Imperial College London

DrPhilippThomas

Faculty of Natural SciencesDepartment of Mathematics

Senior Lecturer in Biomathematics
 
 
 
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Contact

 

+44 (0)20 7594 2647p.thomas

 
 
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Location

 

626Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Publication Type
Year
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31 results found

Thomas P, Shahrezaei V, 2021, Coordination of gene expression noise with cell size: analytical results for agent-based models of growing cell populations, Journal of the Royal Society Interface, Vol: 18, Pages: 1-16, ISSN: 1742-5662

The chemical master equation and the Gillespie algorithm are widely used to model the reaction kinetics inside living cells. It is thereby assumed that cell growth and division can be modelled through effective dilution reactions and extrinsic noise sources. We here re-examine these paradigms through developing an analytical agent-based framework of growing and dividing cells accompanied by an exact simulation algorithm, which allows us to quantify the dynamics of virtually any intracellular reaction network affected by stochastic cell size control and division noise. We find that the solution of the chemical master equation—including static extrinsic noise—exactly agrees with the agent-based formulation when the network under study exhibits stochastic concentration homeostasis, a novel condition that generalizes concentration homeostasis in deterministic systems to higher order moments and distributions. We illustrate stochastic concentration homeostasis for a range of common gene expression networks. When this condition is not met, we demonstrate by extending the linear noise approximation to agent-based models that the dependence of gene expression noise on cell size can qualitatively deviate from the chemical master equation. Surprisingly, the total noise of the agent-based approach can still be well approximated by extrinsic noise models.

Journal article

Kuntz Nussio J, Thomas P, Stan G, Barahona Met al., 2021, Approximations of countably-infinite linear programs over bounded measure spaces, SIAM Journal on Optimization, Vol: 31, Pages: 604-625, ISSN: 1052-6234

We study a class of countably-infinite-dimensional linear programs (CILPs)whose feasible sets are bounded subsets of appropriately defined spaces ofmeasures. The optimal value, optimal points, and minimal points of these CILPscan be approximated by solving finite-dimensional linear programs. We show howto construct finite-dimensional programs that lead to approximations witheasy-to-evaluate error bounds, and we prove that the errors converge to zero asthe size of the finite-dimensional programs approaches that of the originalproblem. We discuss the use of our methods in the computation of the stationarydistributions, occupation measures, and exit distributions of Markov~chains.

Journal article

Kuntz J, Thomas P, Stan G-B, Barahona Met al., 2021, Stationary distributions of continuous-time Markov chains: a review of theory and truncation-based approximations, SIAM Review, ISSN: 0036-1445

Computing the stationary distributions of a continuous-time Markov chaininvolves solving a set of linear equations. In most cases of interest, thenumber of equations is infinite or too large, and cannot be solved analyticallyor numerically. Several approximation schemes overcome this issue by truncatingthe state space to a manageable size. In this review, we first give acomprehensive theoretical account of the stationary distributions and theirrelation to the long-term behaviour of the Markov chain, which is readilyaccessible to non-experts and free of irreducibility assumptions made instandard texts. We then review truncation-based approximation schemes payingparticular attention to their convergence and to the errors they introduce, andwe illustrate their performance with an example of a stochastic reactionnetwork of relevance in biology and chemistry. We conclude by elaborating oncomputational trade-offs associated with error control and some open questions.

Journal article

Tonn M, Thomas P, Barahona M, Oyarzun Det al., 2020, Computation of single-cell metabolite distributions using mixture models, Frontiers in Cell and Developmental Biology, Vol: 8, Pages: 1-11, ISSN: 2296-634X

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.

Journal article

Thomas P, 2020, Stochastic Modeling Approaches for Single-Cell Analyses, Systems Medicine: Integrative, Qualitative and Computational Approaches, Editors: Wolkenhauer, Publisher: Elsevier, Oxford, Pages: 45-55

Single-cell analyses are becoming increasingly important in cell biology and personalized approaches to medicine. Such analyses frequently reveal heterogeneity that exists within and between cells. We give a concise overview of stochastic methods used to analyze non-genetic heterogeneity in models of cell populations and examine several analytical results on the determinants of gene expression noise. We then review models that advanced our understanding of stochastic phenomena in cellular decision making, stem cell differentiation, tissue homoeostasis and cell cycle dynamics.

