [ABSTRACT FROM AUTHOR]

The expression of individual genes into functional protein molecules is a noisy dynamical process. Here we model the protein concentration as a jump-drift process which combines discrete stochastic production bursts (jumps) with continuous deterministic decay (drift). We allow the drift rate, the jump rate, and the jump size to depend on the protein level to implement feedback in protein stability, burst frequency, and burst size. We specifically focus on positive feedback in burst size, while allowing for arbitrary autoregulation in burst frequency and protein stability. Two versions of feedback in burst size are thereby considered: in the first, newly produced molecules instantly participate in feedback, even within the same burst; in the second, within-burst regulation does not occur due to the so-called infinitesimal delay. Without infinitesimal delay, the model is explicitly solvable; with its inclusion, an exact distribution to the model is unavailable, but we are able to construct a WKB approximation that applies in the asymptotic regime of small but frequent bursts. Comparing the asymptotic behaviour of the two model versions, we report that they yield the same WKB quasi-potential but a different exponential prefactor. We illustrate the difference on the case of a bimodal protein distribution sustained by a sigmoid feedback in burst size: we show that the omission of the infinitesimal delay overestimates the weight of the upper mode of the protein distribution. The analytic results are supported by kinetic Monte-Carlo simulations. [ABSTRACT FROM AUTHOR]

Mammals use X chromosome inactivation to compensate for the sex difference in numbers of X chromosomes. A relatively unexplored question is how the active X is protected from inactivation by its own XIST gene, the long non-coding RNA, which initiates silence of the inactive X. Previous studies of autosomal duplications show that human chromosome 19 plays a critical role in protecting the active X. I proposed that it genetically interacts with the X chromosome to repress XIST function on the future active X. Here, I show that the type of chromosome 19 duplication influences the outcome of the interaction: the presence of three chromosome 19s is tolerated whereas duplications affecting only one chromosome 19 are not. The different outcomes have mechanistic implications for how chromosome 19 interacts with the future active X, pointing to a role for stochastic gene expression and possibly physical interaction.

Ribosomes-the primary macromolecular machines responsible for translating the genetic code into proteins-are complexes of precisely folded RNA and proteins. The ways in which their production and assembly are managed by the living cell is of deep biological importance. Here we extend a recent spatially resolved whole-cell model of ribosome biogenesis in a fixed volume [Earnest et al., Biophys J 2015, 109, 1117-1135] to include the effects of growth, DNA replication, and cell division. All biological processes are described in terms of reaction-diffusion master equations and solved stochastically using the Lattice Microbes simulation software. In order to determine the replication parameters, we construct and analyze a series of Escherichia coli strains with fluorescently labeled genes distributed evenly throughout their chromosomes. By measuring these cells' lengths and number of gene copies at the single-cell level, we could fit a statistical model of the initiation and duration of chromosome replication. We found that for our slow-growing (120 min doubling time) E. coli cells, replication was initiated 42 min into the cell cycle and completed after an additional 42 min. While simulations of the biogenesis model produce the correct ribosome and mRNA counts over the cell cycle, the kinetic parameters for transcription and degradation are lower than anticipated from a recent analytical time dependent model of in vivo mRNA production. Describing expression in terms of a simple chemical master equation, we show that the discrepancies are due to the lack of nonribosomal genes in the extended biogenesis model which effects the competition of mRNA for ribosome binding, and suggest corrections to parameters to be used in the whole-cell model when modeling expression of the entire transcriptome. © 2016 Wiley Periodicals, Inc. Biopolymers 105: 735-751, 2016.

Delayed production can substantially alter the qualitative behaviour of feedback systems. Motivated by stochastic mechanisms in gene expression, we consider a protein molecule which is produced in randomly timed bursts, requires an exponentially distributed time to activate and then partakes in positive regulation of its burst frequency. Asymptotically analysing the underlying master equation in the large-delay regime, we provide tractable approximations to time-dependent probability distributions of molecular copy numbers. Importantly, the presented analysis demonstrates that positive feedback systems with large production delays can constitute a stable toggle switch even if they operate with low copy numbers of active molecules. [ABSTRACT FROM AUTHOR]

