Effects of cell-cycle-dependent expression on random fluctuations in protein levels

Cyclo-stationary distributions of mRNA and Protein counts for random cell division times

There is a long history of using experimental and computational approaches to study noise in single-cell levels of mRNA and proteins. The noise originates from a myriad of factors: intrinsic processes of gene expression, partitioning errors during division, and extrinsic effects, such as, random cell-cycle times. Although theoretical methods are well developed to analytically understand full statistics of copy numbers for fixed or Erlang distributed cell cycle times, the general problem of random division times is still open. For any random (but uncorrelated) division time distribution, we present a method to address this challenging problem and obtain exact series representations of the copy number distributions in the cyclo-stationary state. We provide explicit cell age-specific and age-averaged results, and analyze the relative contribution to noise from intrinsic and extrinsic sources. Our analytical approach will aid the analysis of single-cell expression data and help in disentangling the impact of variability in division times.

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 can lead to different levels of noise in a given gene product. We first consider a stable protein, whose concentration is diluted by cellular growth. This protein inhibits growth at high concentrations, establishing a positive feedback loop. Using 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 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.

Modification of Seurat v4 for the Development of a Phase Assignment Tool Able to Distinguish between G2 and Mitotic Cells

Single-cell RNA sequencing (scRNAseq) is a rapidly advancing field enabling the characterisation of heterogeneous gene expression profiles within a population. The cell cycle phase is a major contributor to gene expression variance between cells and computational analysis tools have been developed to assign cell cycle phases to cells within scRNAseq datasets. Whilst these tools can be extremely useful, all have the drawback that they classify cells as only G1, S or G2/M. Existing discrete cell phase assignment tools are unable to differentiate between G2 and M and continuous-phase-assignment tools are unable to identify a region corresponding specifically to mitosis in a pseudo-timeline for continuous assignment along the cell cycle. In this study, bulk RNA sequencing was used to identify differentially expressed genes between mitotic and interphase cells isolated based on phospho-histone H3 expression using fluorescence-activated cell sorting. These gene lists were used to develop a methodology which can distinguish G2 and M phase cells in scRNAseq datasets. The phase assignment tools present in Seurat were modified to allow for cell cycle phase assignment of all stages of the cell cycle to identify a mitotic-specific cell population.

Exact solution of a three-stage model of stochastic gene expression including cell-cycle dynamics

The classical three-stage model of stochastic gene expression predicts the statistics of single cell mRNA and protein number fluctuations as a function of the rates of promoter switching, transcription, translation, degradation and dilution. While this model is easily simulated, its analytical solution remains an unsolved problem. Here we modify this model to explicitly include cell-cycle dynamics and then derive an exact solution for the time-dependent joint distribution of mRNA and protein numbers. We show large differences between this model and the classical model which captures cell-cycle effects implicitly via effective first-order dilution reactions. In particular we find that the Fano factor of protein numbers calculated from a population snapshot measurement are underestimated by the classical model whereas the correlation between mRNA and protein can be either over- or underestimated, depending on the timescales of mRNA degradation and promoter switching relative to the mean cell-cycle duration time.

PyEcoLib: a python library for simulating stochastic cell size dynamics

Recently, there has been an increasing need for tools to simulate cell size regulation due to important applications in cell proliferation and gene expression. However, implementing the simulation usually presents some difficulties, as the division has a cycle-dependent occurrence rate. In this article, we gather a recent theoretical framework inPyEcoLib, a python-based library to simulate the stochastic dynamics of the size of bacterial cells. This library can simulate cell size trajectories with an arbitrarily small sampling period. In addition, this simulator can include stochastic variables, such as the cell size at the beginning of the experiment, the cycle duration timing, the growth rate, and the splitting position. Furthermore, from a population perspective, the user can choose between tracking a single lineage or all cells in a colony. They can also simulate the most common division strategies (adder, timer, and sizer) using the division rate formalism and numerical methods. As an example of PyecoLib applications, we explain how to couple size dynamics with gene expression predicting, from simulations, how the noise in protein levels increases by increasing the noise in division timing, the noise in growth rate and the noise in cell splitting position. The simplicity of this library and its transparency about the underlying theoretical framework yield the inclusion of cell size stochasticity in complex models of gene expression.

