Research Article  Open Access
Shuichi Kajihata, Chikara Furusawa, Fumio Matsuda, Hiroshi Shimizu, "OpenMebius: An Open Source Software for Isotopically Nonstationary ^{13}CBased Metabolic Flux Analysis", BioMed Research International, vol. 2014, Article ID 627014, 10 pages, 2014. https://doi.org/10.1155/2014/627014
OpenMebius: An Open Source Software for Isotopically Nonstationary ^{13}CBased Metabolic Flux Analysis
Abstract
The in vivo measurement of metabolic flux by ^{13}Cbased metabolic flux analysis (^{13}CMFA) provides valuable information regarding cell physiology. Bioinformatics tools have been developed to estimate metabolic flux distributions from the results of tracer isotopic labeling experiments using a ^{13}Clabeled carbon source. Metabolic flux is determined by nonlinear fitting of a metabolic model to the isotopic labeling enrichment of intracellular metabolites measured by mass spectrometry. Whereas ^{13}CMFA is conventionally performed under isotopically constant conditions, isotopically nonstationary ^{13}C metabolic flux analysis (INST^{13}CMFA) has recently been developed for flux analysis of cells with photosynthetic activity and cells at a quasisteady metabolic state (e.g., primary cells or microorganisms under stationary phase). Here, the development of a novel open source software for INST^{13}CMFA on the Windows platform is reported. OpenMebius (Open source software for Metabolic flux analysis) provides the function of autogenerating metabolic models for simulating isotopic labeling enrichment from a userdefined configuration worksheet. Analysis using simulated data demonstrated the applicability of OpenMebius for INST^{13}CMFA. Confidence intervals determined by INST^{13}CMFA were less than those determined by conventional methods, indicating the potential of INST^{13}CMFA for precise metabolic flux analysis. OpenMebius is the open source software for the general application of INST^{13}CMFA.
1. Introduction
The in vivo measurement of metabolic flux by ^{13}Cbased metabolic flux analysis (^{13}CMFA) provides valuable information regarding cell physiology in fields ranging from the metabolic engineering of microorganisms to the analysis of human metabolic diseases [1–3]. Since metabolic fluxes are estimated by a computational analysis of the isotopic labeling data produced by a series of wet experiments [4–7], the development of an open software platform for ^{13}CMFA is desired for further methodology improvement and wider applications for in vivo metabolic flux measurement.
In ^{13}CMFA, after feeding of a ^{13}Clabeled carbon source into a cell culture, amino acids or intermediates are extracted and subjected to mass spectrometric analysis. For the simplest example, [1^{13}C] glucose is converted to pyruvate (PYR) and then alanine (Ala) via two glycolytic pathways including the EmbdenMeyerhofParnas (EMP) pathway and the pentose phosphate (PP) pathway (Figure 1(a)). Whereas one ^{13}Clabeled molecule and one nonlabeled molecule of Ala are generated from one molecule of [1^{13}C] glucose by the EMP pathway, no ^{13}Clabeled Ala is produced via the PP pathway, because the ^{13}C atom is metabolically discarded as CO_{2}. Thus, the metabolic flux ratio between the EMP and PP pathways could be estimated from the relative abundances of ^{13}Clabeled and nonlabeled Ala using mass spectrometry.
In ^{13}CMFA of complex networks of carbon central metabolism, metabolic fluxes are computationally estimated by a nonlinear optimization method since the relationship between metabolic fluxes and isotopic labeling enrichment is usually nonlinear. For that purpose, a metabolic model is constructed based on the metabolic pathway network and the carbon transition network, which represents the transitions of carbon atoms between substrates and products in a metabolic reaction (Figure 1(b)). is a function to calculate isotopic labeling enrichment or the mass distribution vector (MDV) of metabolites from the given metabolic fluxes and isotopic labeling patterns of carbon sources. Consider Here, is a simulated mass spectrum of metabolite . and are the vectors of metabolic flux and isotopic labeling pattern of carbon source, respectively. A vector of metabolic flux is fitted to the observed mass spectrum () by a nonlinear optimization method:
The optimized value is the estimated metabolic flux distribution in the cells to minimize the covarianceweighted sum of squared difference. is the covariance matrix with a measurement standard deviation located on the diagonal. is the stoichiometric matrix. There are several software packages to perform conventional ^{13}CMFA such as 13CFLUX [8], 13CFLUX2 [9], C13 [10], Metran [11], FIA [12], influx_s [13], and OpenFLUX [14].