p53 plays a central role in defending the genomic integrity of our cells. In response to genotoxic stress, this tumour suppressor orchestrates the expression of hundreds of target genes, which induce a variety of cellular outcomes ranging from damage repair to induction of apoptosis. In this review, we examine how the p53 response is regulated on several levels in individual cells to allow precise and context‐specific fate decisions. We discuss that the p53 response is not only controlled by its canonical regulators but also controlled by interconnected signalling pathways that influence the dynamics of p53 accumulation upon damage and modulate its transcriptional activity at target gene promoters. Additionally, we consider how the p53 response is diversified through a variety of mechanisms at the promoter level and beyond to induce context‐specific outcomes in individual cells. These layers of regulation allow p53 to react in a stimulus‐specific manner and fine‐tune its signalling according to the individual needs of a given cell, enabling it to take the right decision on survival or death. [ABSTRACT FROM AUTHOR]

Mammals use X chromosome inactivation to compensate for the sex difference in numbers of X chromosomes. A relatively unexplored question is how the active X is protected from inactivation by its own XIST gene, the long non-coding RNA, which initiates silence of the inactive X. Previous studies of autosomal duplications show that human chromosome 19 plays a critical role in protecting the active X. I proposed that it genetically interacts with the X chromosome to repress XIST function on the future active X. Here, I show that the type of chromosome 19 duplication influences the outcome of the interaction: the presence of three chromosome 19s is tolerated whereas duplications affecting only one chromosome 19 are not. The different outcomes have mechanistic implications for how chromosome 19 interacts with the future active X, pointing to a role for stochastic gene expression and possibly physical interaction. [ABSTRACT FROM AUTHOR]

Cell fate is maintained over long timescales, yet molecular fluctuations can lead to spontaneous loss of this differentiated state. Our simulations identified a possible mechanism that explains life-long maintenance of ASE neuron fate in Caenorhabditis elegans by the terminal selector transcription factor CHE-1. Here, fluctuations in CHE-1 level are buffered by the reservoir of CHE-1 bound at its target promoters, which ensures continued che-1 expression by preferentially binding the che-1 promoter. We provide experimental evidence for this mechanism by showing that che-1 expression was resilient to induced transient CHE-1 depletion, while both expression of CHE-1 targets and ASE function were lost. We identified a 130 bp che-1 promoter fragment responsible for this resilience, with deletion of a homeodomain binding site in this fragment causing stochastic loss of ASE identity long after its determination. Because network architectures that support this mechanism are highly conserved in cell differentiation, it may explain stable cell fate maintenance in many systems.

Development is a self-organized process that builds on cells and their interactions. Cells are heterogeneous in gene expression, growth, and division; yet how development is robust despite such heterogeneity is a fascinating question. Here, we review recent progress on this topic, highlighting how developmental robustness is achieved through self-organization. We will first discuss sources of heterogeneity, including stochastic gene expression, heterogeneity in growth rate and direction, and heterogeneity in division rate and precision. We then discuss cellular mechanisms that buffer against such noise, including Paf1C- and miRNA-mediated denoising, spatiotemporal growth averaging and compensation, mechanisms to improve cell division precision, and coordination of growth rate and developmental timing between different parts of an organ. We also discuss cases where such heterogeneity is not buffered but utilized for development. Finally, we highlight potential directions for future studies of noise and developmental robustness., Competing Interests: Declaration of competing interest The authors declare no competing interests., (Copyright © 2024 Elsevier B.V. All rights reserved.)

• Extended the approach from gene network to general biochemical networks and piecewise deterministic Markov processes models. • Many versions of the push-forward method, including mean-field approximation and symbolic calculations of the ODE solutions. • Included the comparison with the Liouville-master PDE method. • Provided a complete rigorous proof of the convergence of the push-forward scheme to the solution of the PDMP. A biochemical network can be simulated by a set of ordinary differential equations (ODE) under well-stirred reactor conditions, for large numbers of molecules, and frequent reactions. This is no longer a robust representation when some molecular species are in small numbers and reactions changing them are infrequent. In this case, discrete stochastic events trigger changes of the smooth deterministic dynamics of the biochemical network. Piecewise-deterministic Markov processes (PDMP) are well adapted for describing such situations. Although PDMP models are now well established in biology, these models remain computationally challenging. Previously we have introduced the push-forward method to compute how the probability measure is spread by the deterministic ODE flow of PDMPs, through the use of analytic expressions of the corresponding semigroup. In this paper we provide a more general simulation algorithm that works also for non-integrable systems. The method can be used for biochemical simulations with applications in fundamental biology, biotechnology and biocomputing. This work is an extended version of the work presented at the conference CMSB2019. [ABSTRACT FROM AUTHOR]