Making use of noise in biological systems

Disorder and noise are inherent in biological systems. They are required to provide systems with the advantages required for proper functioning. Noise is a part of the flexibility and plasticity of biological systems. It provides systems with increased routes, improves information transfer, and assists in response triggers. This paper reviews recent studies on noise at the genome, cellular, and whole organ levels. We focus on the need to use noise in system engineering. We present some of the challenges faced in studying noise. Optimizing the efficiency of complex systems requires a degree of variability in their functions within certain limits. Constrained noise can be considered a method for improving system robustness by regulating noise levels in continuously dynamic settings. The digital pill-based artificial intelligence (AI)-based platform is the first to implement second-generation AI comprising variability-based signatures. This platform enhances the efficacy of the therapeutic regimens. Systems requiring variability and mechanisms regulating noise are mandatory for understanding biological functions.

Noise suppression in stochastic genetic circuits using PID controllers

Inside individual cells, protein population counts are subject to molecular noise due to low copy numbers and the inherent probabilistic nature of biochemical processes. We investigate the effectiveness of proportional, integral and derivative (PID) based feedback controllers to suppress protein count fluctuations originating from two noise sources: bursty expression of the protein, and external disturbance in protein synthesis. Designs of biochemical reactions that function as PID controllers are discussed, with particular focus on individual controllers separately, and the corresponding closed-loop system is analyzed for stochastic controller realizations. Our results show that proportional controllers are effective in buffering protein copy number fluctuations from both noise sources, but this noise suppression comes at the cost of reduced static sensitivity of the output to the input signal. In contrast, integral feedback has no effect on the protein noise level from stochastic expression, but significantly minimizes the impact of external disturbances, particularly when the disturbance comes at low frequencies. Counter-intuitively, integral feedback is found to amplify external disturbances at intermediate frequencies. Next, we discuss the design of a coupled feedforward-feedback biochemical circuit that approximately functions as a derivate controller. Analysis using both analytical methods and Monte Carlo simulations reveals that this derivative controller effectively buffers output fluctuations from bursty stochastic expression, while maintaining the static input-output sensitivity of the open-loop system. In summary, this study provides a systematic stochastic analysis of biochemical controllers, and paves the way for their synthetic design and implementation to minimize deleterious fluctuations in gene product levels.

In the noisy cellular environment, biochemical species such as genes, RNAs and proteins that often occur at low molecular counts, are subject to considerable stochastic fluctuations in copy numbers over time. How cellular biochemical processes function reliably in the face of such randomness is an intriguing fundamental problem. Increasing evidence suggests that random fluctuations (noise) in protein copy numbers play important functional roles, such as driving genetically identical cells to different cell fates. Moreover, many disease states have been attributed to elevated noise levels in specific proteins. Here we systematically investigate design of biochemical systems that function as proportional, integral and derivative-based feedback controllers to suppress protein count fluctuations arising from bursty expression of the protein and external disturbance in protein synthesis. Our results show that different controllers are effective in buffering different noise components, and identify ranges of feedback gain for minimizing deleterious fluctuations in protein levels.

Contributions to the 'noise floor' in gene expression in a population of dividing cells

Experiments with cells reveal the existence of a lower bound for protein noise, the noise floor, in highly expressed genes. Its origins are still debated. We propose a minimal model of gene expression in a proliferating bacterial cell population. The model predicts the existence of a noise floor and it semi-quantitatively reproduces the curved shape of the experimental noise vs. mean protein concentration plots. When the cell volume increases in a different manner than does the mean protein copy number, the noise floor level is determined by the cell population’s age structure and by the dependence of the mean protein concentration on cell age. Additionally, the noise floor level may depend on a biological limit for the mean number of bursts in the cell cycle. In that case, the noise floor level depends on the burst size distribution width but it is insensitive to the mean burst size. Our model quantifies the contributions of each of these mechanisms to gene expression noise.

Role of ion channels during cell division

Ion channels are transmembrane proteins whose canonical function is the transport of ions across the plasma membrane to regulate cell membrane potential and play an essential role in neural communication, nerve conduction, and muscle contraction. However, over the last few years, non-canonical functions have been identified for many channels, having active roles in phagocytosis, invasiveness, proliferation, among others. The participation of some channels in cell proliferation has raised the question of whether they may play an active role in mitosis. There are several reports showing the participation of channels during interphase, however, the direct participation of ion channels in mitosis has received less attention. In this article, we summarize the current evidence on the participation of ion channels in mitosis. We also summarize some tools that would allow the study of ion channels and cell cycle regulatory molecules in individual cells during mitosis.