Book chapter

Tang W, Bertaux F, Thomas P, Stefanelli C, Saint M, Marguerat S, Shahrezaei Vet al., 2020, bayNorm: Bayesian gene expression recovery, imputation and normalisation for single cell RNA-sequencing data, Bioinformatics, Vol: 36, Pages: 1174-1181, ISSN: 1367-4803

Motivation:Normalisation of single cell RNA sequencing (scRNA-seq) data is a prerequisite to theirinterpretation. The marked technical variability, high amounts of missing observations and batch effecttypical of scRNA-seq datasets make this task particularly challenging. There is a need for an efficient andunified approach for normalisation, imputation and batch effect correction.Results:Here, we introduce bayNorm, a novel Bayesian approach for scaling and inference of scRNA-seq counts. The method’s likelihood function follows a binomial model of mRNA capture, while priorsare estimated from expression values across cells using an empirical Bayes approach. We first validateour assumptions by showing this model can reproduce different statistics observed in real scRNA-seqdata. We demonstrate using publicly-available scRNA-seq datasets and simulated expression data thatbayNorm allows robust imputation of missing values generating realistic transcript distributions that matchsingle molecule FISH measurements. Moreover, by using priors informed by dataset structures, bayNormimproves accuracy and sensitivity of differential expression analysis and reduces batch effect comparedto other existing methods. Altogether, bayNorm provides an efficient, integrated solution for global scalingnormalisation, imputation and true count recovery of gene expression measurements from scRNA-seqdata.Availability:The R package “bayNorm” is available at https://github.com/WT215/bayNorm. The code foranalysing data in this paper is available at https://github.com/WT215/bayNorm_papercode.Contact:samuel.marguerat@imperial.ac.uk or v.shahrezaei@imperial.ac.ukSupplementary information:Supplementary data are available atBioinformaticsonline.

Journal article

Kuntz J, Thomas P, Stan G-B, Barahona Met al., 2019, Bounding the stationary distributions of the chemical master equation via mathematical programming, The Journal of Chemical Physics, Vol: 151, Pages: 034109-034109, ISSN: 0021-9606

Journal article

Kuntz J, Thomas P, Stan G-B, Barahona Met al., 2019, The exit time finite state projection scheme: bounding exit distributions and occupation measures of continuous-time Markov chains, SIAM Journal on Scientific Computing, Vol: 41, Pages: A748-A769, ISSN: 1064-8275

We introduce the exit time finite state projection (ETFSP) scheme, a truncation- based method that yields approximations to the exit distribution and occupation measure associated with the time of exit from a domain (i.e., the time of first passage to the complement of the domain) of time-homogeneous continuous-time Markov chains. We prove that: (i) the computed approximations bound the measures from below; (ii) the total variation distances between the approximations and the measures decrease monotonically as states are added to the truncation; and (iii) the scheme converges, in the sense that, as the truncation tends to the entire state space, the total variation distances tend to zero. Furthermore, we give a computable bound on the total variation distance between the exit distribution and its approximation, and we delineate the cases in which the bound is sharp. We also revisit the related finite state projection scheme and give a comprehensive account of its theoretical properties. We demonstrate the use of the ETFSP scheme by applying it to two biological examples: the computation of the first passage time associated with the expression of a gene, and the fixation times of competing species subject to demographic noise.

Journal article

Tonn M, Thomas P, Barahona M, Oyarzun Det al., 2019, Stochastic modelling reveals mechanisms of metabolic heterogeneity, Communications Biology, Vol: 2, ISSN: 2399-3642

Phenotypic variation is a hallmark of cellular physiology. Metabolic heterogeneity, in particular, underpins single-cell phenomena such as microbial drug tolerance and growth variability. Much research has focussed on transcriptomic and proteomic heterogeneity, yet it remains unclear if such variation permeates to the metabolic state of a cell. Here we propose a stochastic model to show that complex forms of metabolic heterogeneity emerge from fluctuations in enzyme expression and catalysis. The analysis predicts clonal populations to split into two or more metabolically distinct subpopulations. We reveal mechanisms not seen in deterministic models, in which enzymes with unimodal expression distributions lead to metabolites with a bimodal or multimodal distribution across the population. Based on published data, the results suggest that metabolite heterogeneity may be more pervasive than previously thought. Our work casts light on links between gene expression and metabolism, and provides a theory to probe the sources of metabolite heterogeneity.