Advanced metastatic cancer is currently not curable and the major barrier to eliminating the disease in patients is the resistance of subpopulations of tumor cells to drug treatments. These resistant subpopulations can arise stochastically among the billions of tumor cells in a patient or emerge over time during therapy due to adaptive mechanisms and the selective pressures of drug therapies. Epigenetic mechanisms play important roles in tumor cell diversity and adaptability, and are regulated by metabolic pathways. Here, I discuss knowledge from ecology, evolution, infectious disease, species extinction, metabolism and epigenetics to synthesize a roadmap to a clinically feasible approach to help homogenize tumor cells and, in combination with drug treatments, drive their extinction. Specifically, cycles of starvation and hyperthermia could help synchronize tumor cells and constrain epigenetic diversity and adaptability by limiting substrates and impairing the activity of chromatin modifying enzymes. Hyperthermia could also help prevent cancer cells from entering dangerous hibernation-like states. I propose steps to a treatment paradigm to help drive cancer extinction that builds on the successes of fasting, hyperthermia and immunotherapy and is achievable in patients. Finally, I highlight the many unknowns, opportunities for discovery and that stochastic gene and allele level epigenetic mechanisms pose a major barrier to cancer extinction that warrants deeper investigation.

Genetically identical individuals in bacterial populations can display significant phenotypic variability. This variability can be functional, for example by allowing a fraction of stress prepared cells to survive an otherwise lethal stress. The optimal fraction of stress prepared cells depends on environmental conditions. However, how bacterial populations modulate their level of phenotypic variability remains unclear. Here we show that the alternative sigma factor σV circuit in Bacillus subtilis generates functional phenotypic variability that can be tuned by stress level, environmental history and genetic perturbations. Using single‐cell time‐lapse microscopy and microfluidics, we find the fraction of cells that immediately activate σV under lysozyme stress depends on stress level and on a transcriptional memory of previous stress. Iteration between model and experiment reveals that this tunability can be explained by the autoregulatory feedback structure of the sigV operon. As predicted by the model, genetic perturbations to the operon also modulate the response variability. The conserved sigma‐anti‐sigma autoregulation motif is thus a simple mechanism for bacterial populations to modulate their heterogeneity based on their environment. SYNOPSIS: A combination of single‐cell imaging and mathematical modelling reveals a simple mechanism for bacterial populations to modulate their heterogeneity based on their environment and environmental history. The time to activate the alternative sigma factor σV in Bacillussubtilis is found to be heterogeneous under lysozyme stress.This phenotypic variability can be tuned by stress levels, previous stress applications, and genetic perturbations.Modelling and experiments show that this tunability can be explained by the structure of the sigV operon, which consists of a 'mixed' positive and negative feedback loop. [ABSTRACT FROM AUTHOR]

Background: Single-cell RNA sequencing (scRNA-seq) is the most widely used technique to obtain gene expression profiles from complex tissues. Cell subsets and developmental states are often identified via differential gene expression patterns. Most of the single-cell tools utilized highly variable genes to annotate cell subsets and states. However, we have discovered that a group of genes, which sensitively respond to environmental stimuli with high coefficients of variation (CV), might impose overwhelming influences on the cell type annotation. Result: In this research, we developed a method, based on the CV-rank and Shannon entropy, to identify these noise genes, and termed them as "sensitive genes". To validate the reliability of our methods, we applied our tools in 11 single-cell data sets from different human tissues. The results showed that most of the sensitive genes were enriched pathways related to cellular stress response. Furthermore, we noticed that the unsupervised result was closer to the ground-truth cell labels, after removing the sensitive genes detected by our tools. Conclusion: Our study revealed the prevalence of stochastic gene expression patterns in most types of cells, compared the differences among cell marker genes, housekeeping genes (HK genes), and sensitive genes, demonstrated the similarities of functions of sensitive genes in various scRNA-seq data sets, and improved the results of unsupervised clustering towards the ground-truth labels. We hope our method would provide new insights into the reduction of data noise in scRNA-seq data analysis and contribute to the development of better scRNA-seq unsupervised clustering algorithms in the future. [ABSTRACT FROM AUTHOR]