Effects of cell cycle variability on lineage and population measurements of messenger RNA abundance

Many models of gene expression do not explicitly incorporate a cell cycle description. Here, we derive a theory describing how messenger RNA (mRNA) fluctuations for constitutive and bursty gene expression are influenced by stochasticity in the duration of the cell cycle and the timing of DNA replication. Analytical expressions for the moments show that omitting cell cycle duration introduces an error in the predicted mean number of mRNAs that is a monotonically decreasing function of η, which is proportional to the ratio of the mean cell cycle duration and the mRNA lifetime. By contrast, the error in the variance of the mRNA distribution is highest for intermediate values of η consistent with genome-wide measurements in many organisms. Using eukaryotic cell data, we estimate the errors in the mean and variance to be at most 3% and 25%, respectively. Furthermore, we derive an accurate negative binomial mixture approximation to the mRNA distribution. This indicates that stochasticity in the cell cycle can introduce fluctuations in mRNA numbers that are similar to the effect of bursty transcription. Finally, we show that for real experimental data, disregarding cell cycle stochasticity can introduce errors in the inference of transcription rates larger than 10%.

Stochastic models coupling gene expression and partitioning in cell division in Escherichia coli

Regulation of future RNA and protein numbers is a key process by which cells continuously best fit the environment. In bacteria, RNA and proteins exist in small numbers and their regulatory processes are stochastic. Consequently, there is cell-to-cell variability in these numbers, even between sister cells. Traditionally, the two most studied sources of this variability are gene expression and RNA and protein degradation, with evidence suggesting that the latter is subject to little regulation, when compared to the former. However, time-lapse microscopy and single molecule fluorescent tagging have produced evidence that cell division can also be a significant source of variability due to asymmetries in the partitioning of RNA and proteins. Relevantly, the impact of this noise differs from noise in production and degradation since, unlike these, it is not continuous. Rather, it occurs at specific time points, at which moment it can introduce major fluctuations. Several models have now been proposed that integrate noise from cell division, in addition to noise in gene expression, to mimic the dynamics of RNA and protein numbers of cell lineages. This is expected to be particularly relevant in genetic circuits, where significant fluctuations in one component protein, at specific time moments, are expected to perturb near-equilibrium states of the circuits, which can have long-lasting consequences. Here we review stochastic models coupling these processes in Escherichia coli, from single genes to small circuits.

Characterizing and inferring quantitative cell cycle phase in single-cell RNA-seq data analysis

Cellular heterogeneity in gene expression is driven by cellular processes, such as cell cycle and cell-type identity, and cellular environment such as spatial location. The cell cycle, in particular, is thought to be a key driver of cell-to-cell heterogeneity in gene expression, even in otherwise homogeneous cell populations. Recent advances in single-cell RNA-sequencing (scRNA-seq) facilitate detailed characterization of gene expression heterogeneity and can thus shed new light on the processes driving heterogeneity. Here, we combined fluorescence imaging with scRNA-seq to measure cell cycle phase and gene expression levels in human induced pluripotent stem cells (iPSCs). By using these data, we developed a novel approach to characterize cell cycle progression. Although standard methods assign cells to discrete cell cycle stages, our method goes beyond this and quantifies cell cycle progression on a continuum. We found that, on average, scRNA-seq data from only five genes predicted a cell's position on the cell cycle continuum to within 14% of the entire cycle and that using more genes did not improve this accuracy. Our data and predictor of cell cycle phase can directly help future studies to account for cell cycle-related heterogeneity in iPSCs. Our results and methods also provide a foundation for future work to characterize the effects of the cell cycle on expression heterogeneity in other cell types.

Exact solution of stochastic gene expression models with bursting, cell cycle and replication dynamics

The bulk of stochastic gene expression models in the literature do not have an explicit description of the age of a cell within a generation and hence they cannot capture events such as cell division and DNA replication. Instead, many models incorporate the cell cycle implicitly by assuming that dilution due to cell division can be described by an effective decay reaction with first-order kinetics. If it is further assumed that protein production occurs in bursts, then the stationary protein distribution is a negative binomial. Here we seek to understand how accurate these implicit models are when compared with more detailed models of stochastic gene expression. We derive the exact stationary solution of the chemical master equation describing bursty protein dynamics, binomial partitioning at mitosis, age-dependent transcription dynamics including replication, and random interdivision times sampled from Erlang or more general distributions; the solution is different for single lineage and population snapshot settings. We show that protein distributions are well approximated by the solution of implicit models (a negative binomial) when the mean number of mRNAs produced per cycle is low and the cell cycle length variability is large. When these conditions are not met, the distributions are either almost bimodal or else display very flat regions near the mode and cannot be described by implicit models. We also show that for genes with low transcription rates, the size of protein noise has a strong dependence on the replication time, it is almost independent of cell cycle variability for lineage measurements, and increases with cell cycle variability for population snapshot measurements. In contrast for large transcription rates, the size of protein noise is independent of replication time and increases with cell cycle variability for both lineage and population measurements.

