Metabolic flux estimation--a self-adaptive evolutionary algorithm with singular value decomposition

Metabolic flux analysis is important for metabolic system regulation and intracellular pathway identification. A popular approach for intracellular flux estimation involves using ¹³C tracer experiments to label states that can be measured by nuclear magnetic resonance spectrometry or gas chromatography mass spectrometry. However, the bilinear balance equations derived from ¹³C tracer experiments and the noisy measurements require a nonlinear optimization approach to obtain the optimal solution. In this paper, the flux quantification problem is formulated as an error-minimization problem with equality and inequality constraints through the ¹³C balance and stoichiometric equations. The stoichiometric constraints are transformed to a null space by singular value decomposition. Self-adaptive evolutionary algorithms are then introduced for flux quantification. The performance of the evolutionary algorithm is compared with ordinary least squares estimation by the simulation of the central pentose phosphate pathway. The proposed algorithm is also applied to the central metabolism of Corynebacterium glutamicum under lysine-producing conditions. A comparison between the results from the proposed algorithm and data from the literature is given. The complexity of a metabolic system with bidirectional reactions is also investigated by analyzing the fluctuations in the flux estimates when available measurements are varied.     

Automated Bayesian model development for frequency detection in biological time series

Background

The ability to perform quantitative studies using isotope tracers and metabolic flux analysis (MFA) is critical for detecting pathway bottlenecks and elucidating network regulation in biological systems, especially those that have been engineered to alter their native metabolic capacities. Mathematically, MFA models are traditionally formulated using separate state variables for reaction fluxes and isotopomer abundances. Analysis of isotope labeling experiments using this set of variables results in a non-convex optimization problem that suffers from both implementation complexity and convergence problems.

Results

This article addresses the mathematical and computational formulation of ¹³C MFA models using a new set of variables referred to as fluxomers. These composite variables combine both fluxes and isotopomer abundances, which results in a simply-posed formulation and an improved error model that is insensitive to isotopomer measurement normalization. A powerful fluxomer iterative algorithm (FIA) is developed and applied to solve the MFA optimization problem. For moderate-sized networks, the algorithm is shown to outperform the commonly used 13CFLUX cumomer-based algorithm and the more recently introduced OpenFLUX software that relies upon an elementary metabolite unit (EMU) network decomposition, both in terms of convergence time and output variability.

Conclusions

Substantial improvements in convergence time and statistical quality of results can be achieved by applying fluxomer variables and the FIA algorithm to compute best-fit solutions to MFA models. We expect that the fluxomer formulation will provide a more suitable basis for future algorithms that analyze very large scale networks and design optimal isotope labeling experiments.     

A Method to Constrain Genome-Scale Models with 13C Labeling Data

Current limitations in quantitatively predicting biological behavior hinder our efforts to engineer biological systems to produce biofuels and other desired chemicals. Here, we present a new method for calculating metabolic fluxes, key targets in metabolic engineering, that incorporates data from ¹³C labeling experiments and genome-scale models. The data from ¹³C labeling experiments provide strong flux constraints that eliminate the need to assume an evolutionary optimization principle such as the growth rate optimization assumption used in Flux Balance Analysis (FBA). This effective constraining is achieved by making the simple but biologically relevant assumption that flux flows from core to peripheral metabolism and does not flow back. The new method is significantly more robust than FBA with respect to errors in genome-scale model reconstruction. Furthermore, it can provide a comprehensive picture of metabolite balancing and predictions for unmeasured extracellular fluxes as constrained by ¹³C labeling data. A comparison shows that the results of this new method are similar to those found through ¹³C Metabolic Flux Analysis (¹³C MFA) for central carbon metabolism but, additionally, it provides flux estimates for peripheral metabolism. The extra validation gained by matching 48 relative labeling measurements is used to identify where and why several existing COnstraint Based Reconstruction and Analysis (COBRA) flux prediction algorithms fail. We demonstrate how to use this knowledge to refine these methods and improve their predictive capabilities. This method provides a reliable base upon which to improve the design of biological systems.     