Journal article

Thomas P, 2019, Intrinsic and extrinsic noise of gene expression in lineage trees, Scientific Reports, Vol: 9, ISSN: 2045-2322

Cell-to-cell heterogeneity is driven by stochasticity in intracellular reactions and the population dynamics. While these sources are usually studied separately, we develop an agent-based framework that accounts for both factors while tracking every single cell of a growing population. Apart from the common intrinsic variability, the framework also predicts extrinsic noise without the need to introduce fluctuating rate constants. Instead, extrinsic fluctuations are explained by cell cycle fluctuations and differences in cell age. We provide explicit formulas to quantify mean molecule numbers, intrinsic and extrinsic noise statistics in two-colour experiments. We find that these statistics differ significantly depending on the experimental setup used to observe the cells. We illustrate this fact using (i) averages over an isolated cell lineage tracked over many generations as observed in the mother machine, (ii) population snapshots with known cell ages as recorded in time-lapse microscopy, and (iii) snapshots with unknown cell ages as measured from static images or flow cytometry. Applying the method to models of stochastic gene expression and feedback regulation elucidates that isolated lineages, as compared to snapshot data, can significantly overestimate the mean number of molecules, overestimate extrinsic noise but underestimate intrinsic noise and have qualitatively different sensitivities to cell cycle fluctuations.

Journal article

Martins B, Tooke AK, Thomas P, Locke JCWet al., 2018, Cell size control driven by the circadian clock and environment in cyanobacteria, Proceedings of the National Academy of Sciences, Vol: 115, Pages: E11415-E11424, ISSN: 0027-8424

How cells maintain their size has been extensively studied under constant conditions. In the wild, however, cells rarely experience constant environments. Here, we examine how the 24-hour circadian clock and environmental cycles modulate cell size control and division timings in the cyanobacterium Synechococcus elongatus using single-cell time-lapse microscopy. Under constant light, wild type cells follow an apparent sizer-like principle. Closer inspection reveals that the clock generates two subpopulations, with cells born in the subjective day following different division rules from cells born in subjective night. A stochastic model explains how this behaviour emerges from the interaction of cell size control with the clock. We demonstrate that the clock continuously modulates the probability of cell division throughout day and night, rather than solely applying an on-off gate to division as previously proposed. Iterating between modelling and experiments, we go on to identify an effective coupling of the division rate to time of day through the combined effects of the environment and the clock on cell division. Under naturally graded light-dark cycles, this coupling narrows the time window of cell divisions and shifts divisions away from when light levels are low and cell growth is reduced. Our analysis allows us to disentangle, and predict the effects of, the complex interactions between the environment, clock, and cell size control.

Journal article

Thomas P, Terradot G, Danos V, Weisse Aet al., 2018, Sources, propagation and consequences of stochasticity in cellular growth, Nature Communications, Vol: 9, ISSN: 2041-1723

Growth impacts a range of phenotypic responses. Identifying the sources of growth variation and their propagation across the cellular machinery can thus unravel mechanisms that underpin cell decisions. We present a stochastic cell model linking gene expression, metabolism and replication to predict growth dynamics in single bacterial cells. Alongside we provide a theory to analyse stochastic chemical reactions coupled with cell divisions, enabling efficient parameter estimation, sensitivity analysis and hypothesis testing. The cell model recovers population-averaged data on growth-dependence of bacterial physiology and how growth variations in single cells change across conditions. We identify processes responsible for this variation and reconstruct the propagation of initial fluctuations to growth and other processes. Finally, we study drug-nutrient interactions and find that antibiotics can both enhance and suppress growth heterogeneity. Our results provide a predictive framework to integrate heterogeneous data and draw testable predictions with implications for antibiotic tolerance, evolutionary and synthetic biology.