In the case of conventional ^{13}CMFA, isotopic labeling data must be obtained from cell culture under metabolic steady state and isotopically stationary conditions (Figures 1(c) and 1(d)). Here, metabolic steady state indicates the constant flux distribution and pool size of intracellular metabolites that has to be maintained during the isotopic labeling experiment (Figure 1(c)). An isotopically stationary condition means constant isotopic labeling enrichment of metabolites. A long culture period has often been required to achieve isotopically stationary conditions after feeding a ^{13}Clabeled substrate.
In recent years, a novel method has been developed to determine metabolic flux using a time course of isotopic labeling data obtained from an isotopically transient state (Figure 1(d)) [15–17]. For the isotopically nonstationary MFA (INST^{13}CMFA), an expanded metabolic model is used to simulate isotopic labeling dynamics, taking into consideration the metabolite pool size in the cell: where is the time of the th sampling point. is the vector of the pool sizes of all metabolites in the metabolic system. The formulation indicates that the intracellular pool sizes of intermediates in central metabolism must be precisely determined for INST^{13}CMFA [18, 19]. Time course analysis by rapid sampling techniques has also been performed in INST^{13}CMFA to analyze the fast turnover of isotopic labeling enrichment in carbon central metabolism [20, 21]. Despite these technical challenges, INST^{13}CMFA would be essential for the analysis of photoautotrophic organisms using CO_{2} as a carbon source. Metabolic flux cannot be determined by conventional ^{13}CMFA using ^{13}CO_{2} as a carbon source, because all metabolites are uniformly labeled after reaching an isotopically stationary phase [22]. The methodology is also promising for the precise metabolic flux analysis of cells at a quasisteady metabolic state (e.g., primary cells or microorganisms in stationary phase). In order to analyze a time course dataset produced by INST^{13}CMFA, a software package with a graphical user interface has recently been reported (INCA [23]). In addition to these sophisticated tools, open source software packages such as OpenFLUX [14] for conventional ^{13}CMFA are also useful for facilitating the further development of INST^{13}CMFA [24].
Here, a novel open source software package for INST^{13}CMFA, OpenMebius (Open source software for Metabolic flux analysis), is reported. OpenMebius has been developed to perform INST^{13}CMFA and conventional ^{13}CMFA using a userdefined metabolic model. A metabolic model can be automatically generated from a metabolic pathway and a carbon transition network described in text or Microsoft Excel worksheet files. The metabolic flux distribution can be estimated by nonlinear fitting of the metabolic model to the isotopic labeling enrichment data.
2. Materials and Methods
2.1. Model Construction
OpenMebius is implemented in MATLAB (MathWorks, Natick, MA, USA) for the Windows platform. The software consists of two parts: automated model construction and metabolic flux estimation by nonlinear optimization. Functions for processing raw mass spectrum data and the determination of confidence intervals are also included. OpenMebius is designed for conventional ^{13}CMFA and INST^{13}CMFA using mass spectrometry data. Isotopic labeling enrichment of metabolites is described by a mass distribution vector (MDV) [25]: where is the vector of isotopic labeling enrichment of metabolite . indicates the relative abundance of a metabolite in which carbons are labeled with ^{13}C. To obtain the of the carbon skeleton, mass spectrum data are corrected for the presence of naturally occurring isotopes using the correction matrix [26].