A guide to metabolic flux analysis in metabolic engineering: Methods,tools and applications

The field of metabolic engineering is primarily concerned with improving the biological production of value-added chemicals, fuels and pharmaceuticals through the design, construction and optimization of metabolic pathways, redirection of intracellular fluxes, and refinement of cellular properties relevant for industrial bioprocess implementation. Metabolic network models and metabolic fluxes are central concepts in metabolic engineering, as was emphasized in the first paper published in this journal, “Metabolic fluxes and metabolic engineering” (Metabolic Engineering, 1: 1–11, 1999). In the past two decades, a wide range of computational, analytical and experimental approaches have been developed to interrogate the capabilities of biological systems through analysis of metabolic network models using techniques such as flux balance analysis (FBA), and quantify metabolic fluxes using constrained-based modeling approaches such as metabolic flux analysis (MFA) and more advanced experimental techniques based on the use of stable-isotope tracers, i.e. ¹³C-metabolic flux analysis (¹³C-MFA). In this review, we describe the basic principles of metabolic flux analysis, discuss current best practices in flux quantification, highlight potential pitfalls and alternative approaches in the application of these tools, and give a broad overview of pragmatic applications of flux analysis in metabolic engineering practice.     

Quantification of central metabolic fluxes in the facultative methylotroph methylobacterium extorquens AM1 using 13C-label tracing and mass spectrometry

The metabolic fluxes of central carbon metabolism were measured in chemostat-grown cultures of Methylobacterium extorquens AM1 with methanol as the sole organic carbon and energy source and growth-limiting substrate. Label tracing experiments were carried out using 70% (13)C-methanol in the feed, and the steady-state mass isotopomer distributions of amino acids derived from total cell protein were measured by gas chromatography coupled to mass spectrometry. Fluxes were calculated from the isotopomer distribution data using an isotopomer balance model and evolutionary error minimization algorithm. The combination of labeled methanol with unlabeled CO(2), which enters central metabolism in two different reactions, provided the discriminatory power necessary to allow quantification of the unknown fluxes within a reasonably small confidence interval. In wild-type M. extorquens AM1, no measurable flux was detected through pyruvate dehydrogenase or malic enzyme, and very little flux through alpha-ketoglutarate dehydrogenase (1.4% of total carbon). In contrast, the alpha-ketoglutarate dehydrogenase flux was 25.5% of total carbon in the regulatory mutant strain phaR, while the pyruvate dehydrogenase and malic enzyme fluxes remained insignificant. The success of this technique with growth on C(1) compounds suggests that it can be applied to help characterize the effects of other regulatory mutations, and serve as a diagnostic tool in the metabolic engineering of methylotrophic bacteria.     

Bidirectional reaction steps in metabolic networks: II. Flux estimation and statistical analysis

Metabolic carbon labelling experiments enable a large amount of extracellular fluxes and intracellular carbon isotope enrichments to be measured. Since the relation between the measured quantities and the unknown intracellular metabolic fluxes is given by bilinear balance equations, flux determination from this data set requires the numerical solution of a nonlinear inverse problem. To this end, a general algorithm for flux estimation from metabolic carbon labelling experiments based on the least squares approach is developed in this contribution and complemented by appropriate tools for statistical analysis. The linearization technique usually applied for the computation of nonlinear confidence regions is shown to be inappropriate in the case of large exchange fluxes. For this reason a sophisticated compactification transformation technique for nonlinear statistical analysis is developed. Statistical analysis is then performed by computing appropriate statistical quality measures like output sensitivities, parameter sensitivities and the parameter covariance matrix. This allows one to determine the order of magnitude of exchange fluxes in most practical situations. An application study with a large data set from lysine-producing Corynebacterium glutamicum demonstrates the power and limitations of the carbon-labelling technique. It is shown that all intracellular fluxes in central metabolism can be quantitated without assumptions on intracellular energy yields. At the same time several exchange fluxes are determined which is invaluable information for metabolic engineering. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 55: 118-135, 1997.     