Journal article

Voliotis M, Thomas P, Bowsher CG, Grima Ret al., 2018, The Extra Reaction Algorithm for Stochastic Simulation of Biochemical Reaction Systems in Fluctuating Environments, Quantitative Biology: Theory, Computational Methods, and Models, Editors: Munsky, Hlavacek, Tsimring

Book chapter

Thomas P, 2018, Analysis of cell size homeostasis at the single-cell and population level, Frontiers in Physics, Vol: 6, ISSN: 2296-424X

Growth pervades all areas of life from single cells to cell populations to tissues. Cell size often fluctuates significantly from cell to cell and from generation to generation. Here we present a unified framework to predict the statistics of cell size variations within a lineage tree of a proliferating population. We analytically characterize (i) the distributions of cell size snapshots, (ii) the distribution within a population tree, and (iii) the distribution of lineages across the tree. Surprisingly, these size distributions differ significantly from observing single cells in isolation. In populations, cells seemingly grow to different sizes, typically exhibit less cell-to-cell variability and often display qualitatively different sensitivities to cell cycle noise and division errors. We demonstrate the key findings using recent single-cell data and elaborate on the implications for the ability of cells to maintain a narrow size distribution and the emergence of different power laws in these distributions.

Journal article

Thomas P, 2017, Making sense of snapshot data: ergodic principle for clonal cell populations, Journal of the Royal Society Interface, Vol: 14, ISSN: 1742-5662

Population growth is often ignored when quantifying gene expression levels across clonal cell populations. We develop a framework for obtaining the molecule number distributions in an exponentially growing cell population taking into account its age structure. In the presence of generation time variability, the average acquired across a population snapshot does not obey the average of a dividing cell over time, apparently contradicting ergodicity between single cells and the population. Instead, we show that the variation observed across snapshots with known cell age is captured by cell histories, a single-cell measure obtained from tracking an arbitrary cell of the population back to the ancestor from which it originated. The correspondence between cells of known age in a population with their histories represents an ergodic principle that provides a new interpretation of population snapshot data. We illustrate the principle using analytical solutions of stochastic gene expression models in cell populations with arbitrary generation time distributions. We further elucidate that the principle breaks down for biochemical reactions that are under selection, such as the expression of genes conveying antibiotic resistance, which gives rise to an experimental criterion with which to probe selection on gene expression fluctuations.

Journal article

Shahrezaei V, Robertson B, Thomas P, Priestman Met al., 2017, Mycobacteria modify their cell size control under sub-optimal carbon sources, Frontiers in Cell and Developmental Biology, Vol: 5, ISSN: 2296-634X

The decision to divide is the most important one that any cell must make. Recent single cell studies suggest that most bacteria follow an “adder” model of cell size control, incorporating a fixed amount of cell wall material before dividing. Mycobacteria, including the causative agent of tuberculosis Mycobacterium tuberculosis, are known to divide asymmetrically resulting in heterogeneity in growth rate, doubling time, and other growth characteristics in daughter cells. The interplay between asymmetric cell division and adder size control has not been extensively investigated. Moreover, the impact of changes in the environment on growth rate and cell size control have not been addressed for mycobacteria. Here, we utilize time-lapse microscopy coupled with microfluidics to track live Mycobacterium smegmatis cells as they grow and divide over multiple generations, under a variety of growth conditions. We demonstrate that, under optimal conditions, M. smegmatis cells robustly follow the adder principle, with constant added length per generation independent of birth size, growth rate, and inherited pole age. However, the nature of the carbon source induces deviations from the adder model in a manner that is dependent on pole age. Understanding how mycobacteria maintain cell size homoeostasis may provide crucial targets for the development of drugs for the treatment of tuberculosis, which remains a leading cause of global mortality.