Genetic variability, epigenetic variability, and gene expression variability (noise) are generally considered independently in their relationship with phenotypic variation. However, they appear to be intrinsically interconnected and influence it in combination. The study of the interplay between genetic and epigenetic variability has the longest history. This article rather considers the introduction of gene expression variability in its relationships with the two others and reviews for the first time experimental evidences over the four relationships connected to gene expression noise. They show how introducing this third source of variability complicates the way of thinking evolvability and the emergence of biological novelty. Finally, cancer cells are proposed to be an ideal model to decipher the dynamic interplay between genetic, epigenetic, and gene expression variability when one of them is either experimentally increased or therapeutically targeted. This interplay is also discussed in an evolutionary perspective in the context of cancer cell drug resistance. [ABSTRACT FROM AUTHOR]

The circadian clock is an endogenous and self‐sustained oscillator that anticipates daily environmental cycles. While rhythmic gene expression of circadian genes is well‐described in populations of cells, the single‐cell mRNA dynamics of multiple core clock genes remain largely unknown. Here we use single‐molecule fluorescence in situ hybridisation (smFISH) at multiple time points to measure pairs of core clock transcripts, Rev‐erbα (Nr1d1), Cry1 and Bmal1, in mouse fibroblasts. The mean mRNA level oscillates over 24 h for all three genes, but mRNA numbers show considerable spread between cells. We develop a probabilistic model for multivariate mRNA counts using mixtures of negative binomials, which accounts for transcriptional bursting, circadian time and cell‐to‐cell heterogeneity, notably in cell size. Decomposing the mRNA variability into distinct noise sources shows that clock time contributes a small fraction of the total variability in mRNA number between cells. Thus, our results highlight the intrinsic biological challenges in estimating circadian phase from single‐cell mRNA counts and suggest that circadian phase in single cells is encoded post‐transcriptionally. SYNOPSIS: Single‐molecule imaging of transcripts is combined with mathematical modelling to investigate how the mRNA distributions of core‐clock circadian genes evolve over the circadian cycle while considering multiple sources of intrinsic and extrinsic variability. Single molecule fluorescence in‐situ hybridization (smFISH) of core‐clock genes at multiple time points over the circadian cycle shows that the mean mRNA level oscillates but there is significant variability in mRNA counts between cells.Bayesian model selection identifies a mixture model of negative binomials as the preferred model of the mRNA counts.Noise decomposition of the preferred model shows that a small percentage of measured variation in core clock transcript number is attributable to circadian time. [ABSTRACT FROM AUTHOR]

Single-cell variability of growth is a biological phenomenon that has attracted growing interest in recent years. Important progress has been made in the knowledge of the origin of cell-to-cell heterogeneity of growth, especially in microbial cells. To better understand the origins of such heterogeneity at the single-cell level, we developed a new methodological pipeline that coupled cytometry-based cell sorting with automatized microscopy and image analysis to score the growth rate of thousands of single cells. This allowed investigating the influence of the initial amount of proteins of interest on the subsequent growth of the microcolony. As a preliminary step to validate this experimental setup, we referred to previous findings in yeast where the expression level of Tsl1, a member of the Trehalose Phosphate Synthase (TPS) complex, negatively correlated with cell division rate. We unfortunately could not find any influence of the initial TSL1 expression level on the growth rate of the microcolonies. We also analyzed the effect of the natural variations of trehalose-6-phosphate synthase (TPS1) expression on cell-to-cell growth heterogeneity, but we did not find any correlation. However, due to the already known altered growth of the tps1Δ mutants, we tested this strain at the single-cell level on a permissive carbon source. This mutant showed an outstanding lack of reproducibility of growth rate distributions as compared to the wild-type strain, with variable proportions of non-growing cells between cultivations and more heterogeneous microcolonies in terms of individual growth rates. Interestingly, this variable behavior at the single-cell level was reminiscent to the high variability that is also stochastically suffered at the population level when cultivating this tps1Δ strain, even when using controlled bioreactors.

Metabolic heterogeneity is widely recognized as the next challenge in our understanding of non-genetic variation. A growing body of evidence suggests that metabolic heterogeneity may result from the inherent stochasticity of intracellular events. However, metabolism has been traditionally viewed as a purely deterministic process, on the basis that highly abundant metabolites tend to filter out stochastic phenomena. Here we bridge this gap with a general method for prediction of metabolite distributions across single cells. By exploiting the separation of time scales between enzyme expression and enzyme kinetics, our method produces estimates for metabolite distributions without the lengthy stochastic simulations that would be typically required for large metabolic models. The metabolite distributions take the form of Gaussian mixture models that are directly computable from single-cell expression data and standard deterministic models for metabolic pathways. The proposed mixture models provide a systematic method to predict the impact of biochemical parameters on metabolite distributions. Our method lays the groundwork for identifying the molecular processes that shape metabolic heterogeneity and its functional implications in disease.

