Biology subjects

Grima, R.

Publications and source records attributed to Grima, R..

At least 37 records · Page 2Linked to original sources

bioRxiv2022

Accuracy and limitations of extrinsic noise models to describe gene expression in growing cells

The standard model describing the fluctuations of mRNA numbers in single cells is the telegraph model which includes synthesis and degradation of mRNA, and switching of the gene between active and inactive states. While commonly used, this model does not describe how fluctuations are influenced by the cell cycle phase, cellular growth and division, and other crucial aspects of cellular biology. Here we derive the analytical time-dependent solution of an extended telegraph model that explicitly considers the doubling of gene copy numbers upon DNA replication, dependence of the mRNA synthesis rate on cellular volume, gene dosage compensation, partitioning of molecules during cell division, cell-cycle duration variability, and cell-size control strategies. Based on the time-dependent solution, we obtain the analytical distributions of transcript numbers for lineage and population measurements in steady-state growth and also find a linear relation between the Fano factor of mRNA fluctuations and cell volume fluctuations. We show that generally the lineage and population distributions in steady-state growth cannot be accurately approximated by the steady-state solution of extrinsic noise models, i.e. a telegraph model with parameters drawn from probability distributions. This is because the mRNA lifetime is often not small enough compared to the cell cycle duration to erase the memory of division and replication. Accurate approximations are possible when this memory is weak, e.g. for genes with bursty expression and for which there is sufficient gene dosage compensation when replication occurs.

systems biology↗

bioRxiv2022

Approximating Solutions of the Chemical Master Equation using Neural Networks

The Chemical Master Equation (CME) provides an accurate description of stochastic biochemical reaction networks in well-mixed conditions, but it cannot be solved analytically for most systems of practical interest. While Monte Carlo methods provide a principled means to probe the system dynamics, their high computational cost can render the estimation of molecule number distributions and other numerical tasks infeasible due to the large number of repeated simulations typically required. In this paper we aim to leverage the representational power of neural networks to approximate the solutions of the CME and propose a framework for Neural Estimation of Stochastic Simulations for Inference and Exploration (Nessie). Our approach is based on training a neural network to learn the distributions predicted by the CME from a relatively small number of stochastic simulations, thereby accelerating computationally intensive tasks such as parameter exploration and inference. We show on biologically relevant examples that simple neural networks with one hidden layer are able to capture highly complex distributions across parameter space. We provide a detailed discussion of the neural network implementation and code for easy reproducibility.

systems biology↗

bioRxiv2022

A fluctuation-based approach to infer kinetics and topology of cell-state switching

In the noisy cellular environment, RNAs and proteins are subject to considerable stochastic fluctuations in copy numbers over time. As a consequence, single cells within the same isoclonal population can differ in their expression profile and reside in different phenotypic states. The dynamic nature of this intercellular variation, where individual cells can transition between different states over time makes it a particularly hard phenomenon to characterize. Here we propose a novel fluctuation-test approach to infer the kinetics of transitions between cell states. More specifically, single cells are randomly drawn from the population and grown into cell colonies. After growth for a fixed number of generations, the number of cells residing in different states is assayed for each colony. In a simple system with reversible switching between two cell states, our analysis shows that the extent of colony-to-colony fluctuations in the fraction of cells in a given state is monotonically related to the switching kinetics. Several closed-form formulas for inferring the switching rates from experimentally quantified fluctuations are presented. We further extend this approach to multiple cell states where harnessing fluctuation signatures can reveal both the topology and the rates of cell-state switching. In summary, our analysis provides a powerful approach for dissecting cell-state transitions based on a single time point measurement. This is especially important for scenarios where a measurement involves killing the cell (for example, performing single-cell RNA-seq or assaying whether a microbial/cancer cell is in a drug-sensitive or drug-tolerant state), and hence the state of the same cell cannot be measured at different time points.

systems biology↗

bioRxiv2022

Predicting variability in nascent RNA from transcription initiation kinetics

Fluctuations in the number of nascent RNA accurately reflect transcriptional activity. However, mathematical models predicting their distributions are difficult to solve analytically due to their non-Markovian nature stemming from transcriptional elongation. Here we circumvent this problem by deriving an exact relationship between the steady-state distribution of nascent RNA and the distribution of initiation times, which can be computed for any general initiation mechanism described by a set of first-order reactions. We test our theory using simulations and live cell imaging data.

biophysics↗

bioRxiv2022

Modulation of nuclear and cytoplasmic mRNA fluctuations by time-dependent stimuli: analytical distributions

