Michał Komorowski

Sensitivity, robustness, and identifiability in stochastic chemical kinetics models

We present a novel and simple method to numerically calculate Fisher information matrices for stochastic chemical kinetics models. The linear noise approximation is used to derive model equations and a likelihood function that leads to an efficient computational algorithm. Our approach reduces the problem of calculating the Fisher information matrix to solving a set of ordinary differential equations. This is the first method to compute Fisher information for stochastic chemical kinetics models without the need for Monte Carlo simulations. This methodology is then used to study sensitivity, robustness, and parameter identifiability in stochastic chemical kinetics models. We show that significant differences exist between stochastic and deterministic models as well as between stochastic models with time-series and time-point measurements. We demonstrate that these discrepancies arise from the variability in molecule numbers, correlations between species, and temporal correlations and show how this approach can be used in the analysis and design of experiments probing stochastic processes at the cellular level. The algorithm has been implemented as a Matlab package and is available from the authors upon request.

Maximizing the Information Content of Experiments in Systems Biology

Our understanding of most biological systems is in its infancy. Learning their structure and intricacies is fraught with challenges, and often side-stepped in favour of studying the function of different gene products in isolation from their physiological context. Constructing and inferring global mathematical models from experimental data is, however, central to systems biology. Different experimental setups provide different insights into such systems. Here we show how we can combine concepts from Bayesian inference and information theory in order to identify experiments that maximize the information content of the resulting data. This approach allows us to incorporate preliminary information; it is global and not constrained to some local neighbourhood in parameter space and it readily yields information on parameter robustness and confidence. Here we develop the theoretical framework and apply it to a range of exemplary problems that highlight how we can improve experimental investigations into the structure and dynamics of biological systems and their behavior.

Bayesian inference of biochemical kinetic parameters using the linear noise approximation

BackgroundFluorescent and luminescent gene reporters allow us to dynamically quantify changes in molecular species concentration over time on the single cell level. The mathematical modeling of their interaction through multivariate dynamical models requires the deveopment of effective statistical methods to calibrate such models against available data. Given the prevalence of stochasticity and noise in biochemical systems inference for stochastic models is of special interest. In this paper we present a simple and computationally efficient algorithm for the estimation of biochemical kinetic parameters from gene reporter data.ResultsWe use the linear noise approximation to model biochemical reactions through a stochastic dynamic model which essentially approximates a diffusion model by an ordinary differential equation model with an appropriately defined noise process. An explicit formula for the likelihood function can be derived allowing for computationally efficient parameter estimation. The proposed algorithm is embedded in a Bayesian framework and inference is performed using Markov chain Monte Carlo.ConclusionThe major advantage of the method is that in contrast to the more established diffusion approximation based methods the computationally costly methods of data augmentation are not necessary. Our approach also allows for unobserved variables and measurement error. The application of the method to both simulated and experimental data shows that the proposed methodology provides a useful alternative to diffusion approximation based methods.

Reconstruction of transcriptional dynamics from gene reporter data using differential equations

Motivation: Promoter-driven reporter genes, notably luciferase and green fluorescent protein, provide a tool for the generation of a vast array of time-course data sets from living cells and organisms. The aim of this study is to introduce a modeling framework based on stochastic differential equations (SDEs) and ordinary differential equations (ODEs) that addresses the problem of reconstructing transcription time-course profiles and associated degradation rates. The dynamical model is embedded into a Bayesian framework and inference is performed using Markov chain Monte Carlo algorithms. Results: We present three case studies where the methodology is used to reconstruct unobserved transcription profiles and to estimate associated degradation rates. We discuss advantages and limits of fitting either SDEs ODEs and address the problem of parameter identifiability when model variables are unobserved. We also suggest functional forms, such as on/off switches and stimulus response functions to model transcriptional dynamics and present results of fitting these to experimental data. Contact: b.f.finkenstadt@warwick.ac.uk Supplementary Information: Supplementary data are available at Bioinformatics online.