Age‐related and cancer‐related epigenomic modifications have been associated with enhanced cell‐to‐cell gene expression variability that characterizes increased cellular stochasticity. Since gene expression variability appears to be highly reduced by—and epigenetic and phenotypic stability acquired through—direct or long‐range cellular interactions during cell differentiation, we propose a common origin for aging and cancer in the failure to control cellular stochasticity by cell–cell interactions. Tissue‐disruption‐induced cellular stochasticity associated with epigenetic drift would be at the origin of organ dysfunction because of an increase in phenotypic variation among cells, ultimately leading to cell death and organ failure through a loss of coordination in cellular functions, and eventually to cancerization. We propose mechanistic research perspectives to corroborate this hypothesis and explore its evolutionary consequences, highlighting a positive correlation between the median age of mass loss onset (a proxy for the onset of organ aging) and the median age at cancer diagnosis. [ABSTRACT FROM AUTHOR]

Recent advances in fluorescence microscopy have made it possible to measure the fluctuations of nascent (actively transcribed) RNA. These closely reflect transcription kinetics, as opposed to conventional measurements of mature (cellular) RNA, whose kinetics is affected by additional processes downstream of transcription. Here, we formulate a stochastic model which describes promoter switching, initiation, elongation, premature detachment, pausing, and termination while being analytically tractable. We derive exact closed-form expressions for the mean and variance of nascent RNA fluctuations on gene segments, as well as of total nascent RNA on a gene. We also obtain exact expressions for the first two moments of mature RNA fluctuations and approximate distributions for total numbers of nascent and mature RNA. Our results, which are verified by stochastic simulation, uncover the explicit dependence of the statistics of both types of RNA on transcriptional parameters and potentially provide a means to estimate parameter values from experimental data. [ABSTRACT FROM AUTHOR]

When Streptococcus mutans is transferred from a preferred carbohydrate (glucose or fructose) to lactose, initiation of growth can take several hours, and substantial amounts of glucose are released during growth. Here, S. mutans strains UA159 and GS‐5 were examined for stochastic behaviors in transcription of the lac operon. Using a gfp reporter fusion, we demonstrated that induction of the lac operon occurs in only a fraction of the population, with prior exposure to carbohydrate source and strain influencing the magniture of the sub‐population response. Lower glucokinase activity in GS‐5 was associated with release of substantially more glucose than UA159 and significantly lower lac expression. Mutants unable to use lactose grew on lactose as the sole carbohydrate when strains with an intact lac operon were also present in the cultures, indicative of the potential for population cheating. Utilizing a set of engineered obligate cheating and non‐cheating strains, we confirmed that cheating can sustain a heterogeneous population. Futher, obligate cheaters of GS‐5 competed well with the non‐cheaters and showed a high degree of competitive fitness in a human‐derived consortium biofilm model. The results show that bet‐hedging behaviors in carbohydrate metabolism may substantially influence the composition and pathogenic potential of oral biofilms. [ABSTRACT FROM AUTHOR]

Precise determination of a noisy biological oscillator's period from limited experimental data can be challenging. The common practice is to calculate a single number (a point estimate) for the period of a particular time course. Uncertainty is inherent in any statistical estimator applied to noisy data, so our confidence in such point estimates depends on the quality and quantity of the data. Ideally, a period estimation method should both produce an accurate point estimate of the period and measure the uncertainty in that point estimate. A variety of period estimation methods are known, but few assess the uncertainty of the estimates, and a measure of uncertainty is rarely reported in the experimental literature. We compare the accuracy of point estimates using six common methods, only one of which can also produce uncertainty measures. We then illustrate the advantages of a new Bayesian method for estimating period, which outperforms the other six methods in accuracy of point estimates for simulated data and also provides a measure of uncertainty. We apply this method to analyze circadian oscillations of gene expression in individual mouse fibroblast cells and compute the number of cells and sampling duration required to reduce the uncertainty in period estimates to a desired level. This analysis indicates that, due to the stochastic variability of noisy intracellular oscillators, achieving a narrow margin of error can require an impractically large number of cells. In addition, we use a hierarchical model to determine the distribution of intrinsic cell periods, thereby separating the variability due to stochastic gene expression within each cell from the variability in period across the population of cells.