In conventional ^{13}CMFA, a metabolic model is an algebraic equation used to generate from the vector of metabolic flux () and the isotopic labeling pattern of a carbon source , as shown in (1).
Since the metabolic flux is determined in cells at metabolic steady state, follows the stoichiometric equation described by where is the stoichiometric matrix. In OpenMebius, is constructed from a metabolic network described in the “Rxns” column in a userdefined configuration worksheet (Figure 1(b)), taking into consideration the fluxes for biomass syntheses and product excretion. is calculated by the framework of elementary metabolite units (EMU) [27] using the carbon transition information described in the “carbon_transitions” column of the configuration worksheet (Figure 1(b)). In the framework, the carbon transition network is decomposed to cascade networks of EMUs depending on those carbon numbers. The cascade networks of the EMUs with th carbon follow the EMU balance equation [27]: Here, each row in matrix is MDV of corresponding EMU. The matrix includes EMUs of the carbon source or the smaller size EMUs. The element in row and column of matrices and the element of matrix are described, respectively, as follows:
In the case of INST^{13}CMFA, the metabolic model is expanded to describe a transition state of isotopic labeling (Figure 1(d)) by considering the dilution of isotopic labeling enrichment depending on the pool size of intermediates, as shown in (3), where is a vector of the pool size of each metabolite that is constant under metabolic steady state. is the time of the th sampling point. In this study, instead of a direct description of the metabolic model , timedependent changes in the isotopic labeling enrichment of metabolite are described by the differential equation as follows: where and represent the fluxes of the th inflow reaction and the th outflow reaction of metabolite , respectively. The model is automatically constructed by “ConstEMUnetwork.m.” Detailed rules to describe a userdefined metabolic pathway and carbon transition network are provided on the project home page (http://wwwshimizu.ist.osakau.ac.jp/hp/en/software/OpenMebius.html). Euler’s method is implemented to solve the ordinary differential equation (8) without adaptive step size control. Stiff equations can be resolved by carefully selecting the step size. The are standardized for each step to prevent divergence. Moreover, no specific libraries were used to implement the algorithm for solving differential equations.
2.2. Metabolic Flux Estimation
The procedure for estimating metabolic flux is shown in Figure 2. In Step 1, the initial flux distribution is given considering the rates of biomass synthesis, substrate consumption, and product excretion (Figure 2, Step 1). In Step 2, the metabolic flux vector is optimized to minimize the covarianceweighted sum of squared difference (SSD) using the LevenbergMarquardt method [28] (Figure 2, Step 2): Here, is the vector of experimental data at . is the total number of measured metabolites for data fitting. is the total number of sampling points ( in the case of isotopically stationary), and is the measurement covariance matrix with the measurement standard deviation located on the diagonal.
2.3. Calculation of Confidence Interval
Confidence intervals of estimated fluxes are determined by OpenMebius using the grid search method [29, 30]. The metabolic flux of reaction is fixed to and the objective function is reoptimized. Here, is the optimized metabolic flux of reaction and is the perturbation level. The procedure is iterated with increased or decreased . The range of fixed metabolic flux whose SSD is less than the threshold level is the confidence interval. The threshold level is determined by where is the minimized SSD with one fixed flux, is the original minimized SSD, is the number of independent data points used in the fitting, is the degrees of freedom in the original flux fit, is the distribution, and is the confidence level.
3. Results and Discussion
3.1. Implementation
OpenMebius is a toolbox for conventional ^{13}CMFA and INST^{13}CMFA using mass spectrometry data implemented in MATLAB on the Windows platform. Figure 3 shows a representative MATLAB code to perform INST^{13}CMFA on a simplified TCA cycle model mentioned below. A metabolic model is generated by the “ConstEMUnetwork” function from userdefined metabolic network information described in text or Excel worksheet files. After loading related data, a metabolic flux distribution is estimated by the “marquardt_inst” function using a nonlinear optimization (LevenbergMarquardt method). For a routine analysis, a batch execution of metabolic flux estimations is also supported. See Materials and Methods for detailed information.