Identification of metabolic network models from incomplete high-throughput datasets

MOTIVATION: High-throughput measurement techniques for metabolism and gene expression provide a wealth of information for the identification of metabolic network models. Yet, missing observations scattered over the dataset restrict the number of effectively available datapoints and make classical regression techniques inaccurate or inapplicable. Thorough exploitation of the data by identification techniques that explicitly cope with missing observations is therefore of major importance. RESULTS: We develop a maximum-likelihood approach for the estimation of unknown parameters of metabolic network models that relies on the integration of statistical priors to compensate for the missing data. In the context of the linlog metabolic modeling framework, we implement the identification method by an Expectation-Maximization (EM) algorithm and by a simpler direct numerical optimization method. We evaluate performance of our methods by comparison to existing approaches, and show that our EM method provides the best results over a variety of simulated scenarios. We then apply the EM algorithm to a real problem, the identification of a model for the Escherichia coli central carbon metabolism, based on challenging experimental data from the literature. This leads to promising results and allows us to highlight critical identification issues.     

Metabolic isotopomer labeling systems. Part II: structural flux identifiability analysis

Metabolic flux analysis using carbon labeling experiments (CLEs) is an important tool in metabolic engineering where the intracellular fluxes have to be computed from the measured extracellular fluxes and the partially measured distribution of 13C labeling within the intracellular metabolite pools. The relation between unknown fluxes and measurements is described by an isotopomer labeling system (ILS) (see Part I [Math. Biosci. 169 (2001) 173]). Part II deals with the structural flux identifiability of measured ILSs in the steady state. The central question is whether the measured data contains sufficient information to determine the unknown intracellular fluxes. This question has to be decided a priori, i.e. before the CLE is carried out. In structural identifiability analysis the measurements are assumed to be noise-free. A general theory of structural flux identifiability for measured ILSs is presented and several algorithms are developed to solve the identifiability problem. In the particular case of maximal measurement information, a symbolical algorithm is presented that decides the identifiability question by means of linear methods. Several upper bounds of the number of identifiable fluxes are derived, and the influence of the chosen inputs is evaluated. By introducing integer arithmetic this algorithm can even be applied to large networks. For the general case of arbitrary measurement information, identifiability is decided by a local criterion. A new algorithm based on integer arithmetic enables an a priori local identifiability analysis to be performed for networks of arbitrary size. All algorithms have been implemented and flux identifiability is investigated for the network of the central metabolic pathways of a microorganism. Moreover, several small examples are worked out to illustrate the influence of input metabolite labeling and the paradox of information loss due to network simplification.     

Determination of confidence intervals of metabolic fluxes estimated from stable isotope measurements

Metabolic fluxes, estimated from stable isotope studies, provide a key to quantifying physiology in fields ranging from metabolic engineering to the analysis of human metabolic diseases. A serious drawback of the flux estimation method in current use is that it does not produce confidence limits for the estimated fluxes. Without this information it is difficult to interpret flux results and expand the physiological significance of flux studies. To address this shortcoming we derived analytical expressions of flux sensitivities with respect to isotope measurements and measurement errors. These tools allow the determination of local statistical properties of fluxes and relative importance of measurements. Furthermore, we developed an efficient algorithm to determine accurate flux confidence intervals and demonstrated that confidence intervals obtained with this method closely approximate true flux uncertainty. In contrast, confidence intervals approximated from local estimates of standard deviations are inappropriate due to inherent system nonlinearities. We applied these methods to analyze the statistical significance and confidence of estimated gluconeogenesis fluxes from human studies with [U-13C]glucose as tracer and found true limits for flux estimation in specific human isotopic protocols.     

Analysis of flux estimates based on (13)C-labelling experiments.

Modelling of the fluxes in central metabolism can be performed by combining labelling experiments with metabolite balancing. Using this approach, multiple samples from a cultivation of Saccharomyces cerevisiae in metabolic and isotopic steady state were analysed, and the metabolic fluxes in central metabolism were estimated. In the various samples, the estimates of the central metabolic pathways, the tricarboxylic acid cycle, the oxidative pentose phosphate pathway and the anaplerotic pathway, showed an unprecedented reproducibility. The high reproducibility was obtained with fractional labellings of individual carbon atoms as the calculational base, illustrating that the more complex modelling using isotopomers is not necessarily superior with respect to reproducibility of the flux estimates. Based on these results some general difficulties in flux estimation are discussed.     