Translational Repression Contributes Greater Noise to Gene Expression than Transcriptional Repression

Stochastic effects in gene expression may result in different physiological states of individual cells, with consequences for pathogen survival and artificial gene network design. We studied the contributions of a regulatory factor to gene expression noise in four basic mechanisms of negative gene expression control: 1), transcriptional regulation by a protein repressor, 2), translational repression by a protein; 3), transcriptional repression by RNA; and 4), RNA interference with the translation. We investigated a general model of a two-gene network, using the chemical master equation and a moment generating function approach. We compared the expression noise of genes with the same effective transcription and translation initiation rates resulting from the action of different repressors, whereas previous studies compared the noise of genes with the same mean expression level but different initiation rates. Our results show that translational repression results in a higher noise than repression on the promoter level, and that this relationship does not depend on quantitative parameter values. We also show that regulation of protein degradation contributes more noise than regulated degradation of mRNA. These are unexpected results, because previous investigations suggested that translational regulation is more accurate. The relative magnitude of the noise introduced by protein and RNA repressors depends on the protein and mRNA degradation rates, and we derived expressions for the threshold below which the noise introduced by a protein repressor is higher than the noise introduced by an RNA repressor.

Nitrogen and Carbon Status Are Integrated at the Transcriptional Level by the Nitrogen Regulator NtrC In Vivo

ABSTRACT Nitrogen regulation in Escherichia coli is a model system for gene regulation in bacteria. Growth on glutamine as a sole nitrogen source is assumed to be nitrogen limiting, inferred from slow growth and strong NtrB/NtrC-dependent gene activation. However, we show that under these conditions, the intracellular glutamine concentration is not limiting but 5.6-fold higher than in ammonium-replete conditions; in addition, α-ketoglutarate concentrations are elevated. We address this glutamine paradox from a systems perspective. We show that the dominant role of NtrC is to regulate glnA transcription and its own expression, indicating that the glutamine paradox is not due to NtrC-independent gene regulation. The absolute intracellular NtrC and GS concentrations reveal molecular control parameters, where NtrC-specific activities were highest in nitrogen-starved cells, while under glutamine growth, NtrC showed intermediate specific activity. We propose an in vivo model in which α-ketoglutarate can derepress nitrogen regulation despite nitrogen sufficiency. IMPORTANCE Nitrogen is the most important nutrient for cell growth after carbon, and its metabolism is coordinated at the metabolic, transcriptional, and protein levels. We show that growth on glutamine as a sole nitrogen source, commonly assumed to be nitrogen limiting and used as such as a model system for nitrogen limitation, is in fact nitrogen replete. Our integrative quantitative analysis of key molecules involved in nitrogen assimilation and regulation reveal that glutamine is not necessarily the dominant molecule signaling nitrogen sufficiency and that α-ketoglutarate may play a more important role in signaling nitrogen status. NtrB/NtrC integrates α-ketoglutarate and glutamine signaling—sensed by the UTase (glnD) and PII (glnB), respectively—and regulates the nitrogen response through self-regulated expression and phosphorylation-dependent activation of the nitrogen (ntr) regulon. Our findings support α-ketoglutarate acting as a global regulatory metabolite. Nitrogen is the most important nutrient for cell growth after carbon, and its metabolism is coordinated at the metabolic, transcriptional, and protein levels. We show that growth on glutamine as a sole nitrogen source, commonly assumed to be nitrogen limiting and used as such as a model system for nitrogen limitation, is in fact nitrogen replete. Our integrative quantitative analysis of key molecules involved in nitrogen assimilation and regulation reveal that glutamine is not necessarily the dominant molecule signaling nitrogen sufficiency and that α-ketoglutarate may play a more important role in signaling nitrogen status. NtrB/NtrC integrates α-ketoglutarate and glutamine signaling—sensed by the UTase (glnD) and PII (glnB), respectively—and regulates the nitrogen response through self-regulated expression and phosphorylation-dependent activation of the nitrogen (ntr) regulon. Our findings support α-ketoglutarate acting as a global regulatory metabolite.