Journal article

Andreychenko A, Bortolussi L, Grima R, Thomas P, Wolf Vet al., 2017, Distribution approximations for the chemical master equation: comparisonof the method of moments and the system size expansion, Modeling Cellular Systems, Editors: Graw, Matthaus, Pahle, Publisher: Springer, Pages: 39-39, ISBN: 978-3-319-45833-5

The stochastic nature of chemical reactions involving randomly fluctuatingpopulation sizes has lead to a growing research interest in discrete-statestochastic models and their analysis. A widely-used approach is the descriptionof the temporal evolution of the system in terms of a chemical master equation(CME). In this paper we study two approaches for approximating the underlyingprobability distributions of the CME. The first approach is based on anintegration of the statistical moments and the reconstruction of thedistribution based on the maximum entropy principle. The second approach relieson an analytical approximation of the probability distribution of the CME usingthe system size expansion, considering higher-order terms than the linear noiseapproximation. We consider gene expression networks with unimodal andmultimodal protein distributions to compare the accuracy of the two approaches.We find that both methods provide accurate approximations to the distributionsof the CME while having different benefits and limitations in applications.

Book chapter

Fröhlich F, Thomas P, Kazeroonian A, Theis FJ, Grima R, Hasenauer Jet al., 2016, Inference for Stochastic Chemical Kinetics Using Moment Equations and System Size Expansion, PLOS Computational Biology, Vol: 12, ISSN: 1553-734X

Quantitative mechanistic models are valuable tools for disentangling biochemical pathways and for achieving a comprehensive understanding of biological systems. However, to be quantitative the parameters of these models have to be estimated from experimental data. In the presence of significant stochastic fluctuations this is a challenging task as stochastic simulations are usually too time-consuming and a macroscopic description using reaction rate equations (RREs) is no longer accurate. In this manuscript, we therefore consider moment-closure approximation (MA) and the system size expansion (SSE), which approximate the statistical moments of stochastic processes and tend to be more precise than macroscopic descriptions. We introduce gradient-based parameter optimization methods and uncertainty analysis methods for MA and SSE. Efficiency and reliability of the methods are assessed using simulation examples as well as by an application to data for Epo-induced JAK/STAT signaling. The application revealed that even if merely population-average data are available, MA and SSE improve parameter identifiability in comparison to RRE. Furthermore, the simulation examples revealed that the resulting estimates are more reliable for an intermediate volume regime. In this regime the estimation error is reduced and we propose methods to determine the regime boundaries. These results illustrate that inference using MA and SSE is feasible and possesses a high sensitivity.

Journal article

Voliotis M, Thomas P, Grima R, Bowsher CGet al., 2016, Stochastic simulation of biomolecular networks in dynamic environments, PLOS Computational Biology, Vol: 12, ISSN: 1553-734X

Simulation of biomolecular networks is now indispensable for studying biological systems, from small reaction networks to large ensembles of cells. Here we present a novel approach for stochastic simulation of networks embedded in the dynamic environment of the cell and its surroundings. We thus sample trajectories of the stochastic process described by the chemical master equation with time-varying propensities. A comparative analysis shows that existing approaches can either fail dramatically, or else can impose impractical computational burdens due to numerical integration of reaction propensities, especially when cell ensembles are studied. Here we introduce the Extrande method which, given a simulated time course of dynamic network inputs, provides a conditionally exact and several orders-of-magnitude faster simulation solution. The new approach makes it feasible to demonstrate-using decision-making by a large population of quorum sensing bacteria-that robustness to fluctuations from upstream signaling places strong constraints on the design of networks determining cell fate. Our approach has the potential to significantly advance both understanding of molecular systems biology and design of synthetic circuits.

Journal article

Thomas P, Grima R, 2015, Approximate probability distributions of the master equation, Physical Review E, Vol: 92, Pages: 012120-012120-12, ISSN: 1539-3755

Master equations are common descriptions of mesoscopic systems. Analytical solutions to these equations can rarely be obtained. We here derive an analytical approximation of the time-dependent probability distribution of the master equation using orthogonal polynomials. The solution is given in two alternative formulations: a series with continuous and a series with discrete support, both of which can be systematically truncated. While both approximations satisfy the system size expansion of the master equation, the continuous distribution approximations become increasingly negative and tend to oscillations with increasing truncation order. In contrast, the discrete approximations rapidly converge to the underlying non-Gaussian distributions. The theory is shown to lead to particularly simple analytical expressions for the probability distributions of molecule numbers in metabolic reactions and gene expression systems.