Quantifying heterogeneity of stochastic gene expression

The heterogeneity of stochastic gene expression, which refers to the temporal fluctuation in a gene product and its cell-to-cell variation, has attracted considerable interest from biologists, physicists, and mathematicians. The dynamics of protein production and degradation have been modeled as random processes with transition probabilities. However, there is a gap between theory and phenomena, particularly in terms of analytical formulation and parameter estimation. In this study, we propose a theoretical framework in which we present a basic model of a gene regulatory system, derive a steady-state solution, and provide a Bayesian approach for estimating the model parameters from single-cell experimental data. The proposed framework is demonstrated to be applicable for various scales of single-cell experiments at both the mRNA and protein levels and is useful for comparing kinetic parameters across species, genomes, and cell strains.

Multicellular Models Bridging Intracellular Signaling and Gene Transcription to Population Dynamics

Cell signaling and gene transcription occur at faster time scales compared to cellular death, division, and evolution. Bridging these multiscale events in a model is computationally challenging. We introduce a framework for the systematic development of multiscale cell population models. Using message passing interface (MPI) parallelism, the framework creates a population model from a single-cell biochemical network model. It launches parallel simulations on a single-cell model and treats each stand-alone parallel process as a cell object. MPI mediates cell-to-cell and cell-to-environment communications in a server-client fashion. In the framework, model-specific higher level rules link the intracellular molecular events to cellular functions, such as death, division, or phenotype change. Cell death is implemented by terminating a parallel process, while cell division is carried out by creating a new process (daughter cell) from an existing one (mother cell). We first demonstrate these capabilities by creating two simple example models. In one model, we consider a relatively simple scenario where cells can evolve independently. In the other model, we consider interdependency among the cells, where cellular communication determines their collective behavior and evolution under a temporally evolving growth condition. We then demonstrate the framework’s capability by simulating a full-scale model of bacterial quorum sensing, where the dynamics of a population of bacterial cells is dictated by the intercellular communications in a time-evolving growth environment.

Single-cell heterogeneity and cell-cycle-related viral gene bursts in the human leukaemia virus HTLV-1

Background: The human leukaemia virus HTLV-1 expresses essential accessory genes that manipulate the expression, splicing and transport of viral mRNAs.  Two of these genes,taxandhbz, also promote proliferation of the infected cell, and both genes are thought to contribute to oncogenesis in adult T-cell leukaemia/lymphoma.  The regulation of HTLV-1 proviral latency is not understood. tax,on the proviral plus strand, is usually silent in freshly-isolated cells, whereas the minus-strand-encodedhbzgene is persistently expressed at a low level.  However, the persistently activated host immune response to Tax indicates frequent expression oftaxin vivo. Methods: We used single-molecule RNA-FISH to quantify the expression of HTLV-1 transcripts at the single-cell level in a total of >19,000 cells from five T-cell clones, naturally infected with HTLV-1, isolated by limiting dilution from peripheral blood of HTLV-1-infected subjects. Results: We found strong heterogeneity both within and between clones in the expression of the proviral plus-strand (detected by hybridization to thetaxgene) and the minus-strand (hbzgene). Both genes are transcribed in bursts;taxexpression is enhanced in the absence ofhbz, whilehbzexpression increased in cells with hightaxexpression. Surprisingly, we found thathbzexpression is strongly associated with the S and G2/M phases of the cell cycle, independent oftaxexpression.  Contrary to current belief,hbzis not expressed in all cells at all times, even within one clone.  Inhbz-positive cells, the abundance ofhbztranscripts showed a very strong positive linear correlation with nuclear volume.Conclusions: The occurrence of intense, intermittent plus-strand gene bursts in independent primary HTLV-1-infected T-cell clones from unrelated individuals strongly suggests that the HTLV-1 plus-strand is expressed in bursts in vivo.  Our results offer an explanation for the paradoxical correlations observed between the host immune response and HTLV-1 transcription. Collapse

Single-cell heterogeneity and cell-cycle-related viral gene bursts in the human leukaemia virus HTLV-1

Background: The human leukaemia virus HTLV-1 expresses essential accessory genes that manipulate the expression, splicing and transport of viral mRNAs.  Two of these genes, tax and hbz, also promote proliferation of the infected cell, and both genes are thought to contribute to oncogenesis in adult T-cell leukaemia/lymphoma.  The regulation of HTLV-1 proviral latency is not understood.  tax, on the proviral plus strand, is usually silent in freshly-isolated cells, whereas the minus-strand-encoded hbz gene is persistently expressed at a low level.  However, the persistently activated host immune response to Tax indicates frequent expression of taxin vivo. 