(a)
(b)
3.2. Test Case of Isotopically Stationary MFA: Simplified TCA Cycle Model
The performance of OpenMebius for conventional ^{13}CMFA was tested with the simplified metabolic network used in the previous study [14] (Figure 4). The metabolic network consisted of the 16 reactions of the TCA cycle using pyruvate and glutamate as substrates described by Table 1. Among 16 metabolic fluxes, one influx (R1) and six effluxes (R8–R13) were predetermined. The metabolic model was successfully constructed from the metabolic pathway and carbon transition networks. Here, the vector of experimental mass spectra () of valine, lysine, aspartate, and succinate was artificially created using the metabolic model, the flux distribution described in the previous research [14], and the isotopic labeling of pyruvate (mixture of 50% 1^{13}C and 50% U^{13}C) and glutamate (100% 1^{13}C). Considering the simulated data as the measured MDV, the metabolic flux distribution was determined by the conventional ^{13}CMFA function of OpenMebius. The estimated flux distribution was essentially identical to that of simulated distribution, which was consistent with the results of 13CFLUX [8] and OpenFLUX [14] (Figure 4). The total computation time was 6 seconds for 10 cycles of optimization (Intel Core i7 2.80 GHz), which was the same as in OpenFLUX.

3.3. Test Case of Isotopically Nonstationary MFA: Simplified TCA Cycle Model
To simulate an isotopic labeling experiment during an isotopically nonstationary period, the pool size information of six intermediates was arbitrarily added to the above TCA metabolic network. A metabolic model for INST^{13}CMFA was successfully constructed by OpenMebius. To prepare simulated experimental data, time course data of isotopic labeling dynamics of oxaloacetate and succinate were created using the differential equation (8) combined with the pool size information . The current version of OpenMebius uses the pool size information as constant values, although should be estimated with an optimization procedure since the pool size data are less reliable than isotopic labeling measurements. That function will be supported in a future version of OpenMebius. The flux distribution and isotopic labeling patterns of substrate were identical to those of the previous section. The MDVs of oxaloacetate and succinate were sampled 17 times at 5second intervals in silico, to which Gaussian noise (1%) was added to imitate actual measurements. Considering the simulated data as measured MDVs (), the metabolic flux distribution was estimated using OpenMebius. The step size was set to 0.01 seconds to compute the simulated MDVs. Although only two intracellular metabolites were used for data fitting, the fitted isotopic labeling dynamics and a flux distribution were consistent with the simulated data (Figure 5). The total computational time for one cycle of optimization was around 10 minutes (Intel Core i7 2.80 GHz).
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(b)
For a performance comparison between conventional ^{13}CMFA and INST^{13}CMFA, the 95% confidence intervals of four representative reactions were determined by the grid search method (Figure 6(a)). For INST^{13}CMFA, confidence intervals were estimated using the simulated data with the 17 time points prepared above. In the case of conventional ^{13}CMFA, a novel simulated dataset was prepared by the following procedure. From the of oxaloacetate and succinate calculated using (1), 17 sets of simulated mass spectra () data were produced with the addition of Gaussian noise (1%). While an identical number of data points was used, the confidence intervals determined by INST^{13}CMFA were approximately 22% that of conventional ^{13}CMFA (Figure 6(a)). The sharply curved parabolas were observed for INST^{13}CMFA, suggesting that the time course MDV data includes information for a more precise estimation of metabolic flux (Figure 6(b)). These results suggest that INST^{13}CMFA could be a reliable method to determine in vivo metabolic flux with narrow confidence intervals.