Temporal variation of environmental stimuli leads to changes in gene expression. Since the latter is noisy and since many reaction events occur between the birth and death of an mRNA molecule, it is of interest to understand how a stimulus affects the transcript numbers measured at various sub-cellular locations. Here, we construct a stochastic model describing the dynamics of signal-dependent gene expression and its propagation downstream of transcription. For any time-dependent stimulus and assuming bursty gene expression, we devise a procedure which allows us to obtain time-dependent distributions of mRNA numbers at various stages of its life-cycle, e.g. in its nascent form at the transcription site, post-splicing in the nucleus, and after it is exported to the cytoplasm. We also derive an expression for the error in the approximation whose accuracy is verified via stochastic simulation. We find that, depending on the frequency of oscillation and the time of measurement, a stimulus can lead to cytoplasmic amplification or attenuation of transcriptional noise.

systems biology↗

bioRxiv2022

Inference and Uncertainty Quantification of Stochastic Gene Expression via Synthetic Models

AO_SCPLOWBSTRACTC_SCPLOWEstimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of nontrivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression.

systems biology↗

bioRxiv2022

DelaySSAToolkit.jl: stochastic simulation of reaction systems with time delays in Julia

SummaryDelaySSAToolkit.jl is a Julia package for modelling reaction systems with non-Markovian dynamics, specifically those with time delays. These delays implicitly capture multiple intermediate reaction steps and hence serve as an effective model reduction technique for complex systems in biology, chemistry, ecology and genetics. The package implements a variety of exact formulations of the delay stochastic simulation algorithm. Contactxmfu@ecust.edu.cn, zcao@ecust.edu.cn, or ramon.grima@ed.ac.uk Availability and ImplementationThe source code and documentation of DelaySSAToolkit.jl are available at https://github.com/palmtree2013/DelaySSAToolkit.jl.

systems biology↗

bioRxiv2021

Accurate inference of stochastic gene expression from nascent transcript heterogeneity

Transcriptional rates are often estimated by fitting the distribution of mature mRNA numbers measured using smFISH (single molecule fluorescence in situ hybridization) with the distribution predicted by the telegraph model of gene expression, which defines two promoter states of activity and inactivity. However, fluctuations in mature mRNA numbers are strongly affected by processes downstream of transcription. In addition, the telegraph model assumes one gene copy, but in experiments cells may have two gene copies as cells replicate their genome during the cell cycle. Whilst it is often presumed that post-transcriptional noise and gene copy number variation affect transcriptional parameter estimation, the size of the error introduced remains unclear. To address this issue, here we measure both mature and nascent mRNA distributions of GAL10 in yeast cells using smFISH and classify each cell according to its cell cycle phase. We infer transcriptional parameters from mature and nascent mRNA distributions, with and without accounting for cell cycle phase and compare the results to live-cell transcription measurements of the same gene. We find that: (i) correcting for cell cycle dynamics decreases the promoter switching rates and the initiation rate, and increases the fraction of time spent in the active state, as well as the burst size; (ii) additional correction for post-transcriptional noise leads to further increases in the burst size and to a large reduction in the errors in parameter estimation. Furthermore, we outline how to correctly adjust for measurement noise in smFISH due to uncertainty in transcription site localisation when introns cannot be labelled. Simulations with parameters estimated from nascent smFISH data, which is corrected for cell cycle phases and measurement noise, leads to autocorrelation functions that agree with those obtained from live-cell imaging.

systems biology↗

bioRxiv2021

Concentration fluctuations due to size-dependent gene expression and cell-size control mechanisms

Intracellular reaction rates depend on concentrations and hence their levels are often regulated. However classical models of stochastic gene expression lack a cell size description and cannot be used to predict noise in concentrations. Here, we construct a model of gene product dynamics that includes a description of cell growth, cell division, size-dependent gene expression, gene dosage compensation, and size control mechanisms that can vary with the cell cycle phase. We obtain expressions for the approximate distributions and power spectra of concentration fluctuations which lead to insight into the emergence of concentration homeostasis. Furthermore, we find that (i) the conditions necessary to suppress cell division-induced concentration oscillations are difficult to achieve; (ii) mRNA concentration and number distributions can have different number of modes; (iii) certain size control strategies are ideal because they maintain constant mean concentrations whilst minimising concentration noise. Predictions are confirmed using lineage data for E. coli, fission yeast and budding yeast.

systems biology↗

bioRxiv2021

Cluster mean-field theory accurately predicts statistical properties of large-scale DNA methylation patterns