Journal article

Thomas P, Fleck C, Grima R, Popovic Net al., 2014, System size expansion using Feynman rules and diagrams, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 47, ISSN: 1751-8113

Few analytical methods exist for quantitative studies of large fluctuations in stochastic systems. In this article, we develop a simple diagrammatic approach to the chemical master equation that allows us to calculate multi-time correlation functions which are accurate to any desired order in van Kampenʼs system size expansion. Specifically, we present a set of Feynman rules from which this diagrammatic perturbation expansion can be constructed algorithmically. We then apply the methodology to derive in closed form the leading order corrections to the linear noise approximation of the intrinsic noise power spectrum for general biochemical reaction networks. Finally, we illustrate our results by describing noise-induced oscillations in the Brusselator reaction scheme which are not captured by the common linear noise approximation.

Journal article

Thomas P, Popovic N, Grima R, 2014, Phenotypic switching in gene regulatory networks, Proceedings of the National Academy of Sciences of the United States of America, Vol: 111, Pages: 6994-6999, ISSN: 0027-8424

Noise in gene expression can lead to reversible phenotypic switching. Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. Here, we devise a methodology which allows us to quantify multimodal gene expression distributions and single-cell power spectra in gene regulatory networks. Extending the commonly used linear noise approximation, we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components in a wide class of networks. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. We demonstrate the applicability of our approach in a number of genetic networks, uncovering previously unidentified dynamical characteristics associated with phenotypic switching. Specifically, we elucidate how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks. We demonstrate how phenotypic switching leads to birhythmical expression in a genetic oscillator, and to hysteresis in phenotypic induction, thus highlighting the ability of regulatory networks to retain memory.

Journal article

Thomas P, Straube AV, Timmer J, Fleck C, Grima Ret al., 2013, Signatures of nonlinearity in single cell noise-induced oscillations, JOURNAL OF THEORETICAL BIOLOGY, Vol: 335, Pages: 222-234, ISSN: 0022-5193

Journal article

Thomas P, Matuschek H, Grima R, 2013, How reliable is the linear noise approximation of gene regulatory networks?, BMC Genomics, Vol: 14, ISSN: 1471-2164

BackgroundThe linear noise approximation (LNA) is commonly used to predict how noise is regulated and exploited at the cellular level. These predictions are exact for reaction networks composed exclusively of first order reactions or for networks involving bimolecular reactions and large numbers of molecules. It is however well known that gene regulation involves bimolecular interactions with molecule numbers as small as a single copy of a particular gene. It is therefore questionable how reliable are the LNA predictions for these systems.ResultsWe implement in the software package intrinsic Noise Analyzer (iNA), a system size expansion based method which calculates the mean concentrations and the variances of the fluctuations to an order of accuracy higher than the LNA. We then use iNA to explore the parametric dependence of the Fano factors and of the coefficients of variation of the mRNA and protein fluctuations in models of genetic networks involving nonlinear protein degradation, post-transcriptional, post-translational and negative feedback regulation. We find that the LNA can significantly underestimate the amplitude and period of noise-induced oscillations in genetic oscillators. We also identify cases where the LNA predicts that noise levels can be optimized by tuning a bimolecular rate constant whereas our method shows that no such regulation is possible. All our results are confirmed by stochastic simulations.ConclusionThe software iNA allows the investigation of parameter regimes where the LNA fares well and where it does not. We have shown that the parametric dependence of the coefficients of variation and Fano factors for common gene regulatory networks is better described by including terms of higher order than LNA in the system size expansion. This analysis is considerably faster than stochastic simulations due to the extensive ensemble averaging needed to obtain statistically meaningful results. Hence iNA is well suited for performing computationall

Journal article

Thomas P, Grima R, Straube AV, 2012, Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators, PHYSICAL REVIEW E, Vol: 86, ISSN: 1539-3755

Journal article

Thomas P, Matuschek H, Grima R, 2012, Intrinsic Noise Analyzer: A Software Package for the Exploration of Stochastic Biochemical Kinetics Using the System Size Expansion, PLOS One, Vol: 7, ISSN: 1932-6203