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(b)
3.4. Test Case of Isotopically Nonstationary MFA: Escherichia coli Model
INST^{13}CMFA was also performed using simulated data produced from the central metabolic model of E. coli with 54 reactions and 22 intermediates. A simulated experimental dataset was prepared based on the literaturereported metabolic flux distribution and metabolite pool size data [31]. Pool sizes of unmeasured metabolites (GAP, PYR, Xu5P, E4P, IsoCit, αKG, and glyoxylate) were arbitrarily set at 0.1 μmol gDCW^{−1}. Simulated MDVs were sampled 11 times at 1second intervals using 100% [1^{13}C] glucose as a carbon source. Considering the simulated dataset as experimental data, metabolic fluxes were estimated using the INST^{13}CMFA function of OpenMebius. The step size was set to 0.001 seconds to compute the simulated MDVs. Although the computation time took 7 h 42 min (Intel Xeon X5670 2.93 GHz), the estimated flux distribution was essentially identical to that of the simulated data (Figure 7). The result indicates that OpenMebius could deal with INST^{13}CMFA using a realistic metabolic model of E. coli.
4. Conclusions
OpenMebius is the first open source software for metabolic flux analyses under both isotopically stationary and nonstationary conditions. The software supports the automatic construction of a metabolic model for INST^{13}CMFA from a userdefined metabolic network. Analysis using simulated data demonstrated not only the utility of OpenMebius for INST^{13}CMFA, but also its potential for use in metabolic flux analysis with reduced confidence intervals. OpenMebius provides an essential bioinformatics tool for INST^{13}CMFA to analyze metabolic flux in cells with slower metabolism (i.e., mammalian) [17] and cultivation with single carbon substrates (i.e., cyanobacteria) [15].
Abbreviations
MDV:  Mass distribution vector 
^{13}CMFA:  ^{13}Cbased metabolic flux analysis 
INST^{13}CMFA:  Isotopically nonstationary ^{13}C metabolic flux analysis 
KG:  Ketoglutarate 
ACCOA:  AcetylCoA 
Cit:  Citrate 
DHAP:  Dihydroxyacetone phosphate 
E4P:  Erythrose4phosphate 
F6P:  Fructose6phosphate 
FBP:  Fructose1,6bisphosphate 
FUM:  Fumarate 
G6P:  Glucose6phosphate 
GAP:  Glyceraldehyde3phosphate 
GLX:  Glyoxylate 
IsoCit:  Isocitrate 
MAL:  Malate 
OAA:  Oxaloacetate 
PEP:  Phosphoenolpyruvate 
6PG:  6Phosphoglycerate 
3PG:  3Phosphoglycerate 
PYR:  Pyruvate 
R5P:  Ribose5phosphate 
Ru5P:  Ribulose5phosphate 
S7P:  Sedoheptulose7phosphate 
SUC:  Succinate 
SUC_FUM:  Sum of metabolite pool of succinate and fumarate 
Xu5P:  Xylulose5phosphate 
VAL:  Valine 
LYS:  Lysine. 
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank Dr. Yoshihiro Toya, Dr. Katsunori Yoshikawa, Dr. Tomokazu Shirai, and all members of the Shimizu Lab for their help with the software development. This research was partially supported by JST, Strategic International Collaborative Research Program, SICORP for JPUS Metabolomics.