Metabolic flux analysis in Escherichia coli by integrating isotopic dynamic and isotopic stationary 13C labeling data

The novel concept of isotopic dynamic 13C metabolic flux analysis (ID-13C MFA) enables integrated analysis of isotopomer data from isotopic transient and/or isotopic stationary phase of a 13C labeling experiment, short-time experiments, and an extended range of applications of 13C MFA. In the presented work, an experimental and computational framework consisting of short-time 13C labeling, an integrated rapid sampling procedure, a LC-MS analytical method, numerical integration of the system of isotopomer differential equations, and estimation of metabolic fluxes was developed and applied to determine intracellular fluxes in glycolysis, pentose phosphate pathway (PPP), and citric acid cycle (TCA) in Escherichia coli grown in aerobic, glucose-limited chemostat culture at a dilution rate of D = 0.10 h(-1). Intracellular steady state concentrations were quantified for 12 metabolic intermediates. A total of 90 LC-MS mass isotopomers were quantified at sampling times t = 0, 91, 226, 346, 589 s and at isotopic stationary conditions. Isotopic stationarity was reached within 10 min in glycolytic and PPP metabolites. Consistent flux solutions were obtained by ID-13C MFA using isotopic dynamic and isotopic stationary 13C labeling data and by isotopic stationary 13C MFA (IS-13C MFA) using solely isotopic stationary data. It is demonstrated that integration of dynamic 13C labeling data increases the sensitivity of flux estimation, particularly at the glucose-6-phosphate branch point. The identified split ratio between glycolysis and PPP was 55%:44%. These results were confirmed by IS-13C MFA additionally using labeling data in proteinogenic amino acids (GC-MS) obtained after 5 h from sampled biomass.     

Metabolic flux analysis of Shewanella spp. reveals evolutionary robustness in central carbon metabolism

Shewanella spp. are a group of facultative anaerobic bacteria widely distributed in marine and freshwater environments. In this study, we profiled the central metabolic fluxes of eight recently sequenced Shewanella species grown under the same condition in minimal medium with [3‐¹³C] lactate. Although the tested Shewanella species had slightly different growth rates (0.23–0.29 h^?1) and produced different amounts of acetate and pyruvate during early exponential growth (pseudo‐steady state), the relative intracellular metabolic flux distributions were remarkably similar. This result indicates that Shewanella species share similar regulation in regard to central carbon metabolic fluxes under steady growth conditions: the maintenance of metabolic robustness is not only evident in a single species under genetic perturbations (Fischer and Sauer, 2005; Nat Genet 37(6):636–640), but also observed through evolutionary related microbial species. This remarkable conservation of relative flux profiles through phylogenetic differences prompts us to introduce the concept of metabotype as an alternative scheme to classify microbial fluxomics. On the other hand, Shewanella spp. display flexibility in the relative flux profiles when switching their metabolism from consuming lactate to consuming pyruvate and acetate. Biotechnol. Bioeng. 2009;102: 1161–1169. © 2008 Wiley Periodicals, Inc.     

Evaluating (13)C enrichment data of free amino acids for precise metabolic flux analysis

Metabolic flux analysis using (13)C enrichment data of intracellular free amino acids (FAAs) can improve the time resolution of flux estimation compared to analysis of proteinogenic amino acid data owing to the faster turnover times of FAAs. The nature of the (13)C enrichment dynamics of FAAs remains obscure, however, especially with regard to its dependence on culture conditions, even though an understanding of dynamic behavior is important for precise metabolic flux estimation. In this study, we analyzed the (13)C enrichment dynamics of free and proteinogenic amino acids in a series of continuous culture experiments with Escherichia coli. The results indicated that the effect of protein degradation on the (13)C enrichment of FAAs was negligible under cellular growth conditions. Furthermore, they showed that the time scale necessary for (13)C enrichment dynamics of FAAs to reach a steady state depends on culture conditions such as oxygen uptake rate, which was likely due to different pool sizes of intracellular metabolites. The results demonstrate the importance of analyzing (13)C enrichment dynamics for the precise estimation of metabolic fluxes using FAA data.     