The accepted stochastic descriptions of biochemical dynamics under well-mixed conditions are given by the Chemical Master Equation and the Stochastic Simulation Algorithm, which are equivalent. The latter is a Monte-Carlo method, which, despite enjoying broad availability in a large number of existing software packages, is computationally expensive due to the huge amounts of ensemble averaging required for obtaining accurate statistical information. The former is a set of coupled differential-difference equations for the probability of the system being in any one of the possible mesoscopic states; these equations are typically computationally intractable because of the inherently large state space. Here we introduce the software package intrinsic Noise Analyzer (iNA), which allows for systematic analysis of stochastic biochemical kinetics by means of van Kampen's system size expansion of the Chemical Master Equation. iNA is platform independent and supports the popular SBML format natively. The present implementation is the first to adopt a complementary approach that combines state-of-the-art analysis tools using the computer algebra system Ginac with traditional methods of stochastic simulation. iNA integrates two approximation methods based on the system size expansion, the Linear Noise Approximation and effective mesoscopic rate equations, which to-date have not been available to non-expert users, into an easy-to-use graphical user interface. In particular, the present methods allow for quick approximate analysis of time-dependent mean concentrations, variances, covariances and correlations coefficients, which typically outperforms stochastic simulations. These analytical tools are complemented by automated multi-core stochastic simulations with direct statistical evaluation and visualization. We showcase iNA's performance by using it to explore the stochastic properties of cooperative and non-cooperative enzyme kinetics and a gene network associated with circad

Journal article

Thomas P, Straube AV, Grima R, 2012, The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions, BMC Systems Biology, Vol: 6, ISSN: 1752-0509

BACKGROUND: It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions. RESULTS: We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations. CONCLUSIONS: A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a sim

Journal article

Thomas P, Matuschek H, Grima R, 2012, Computation of biochemical pathway fluctuations beyond the linear noise approximation using iNA, Pages: 1-5

Conference paper

Thomas P, Straube AV, Grima R, 2011, Communication: Limitations of the stochastic quasi-steady-state approximation in open biochemical reaction networks, Journal of Chemical Physics, Vol: 135, ISSN: 1089-7690

It is commonly believed that, whenever timescale separation holds, the predictions of reduced chemical master equations obtained using the stochastic quasi-steady-state approximation are in very good agreement with the predictions of the full master equations. We use the linear noise approximation to obtain a simple formula for the relative error between the predictions of the two master equations for the Michaelis-Menten reaction with substrate input. The reduced approach is predicted to overestimate the variance of the substrate concentration fluctuations by as much as 30%. The theoretical results are validated by stochastic simulations using experimental parameter values for enzymes involved in proteolysis, gluconeogenesis, and fermentation.

Journal article

Grima R, Thomas P, Straube AV, 2011, How accurate are the nonlinear chemical Fokker-Planck and chemical Langevin equations?, Journal of Chemical Physics, Vol: 135, ISSN: 1089-7690

The chemical Fokker-Planck equation and the corresponding chemical Langevin equation are commonly used approximations of the chemical master equation. These equations are derived from an uncontrolled, second-order truncation of the Kramers-Moyal expansion of the chemical master equation and hence their accuracy remains to be clarified. We use the system-size expansion to show that chemical Fokker-Planck estimates of the mean concentrations and of the variance of the concentration fluctuations about the mean are accurate to order Ω(-3∕2) for reaction systems which do not obey detailed balance and at least accurate to order Ω(-2) for systems obeying detailed balance, where Ω is the characteristic size of the system. Hence, the chemical Fokker-Planck equation turns out to be more accurate than the linear-noise approximation of the chemical master equation (the linear Fokker-Planck equation) which leads to mean concentration estimates accurate to order Ω(-1∕2) and variance estimates accurate to order Ω(-3∕2). This higher accuracy is particularly conspicuous for chemical systems realized in small volumes such as biochemical reactions inside cells. A formula is also obtained for the approximate size of the relative errors in the concentration and variance predictions of the chemical Fokker-Planck equation, where the relative error is defined as the difference between the predictions of the chemical Fokker-Planck equation and the master equation divided by the prediction of the master equation. For dimerization and enzyme-catalyzed reactions, the errors are typically less than few percent even when the steady-state is characterized by merely few tens of molecules.

Journal article

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