Methods: We used single-molecule RNA-FISH to quantify the expression of HTLV-1 transcripts at the single-cell level in a total of >19,000 cells from five T-cell clones, naturally infected with HTLV-1, isolated by limiting dilution from peripheral blood of HTLV-1-infected subjects. 

Results: We found strong heterogeneity both within and between clones in the expression of the proviral plus-strand (detected by hybridization to the tax gene) and the minus-strand ( hbz gene). Both genes are transcribed in bursts; tax expression is enhanced in the absence of hbz, while hbz expression increased in cells with high tax expression. Surprisingly, we found that hbz expression is strongly associated with the S and G ₂/M phases of the cell cycle, independent of tax expression.  Contrary to current belief, hbz is not expressed in all cells at all times, even within one clone.  In hbz-positive cells, the abundance of hbz transcripts showed a very strong positive linear correlation with nuclear volume.

Conclusions: The occurrence of intense, intermittent plus-strand gene bursts in independent primary HTLV-1-infected T-cell clones from unrelated individuals strongly suggests that the HTLV-1 plus-strand is expressed in bursts in vivo.  Our results offer an explanation for the paradoxical correlations observed between the host immune response and HTLV-1 transcription.

Analysis of Noise Mechanisms in Cell-Size Control

At the single-cell level, noise arises from multiple sources, such as inherent stochasticity of biomolecular processes, random partitioning of resources at division, and fluctuations in cellular growth rates. How these diverse noise mechanisms combine to drive variations in cell size within an isoclonal population is not well understood. Here, we investigate the contributions of different noise sources in well-known paradigms of cell-size control, such as adder (division occurs after adding a fixed size from birth), sizer (division occurs after reaching a size threshold), and timer (division occurs after a fixed time from birth). Analysis reveals that variation in cell size is most sensitive to errors in partitioning of volume among daughter cells, and not surprisingly, this process is well regulated among microbes. Moreover, depending on the dominant noise mechanism, different size-control strategies (or a combination of them) provide efficient buffering of size variations. We further explore mixer models of size control, where a timer phase precedes/follows an adder, as has been proposed in Caulobacter crescentus. Although mixing a timer and an adder can sometimes attenuate size variations, it invariably leads to higher-order moments growing unboundedly over time. This results in a power-law distribution for the cell size, with an exponent that depends inversely on the noise in the timer phase. Consistent with theory, we find evidence of power-law statistics in the tail of C. crescentus cell-size distribution, although there is a discrepancy between the observed power-law exponent and that predicted from the noise parameters. The discrepancy, however, is removed after data reveal that the size added by individual newborns in the adder phase itself exhibits power-law statistics. Taken together, this study provides key insights into the role of noise mechanisms in size homeostasis, and suggests an inextricable link between timer-based models of size control and heavy-tailed cell-size distributions.

Careful accounting of extrinsic noise in protein expression reveals correlations among its sources

In order to grow and replicate, living cells must express a diverse array of proteins, but the process by which proteins are made includes a great deal of inherent randomness. Understanding this randomness-whether it arises from the discrete stochastic nature of chemical reactivity ("intrinsic" noise), or from cell-to-cell variability in the concentrations of molecules involved in gene expression, or from the timings of important cell-cycle events like DNA replication and cell division ("extrinsic" noise)-remains a challenge. In this article we analyze a model of gene expression that accounts for several extrinsic sources of noise, including those associated with chromosomal replication, cell division, and variability in the numbers of RNA polymerase, ribonuclease E, and ribosomes. We then attempt to fit our model to a large proteomics and transcriptomics data set and find that only through the introduction of a few key correlations among the extrinsic noise sources can we accurately recapitulate the experimental data. These include significant correlations between the rate of mRNA degradation (mediated by ribonuclease E) and the rates of both transcription (RNA polymerase) and translation (ribosomes) and, strikingly, an anticorrelation between the transcription and the translation rates themselves.