Markov Chain Monte Carlo Algorithm based metabolic flux distribution analysis on Corynebacterium glutamicum

MOTIVATION: Metabolic flux analysis via a (13)C tracer experiment has been achieved using a Monte Carlo method with the assumption of system noise as Gaussian noise. However, an unbiased flux analysis requires the estimation of fluxes and metabolites jointly without the restriction on the assumption of Gaussian noise. The flux distributions under such a framework can be freely obtained with various system noise and uncertainty models. RESULTS: In this paper, a stochastic generative model of the metabolic system is developed. Following this, the Markov Chain Monte Carlo (MCMC) approach is applied to flux distribution analysis. The disturbances and uncertainties in the system are simplified as truncated Gaussian multiplicative models. The performance in a real metabolic system is illustrated by the application to the central metabolism of Corynebacterium glutamicum. The flux distributions are illustrated and analyzed in order to understand the underlying flux activities in the system. AVAILABILITY: Algorithms are available upon request.     

Metabolic flux analysis and visualization

One of the ultimate goals of systems biology research is to obtain a comprehensive understanding of the control mechanisms of complex cellular metabolisms. Metabolic Flux Analysis (MFA) is a important method for the quantitative estimation of intracellular metabolic flows through metabolic pathways and the elucidation of cellular physiology. The primary challenge in the use of MFA is that many biological networks are underdetermined systems; it is therefore difficult to narrow down the solution space from the stoichiometric constraints alone. In this tutorial, we present an overview of Flux Balance Analysis (FBA) and (13)C-Metabolic Flux Analysis ((13)C-MFA), both of which are frequently used to solve such underdetermined systems, and we demonstrate FBA and (13)C-MFA using the genome-scale model and the central carbon metabolism model, respectively. Furthermore, because such comprehensive study of intracellular fluxes is inherently complex, we subsequently introduce various pathway mapping and visualization tools to facilitate understanding of these data in the context of the pathways. Specific visualization of MFA results using the BioCyc Omics Viewer and Pathway Projector are shown as illustrative examples.     

Network identification and flux quantification in the central metabolism of Saccharomyces cerevisiae under different conditions of glucose repression

The network structure and the metabolic fluxes in central carbon metabolism were characterized in aerobically grown cells of Saccharomyces cerevisiae. The cells were grown under both high and low glucose concentrations, i.e., either in a chemostat at steady state with a specific growth rate of 0.1 h(-1) or in a batch culture with a specific growth rate of 0.37 h(-1). Experiments were carried out using [1-(13)C]glucose as the limiting substrate, and the resulting summed fractional labelings of intracellular metabolites were measured by gas chromatography coupled to mass spectrometry. The data were used as inputs to a flux estimation routine that involved appropriate mathematical modelling of the central carbon metabolism of S. cerevisiae. The results showed that the analysis is very robust, and it was possible to quantify the fluxes in the central carbon metabolism under both growth conditions. In the batch culture, 16.2 of every 100 molecules of glucose consumed by the cells entered the pentose-phosphate pathway, whereas the same relative flux was 44.2 per 100 molecules in the chemostat. The tricarboxylic acid cycle does not operate as a cycle in batch-growing cells, in contrast to the chemostat condition. Quantitative evidence was also found for threonine aldolase and malic enzyme activities, in accordance with published data. Disruption of the MIG1 gene did not cause changes in the metabolic network structure or in the flux pattern.     

Modeling and optimization of a multi-product biosynthesis factory for multiple objectives

Genetic algorithms and optimization in general, enable us to probe deeper into the metabolic pathway recipe for multi-product biosynthesis. An augmented model for optimizing serine and tryptophan flux ratios simultaneously in Escherichia coli, was developed by linking the dynamic tryptophan operon model and aromatic amino acid-tryptophan biosynthesis pathways to the central carbon metabolism model. Six new kinetic parameters of the augmented model were estimated with considerations of available experimental data and other published works. Major differences between calculated and reference concentrations and fluxes were explained. Sensitivities and underlying competition among fluxes for carbon sources were consistent with intuitive expectations based on metabolic network and previous results. Biosynthesis rates of serine and tryptophan were simultaneously maximized using the augmented model via concurrent gene knockout and manipulation. The optimization results were obtained using the elitist non-dominant sorting genetic algorithm (NSGA-II) supported by pattern recognition heuristics. A range of Pareto-optimal enzyme activities regulating the amino acids biosynthesis was successfully obtained and elucidated wherever possible vis-à-vis fermentation work based on recombinant DNA technology. The predicted potential improvements in various metabolic pathway recipes using the multi-objective optimization strategy were highlighted and discussed in detail.