References
 Y. L. Sang, D.Y. Lee, and Y. K. Tae, “Systems biotechnology for strain improvement,” Trends in Biotechnology, vol. 23, no. 7, pp. 349–358, 2005. View at: Publisher Site  Google Scholar
 J. F. Moxley, M. C. Jewett, M. R. Antoniewicz et al., “Linking highresolution metabolic flux phenotypes and transcriptional regulation in yeast modulated by the global regulator Gcn4p,” Proceedings of the National Academy of Sciences of the United States of America, vol. 106, no. 16, pp. 6477–6482, 2009. View at: Publisher Site  Google Scholar
 C. S. Duckwall, T. A. Murphy, and J. D. Young, “Mapping cancer cell metabolism with ^{13}C flux analysis: recent progress and future challenges,” Journal of Carcinogenesis, vol. 12, p. 13, 2013. View at: Google Scholar
 O. Frick and C. Wittmann, “Characterization of the metabolic shift between oxidative and fermentative growth in Saccharomyces cerevisiae by comparative ^{13}C flux analysis,” Microbial Cell Factories, vol. 4, article 30, 2005. View at: Publisher Site  Google Scholar
 Y. Toya, N. Ishii, K. Nakahigashi et al., “^{13}CMetabolic flux analysis for batch culture of Escherichia coli and its pyk and pgi gene knockout mutants based on mass isotopomer distribution of intracellular metabolites,” Biotechnology Progress, vol. 26, no. 4, pp. 975–992, 2010. View at: Publisher Site  Google Scholar
 E. Mori, C. Furusawa, S. Kajihata, T. Shirai, and H. Shimizu, “Evaluating^{13}C enrichment data of free amino acids for precise metabolic flux analysis,” Biotechnology Journal, vol. 6, no. 11, pp. 1377–1387, 2011. View at: Publisher Site  Google Scholar
 T. Shirai, K. Fujimura, C. Furusawa, K. Nagahisa, S. Shioya, and H. Shimizu, “Study on roles of anaplerotic pathways in glutamate overproduction of Corynebacterium glutamicum by metabolic flux analysis,” Microbial Cell Factories, vol. 6, article 19, 2007. View at: Publisher Site  Google Scholar
 W. Wiechert, M. Möllney, S. Petersen, and A. A. De Graaf, “A universal framework for ^{13}C metabolic flux analysis,” Metabolic Engineering, vol. 3, no. 3, pp. 265–283, 2001. View at: Publisher Site  Google Scholar
 M. Weitzel, K. Nöh, T. Dalman, S. Niedenführ, B. Stute, and W. Wiechert, “^{13}CFLUX2—highperformance software suite for ^{13}Cmetabolic flux analysis,” Bioinformatics, vol. 29, no. 1, pp. 143–145, 2013. View at: Publisher Site  Google Scholar
 M. Cvijovic, R. OlivaresHernandez, R. Agren et al., “BioMet Toolbox: genomewide analysis of metabolism,” Nucleic Acids Research, vol. 38, no. 2, Article ID gkq404, pp. W144–W149, 2010. View at: Publisher Site  Google Scholar
 H. Yoo, M. R. Antoniewicz, G. Stephanopoulos, and J. K. Kelleher, “Quantifying reductive carboxylation flux of glutamine to lipid in a brown adipocyte cell line,” Journal of Biological Chemistry, vol. 283, no. 30, pp. 20621–20627, 2008. View at: Publisher Site  Google Scholar
 O. Srour, J. D. Young, and Y. C. Eldar, “Fluxomers: a new approach for ^{13}C metabolic flux analysis,” BMC Systems Biology, vol. 5, article 129, 2011. View at: Publisher Site  Google Scholar
 S. Sokol, P. Millard, and J.C. Portais, “Influx_s: increasing numerical stability and precision for metabolic flux analysis in isotope labelling experiments,” Bioinformatics, vol. 28, no. 5, Article ID btr716, pp. 687–693, 2012. View at: Publisher Site  Google Scholar
 L.E. Quek, C. Wittmann, L. K. Nielsen, and J. O. Krömer, “OpenFLUX: efficient modelling software for ^{13}Cbased metabolic flux analysis,” Microbial Cell Factories, vol. 8, article 25, 2009. View at: Publisher Site  Google Scholar
 J. D. Young, A. A. Shastri, G. Stephanopoulos, and J. A. Morgan, “Mapping photoautotrophic metabolism with isotopically nonstationary^{13}C flux analysis,” Metabolic Engineering, vol. 13, no. 6, pp. 656–665, 2011. View at: Publisher Site  Google Scholar