The accurate establishment and maintenance of DNA methylation patterns is vital for mammalian development and disruption to these processes causes human disease. Our understanding of DNA methylation mechanisms has been facilitated by mathematical modelling, particularly stochastic simulations. Mega-base scale variation in DNA methylation patterns is observed in development, cancer and ageing and the mechanisms generating these patterns are little understood. However, the computational cost of stochastic simulations prevents them from modelling such large genomic regions. Here we test the utility of three different mean-field models to predict large-scale DNA methylation patterns. By comparison to stochastic simulations, we show that a cluster mean-field model accurately predicts the statistical properties of steady-state DNA methylation patterns, including the mean and variance of methylation levels calculated across a system of CpG sites, as well as the covariance and correlation of methylation levels between neighbouring sites. We also demonstrate that a cluster mean-field model can be used within an approximate Bayesian computation framework to accurately infer model parameters from data. As mean-field models can be solved numerically in a few seconds, our work demonstrates their utility for understanding the processes underpinning large-scale DNA methylation patterns.

systems biology↗

bioRxiv2021

Mean-field theory accurately captures the variation of copy number distributions across the mRNA's life cycle

We consider a stochastic model where a gene switches between two states, an mRNA transcript is released in the active state and subsequently it undergoes an arbitrary number of sequential unimolecular steps before being degraded. The reactions effectively describe various stages of the mRNA life cycle such as initiation, elongation, termination, splicing, export and degradation. We construct a novel mean-field approach that leads to closed-form steady-state distributions for the number of transcript molecules at each stage of the mRNA life cycle. By comparison with stochastic simulations, we show that the approximation is highly accurate over all of parameter space, independent of the type of expression (constitutive or bursty) and of the shape of the distribution (unimodal, bimodal and nearly bimodal). The theory predicts that in a population of identical cells, any bimodality is gradually washed away as the mRNA progresses through its life cycle.

systems biology↗

bioRxiv2021

Characterizing non-exponential growth and bimodal cell size distributions in Schizosaccharomyces pombe: an analytical approach

Unlike many single-celled organisms, the growth of fission yeast cells within a cell cycle is not exponential. It is rather characterized by three distinct phases (elongation, septation and fission), each with a different growth rate. Experiments also show that the distribution of cell size in a lineage is often bimodal, unlike the unimodal distributions measured for the bacterium Escherichia coli. Here we construct a detailed stochastic model of cell size dynamics in fission yeast. The theory leads to analytic expressions for the cell size and the birth size distributions, and explains the origin of bimodality seen in experiments. In particular our theory shows that the left peak in the bimodal distribution is associated with cells in the elongation phase while the right peak is due to cells in the septation and fission phases. We show that the size control strategy, the variability in the added size during a cell cycle and the fraction of time spent in each of the three cell growth phases have a strong bearing on the shape of the cell size distribution. Furthermore we infer all the parameters of our model by matching the theoretical cell size and birth size distributions to those from experimental single cell time-course data for seven different growth conditions. Our method provides a much more accurate means of determining the cell size control strategy (timer, adder or sizer) than the standard method based on the slope of the best linear fit between the birth and division sizes. We also show that the variability in added size and the strength of cell size control of fission yeast depend weakly on the temperature but strongly on the culture medium. Author summaryAdvances in microscopy enable us to follow single cells over long timescales from which we can understand how their size varies with time and the nature of innate strategies developed to control cell size. This data shows that in many cell types growth is exponential and the distribution of cell sizes has one peak, namely there is a single characteristic cell size. However data for fission yeast shows remarkable differences: growth is non-exponential and the distribution of cell sizes has two peaks, meaning two characteristic cell sizes exist. Here we construct the first mathematical model of this organism; by solving the model analytically we show that it is able to predict the two peaked distributions of cell size seen in data and provides an explanation for each peak in terms of the various growth phases of the single-celled organism. Furthermore by fitting the model to the data, we infer values for the rates of all microscopic processes in our model. This method is shown to provide a much more reliable inference than current methods and sheds light on how the strategy used by fission yeast cells to control their size varies with external conditions.

systems biology↗

bioRxiv2021

Distinguishing between models of mammalian gene expression: telegraph-like models versus mechanistic models

Two-state models (telegraph-like models) have a successful history of predicting distributions of cellular and nascent mRNA numbers that can well fit experimental data. These models exclude key rate limiting steps, and hence it is unclear why they are able to accurately predict the number distributions. To answer this question, here we compare these models to a novel stochastic mechanistic model of transcription in mammalian cells that presents a unified description of transcriptional factor, polymerase and mature mRNA dynamics. We show that there is a large region of parameter space where the first, second and third moments of the distributions of the waiting times between two consecutively produced transcripts (nascent or mature) of two-state and mechanistic models exactly match. In this region, (i) one can uniquely express the two-state model parameters in terms of those of the mechanistic model, (ii) the models are practically indistinguishable by comparison of their transcript numbers distributions, and (iii) they are distinguishable from the shape of their waiting time distributions. Our results clarify the relationship between different gene expression models and identify a means to select between them from experimental data.