Decomposing Noise in Biochemical Signaling Systems Highlights the Role of Protein Degradation

Stochasticity is an essential aspect of biochemical processes at the cellular level. We now know that living cells take advantage of stochasticity in some cases and counteract stochastic effects in others. Here we propose a method that allows us to calculate contributions of individual reactions to the total variability of a systems output. We demonstrate that reactions differ significantly in their relative impact on the total noise and we illustrate the importance of protein degradation on the overall variability for a range of molecular processes and signaling systems. With our flexible and generally applicable noise decomposition method, we are able to shed new, to our knowledge, light on the sources and propagation of noise in biochemical reaction networks; in particular, we are able to show how regulated protein degradation can be employed to reduce the noise in biochemical systems.

Using a Single Fluorescent Reporter Gene to Infer Half-Life of Extrinsic Noise and Other Parameters of Gene Expression

Fluorescent and luminescent proteins are often used as reporters of transcriptional activity. Given the prevalence of noise in biochemical systems, the time-series data arising from these is of significant interest in efforts to calibrate stochastic models of gene expression and obtain information about sources of nongenetic variability. We present a statistical inference framework that can be used to estimate kinetic parameters of gene expression, as well as the strength and half-life of extrinsic noise from single fluorescent-reporter-gene time-series data. The method takes into account stochastic variability in a fluorescent signal resulting from intrinsic noise of gene expression, kinetics of fluorescent protein maturation, and extrinsic noise, which is assumed to arise at transcriptional level. We use the linear noise approximation and derive an explicit formula for the likelihood of observed fluorescent data. The method is embedded in a Bayesian paradigm, so that certain parameters can be informed from other experiments allowing portability of results across different studies. Inference is performed using Markov chain Monte Carlo. Fluorescent reporters are primary tools to observe dynamics of gene expression and the correct interpretation of fluorescent data is crucial to investigating these fundamental processes of cellular life. As both magnitude and frequency of the noise may have a dramatic effect on the cell fitness, the quantification of stochastic fluctuation is essential to the understanding of how genes are regulated. Our method provides a framework that addresses this important question.

Quantifying intrinsic and extrinsic noise in gene transcription using the linear noise approximation: An application to single cell data

A central challenge in computational modeling of dynamic biological systems is parameter inference from experimental time course measurements. However, one would not only like to infer kinetic parameters but also study their variability from cell to cell. Here we focus on the case where single-cell fluorescent protein imaging time series data are available for a population of cells. Based on van Kampen?s linear noise approximation, we derive a dynamic state space model for molecular populations which is then extended to a hierarchical model. This model has potential to address the sources of variability relevant to single-cell data, namely, intrinsic noise due to the stochastic nature of the birth and death processes involved in reactions and extrinsic noise arising from the cell-to-cell variation of kinetic parameters. In order to infer such a model from experimental data, one must also quantify the measurement process where one has to allow for nonmeasurable molecular species as well as measurement noise of unknown level and variance. The availability of multiple single-cell time series data here provides a unique testbed to fit such a model and quantify these different sources of variation from experimental data.

StochSens—matlab package for sensitivity analysis of stochastic chemical systems

MOTIVATION The growing interest in the role of stochasticity in biochemical systems drives the demand for tools to analyse stochastic dynamical models of chemical reactions. One powerful tool to elucidate performance of dynamical systems is sensitivity analysis. Traditionally, however, the concept of sensitivity has mainly been applied to deterministic systems, and the difficulty to generalize these concepts for stochastic systems results from necessity of extensive Monte Carlo simulations. RESULTS Here we present a Matlab package, StochSens, that implements sensitivity analysis for stochastic chemical systems using the concept of the Fisher Information Matrix (FIM). It uses the linear noise approximation to represent the FIM in terms of solutions of ordinary differential equations. This is the first computational tool that allows for quick computation of the Information Matrix for stochastic systems without the need for Monte Carlo simulations. AVAILABILITY http://www.theosysbio.bio.ic.ac.uk/resources/stns SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.