 J. Schaub, K. Mauch, and M. Reuss, “Metabolic flux analysis in Escherichia coli by integrating isotopic dynamic and isotopic stationary ^{13}C labeling data,” Biotechnology and Bioengineering, vol. 99, no. 5, pp. 1170–1185, 2008. View at: Publisher Site  Google Scholar
 T. A. Murphy, C. V. Dang, and J. D. Young, “Isotopically nonstationary ^{13}C flux analysis of Mycinduced metabolic reprogramming in Bcells,” Metabolic Engineering, vol. 15, no. 1, pp. 206–217, 2013. View at: Publisher Site  Google Scholar
 B. Zhou, J. F. Xiao, L. Tuli, and H. W. Ressom, “LCMSbased metabolomics,” Molecular BioSystems, vol. 8, no. 2, pp. 470–481, 2012. View at: Publisher Site  Google Scholar
 D. Vuckovic, “Current trends and challenges in sample preparation for global metabolomics using liquid chromatographymass spectrometry,” Analytical and Bioanalytical Chemistry, vol. 403, no. 6, pp. 1523–1548, 2012. View at: Publisher Site  Google Scholar
 F. Schädel and E. FrancoLara, “Rapid sampling devices for metabolic engineering applications,” Applied Microbiology and Biotechnology, vol. 83, no. 2, pp. 199–208, 2009. View at: Publisher Site  Google Scholar
 W. M. van Gulik, “Fast sampling for quantitative microbial metabolomics,” Current Opinion in Biotechnology, vol. 21, no. 1, pp. 27–34, 2010. View at: Publisher Site  Google Scholar
 A. A. Shastri and J. A. Morgan, “A transient isotopic labeling methodology for ^{13}C metabolic flux analysis of photoautotrophic microorganisms,” Phytochemistry, vol. 68, no. 1618, pp. 2302–2312, 2007. View at: Publisher Site  Google Scholar
 J. D. Young, “INCA: a computational platform for isotopically nonstationary metabolic flux analysis,” Bioinformatics, vol. 30, no. 9, pp. 1333–1335, 2014. View at: Publisher Site  Google Scholar
 W. Wiechert and K. Nöh, “Isotopically nonstationary metabolic flux analysis: complex yet highly informative,” Current Opinion in Biotechnology, vol. 24, no. 6, pp. 319–3332, 2013. View at: Publisher Site  Google Scholar
 C. Wittmann and E. Heinzle, “Mass spectrometry for metabolic flux analysis,” Biotechnology and Bioengineering, vol. 62, pp. 739–750, 1999. View at: Google Scholar
 W. A. Van Winden, C. Wittmann, E. Heinzle, and J. J. Heijnen, “Correcting mass isotopomer distributions for naturally occurring isotopes,” Biotechnology and Bioengineering, vol. 80, no. 4, pp. 477–479, 2002. View at: Publisher Site  Google Scholar
 M. R. Antoniewicz, J. K. Kelleher, and G. Stephanopoulos, “Elementary metabolite units (EMU): a novel framework for modeling isotopic distributions,” Metabolic Engineering, vol. 9, no. 1, pp. 68–86, 2007. View at: Publisher Site  Google Scholar
 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing, Cambridge University Press, Cambridge, Mass, USA, 2007.
 M. R. Antoniewicz, J. K. Kelleher, and G. Stephanopoulos, “Determination of confidence intervals of metabolic fluxes estimated from stable isotope measurements,” Metabolic Engineering, vol. 8, no. 4, pp. 324–337, 2006. View at: Publisher Site  Google Scholar
 R. Costenoble, D. Müller, T. Barl et al., “^{13}CLabeled metabolic flux analysis of a fedbatch culture of elutriated Saccharomyces cerevisiae,” FEMS Yeast Research, vol. 7, no. 4, pp. 511–526, 2007. View at: Publisher Site  Google Scholar
 Y. Toya, K. Nakahigashi, M. Tomita, and K. Shimizu, “Metabolic regulation analysis of wildtype and arcA mutant Escherichia coli under nitrate conditions using different levels of omics data,” Molecular BioSystems, vol. 8, no. 10, pp. 2593–2604, 2012. View at: Publisher Site  Google Scholar
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Copyright © 2014 Shuichi Kajihata et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.