systems biology↗

bioRxiv2020

Cell size distribution of lineage data: analytic results and parameter inference

Recent advances in single-cell technologies have enabled time-resolved measurements of the cell size over several cell cycles. This data encodes information on how cells correct size aberrations so that they do not grow abnormally large or small. Here we formulate a piecewise deterministic Markov model describing the evolution of the cell size over many generations, for all three cell size homeostasis strategies (timer, sizer, and adder). The model is solved to obtain an analytical expression for the non-Gaussian cell size distribution in a cell lineage; the theory is used to understand how the shape of the distribution is influenced by the parameters controlling the dynamics of the cell cycle and by the choice of cell tracking protocol. The theoretical cell size distribution is found to provide an excellent match to the experimental cell size distribution of E. coli lineage data collected under various growth conditions.

systems biology↗

bioRxiv2020

Neural network aided approximation and parameterinference of stochastic models of gene expression

Non-Markov models of stochastic biochemical kinetics often incorporate explicit time delays to effectively model large numbers of intermediate biochemical processes. Analysis and simulation of these models, as well as the inference of their parameters from data, are fraught with difficulties because the dynamics depends on the systems history. Here we use an artificial neural network to approximate the time-dependent distributions of non-Markov models by the solutions of much simpler time-inhomogeneous Markov models; the approximation does not increase the dimensionality of the model and simultaneously leads to inference of the kinetic parameters. The training of the neural network uses a relatively small set of noisy measurements generated by experimental data or stochastic simulations of the non-Markov model. We show using a variety of models, where the delays stem from transcriptional processes and feedback control, that the Markov models learnt by the neural network accurately reflect the stochastic dynamics across parameter space.

systems biology↗

bioRxiv2020

Frequency domain analysis of fluctuations of mRNA and protein copy numbers within a cell lineage: theory and experimental validation

The stochasticity of gene expression is manifested in the fluctuations of mRNA and protein copy numbers within a cell lineage over time. While data of this type can be obtained for many generations, most mathematical models are unsuitable to interpret such data since they assume non-growing cells. Here we develop a theoretical approach that quantitatively links the frequency content of lineage data to subcellular dynamics. We elucidate how the position, height, and width of the peaks in the power spectrum provide a distinctive fingerprint that encodes a wealth of information about mechanisms controlling transcription, translation, replication, degradation, bursting, promoter switching, cell cycle duration, cell division, and gene dosage compensation. Predictions are confirmed by analysis of single-cell Escherichia coli data obtained using fluorescence microscopy. Furthermore, by matching the experimental and theoretical power spectra, we infer the temperature-dependent gene expression parameters, without the need of measurements relating fluorescence intensities to molecule numbers.

systems biology↗

bioRxiv2020

Stochastic time-dependent enzyme kinetics: closed-form solution and transient bimodality

We derive an approximate closed-form solution to the chemical master equation describing the Michaelis-Menten reaction mechanism of enzyme action. In particular, assuming that the probability of a complex dissociating into enzyme and substrate is significantly larger than the probability of a product formation event, we obtain expressions for the time-dependent marginal probability distributions of the number of substrate and enzyme molecules. For delta function initial conditions, we show that the substrate distribution is either unimodal at all times or else becomes bimodal at intermediate times. This transient bimodality, which has no deterministic counterpart, manifests when the initial number of substrate molecules is much larger than the total number of enzyme molecules and if the frequency of enzyme-substrate binding events is large enough. Furthermore, we show that our closed-form solution is different from the solution of the chemical master equation reduced by means of the widely used discrete stochastic Michaelis-Menten approximation, where the propensity for substrate decay has a hyperbolic dependence on the number of substrate molecules. The differences arise because the latter does not take into account enzyme number fluctuations while our approach includes them. We confirm by means of stochastic simulation of all the elementary reaction steps in the Michaelis-Menten mechanism that our closed-form solution is accurate over a larger region of parameter space than that obtained using the discrete stochastic Michaelis-Menten approximation.

systems biology↗

bioRxiv2020

Statistics of nascent and mature RNA fluctuations in a stochastic model of transcriptional initiation, elongation, pausing, and termination

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. By computational binning of the gene into smaller segments, we derive exact closed-form expressions for the mean and variance of nascent RNA fluctuations in each of these segments, as well as for the total nascent RNA on a gene. We also derive 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.

systems biology↗