Antimicrobial resistance (AMR) is an emerging global health crisis that is undermining advances in modern medicine and, if unmitigated, threatens to kill 10 million people per year worldwide by 2050. Research over the last decade has demonstrated that the differences between genetically identical cells in the same environment can lead to drug resistance. Fluctuations in gene expression, modulated by gene regulatory networks, can lead to non-genetic heterogeneity that results in the fractional killing of microbial populations causing drug therapies to fail; this non-genetic drug resistance can enhance the probability of acquiring genetic drug resistance mutations. Mathematical models of gene networks can elucidate general principles underlying drug resistance, predict the evolution of resistance, and guide drug resistance experiments in the laboratory. Cells genetically engineered to carry synthetic gene networks regulating drug resistance genes allow for controlled, quantitative experiments on the role of non-genetic heterogeneity in the development of drug resistance. In this perspective article, we emphasize the contributions that mathematical, computational, and synthetic gene network models play in advancing our understanding of AMR to discover effective therapies against drug-resistant infections.

Single-cell analysis allows biologists to gain huge insight into cell differentiation and tissue structuration. Randomness of differentiation, both in vitro and in vivo, of pluripotent (multipotent) stem cells is now demonstrated to be mainly based on stochastic gene expression. Nevertheless, it remains necessary to incorporate this inherent stochasticity of developmental processes within a coherent scheme. We argue here that the theory called ontophylogenesis is more relevant and better fits with experimental data than alternative theories which have been suggested based on the notions of self-organization and attractor states. The ontophylogenesis theory considers the generation of a differentiated state as a constrained random process: randomness is provided by the stochastic dynamics of biochemical reactions while the environmental constraints, including cell inner structures and cell-cell interactions, drive the system toward a stabilized state of equilibrium. In this conception, biological organization during development can be seen as the result of multiscale constraints produced by the dynamical organization of the biological system which retroacts on the stochastic dynamics at lower scales. This scheme makes it possible to really understand how the generation of reproducible structures at higher organization levels can be fully compatible with probabilistic behavior at the lower levels. It is compatible with the second law of thermodynamics but allows the overtaking of the limitations exhibited by models only based on entropy exchanges which cannot cope with the description nor the dynamics of the mesoscopic and macroscopic organization of biological systems.

Fractional killing, which is a significant impediment to successful chemotherapy, is observed even in a population of genetically identical cancer cells exposed to apoptosis-inducing agents. This phenomenon arises not from genetic mutation but from cell-to-cell variation in the activation timing and level of the proteins that regulates apoptosis. To understand the mechanism behind the phenomenon, we formulate complex fractional killing processes as a first-passage time (FPT) problem with a stochastically fluctuating boundary. Analytical calculations are performed for the FPT distribution in a toy model of stochastic p53 gene expression, where the cancer cell is killed only when the p53 expression level crosses an active apoptotic threshold. Counterintuitively, we find that threshold fluctuations can effectively enhance cellular killing by significantly decreasing the mean time that the p53 protein reaches the threshold level for the first time. Moreover, faster fluctuations lead to the killing of more cells. These qualitative results imply that fluctuations in threshold are a non-negligible stochastic source, and can be taken as a strategy for combating fractional killing of cancer cells.

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. [ABSTRACT FROM AUTHOR]

Single-cell RNA sequencing data have complex features such as dropout events, overdispersion, and high-magnitude outliers, resulting in complicated probability distributions of mRNA abundances that are statistically characterized in terms of a zero-inated negative binomial (ZINB) model. Here we provide a mesoscopic kinetic foundation for the widely used ZINB model based on the biochemical reaction kinetics underlying transcription. Using multiscale modeling and simplification techniques, we show that the ZINB distribution of mRNA abundance and the related phenomenon of transcriptional bursting naturally emerge from a three-state stochastic transcription model. We further reveal a nontrivial quantitative relationship between dropout events and transcriptional bursting, which provides novel insights into how the burst size and burst frequency affect the dropout rate. Two different biophysical origins of overdispersion are also clarified at the single-cell level. [ABSTRACT FROM AUTHOR]