OpenMebius: An Open Source Software for Isotopically Nonstationary 13C-Based Metabolic Flux Analysis

The in vivo measurement of metabolic flux by 13C-based metabolic flux analysis (13C-MFA) 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 13C-labeled 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 13C-MFA is conventionally performed under isotopically constant conditions, isotopically nonstationary 13C metabolic flux analysis (INST-13C-MFA) has recently been developed for flux analysis of cells with photosynthetic activity and cells at a quasi-steady metabolic state (e.g., primary cells or microorganisms under stationary phase). Here, the development of a novel open source software for INST-13C-MFA 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 user-defined configuration worksheet. Analysis using simulated data demonstrated the applicability of OpenMebius for INST-13C-MFA. Confidence intervals determined by INST-13C-MFA were less than those determined by conventional methods, indicating the potential of INST-13C-MFA for precise metabolic flux analysis. OpenMebius is the open source software for the general application of INST-13C-MFA.


Introduction
The in vivo measurement of metabolic flux by 13 C-based metabolic flux analysis ( 13 C-MFA) provides valuable information regarding cell physiology in fields ranging from the metabolic engineering of microorganisms to the analysis of human metabolic diseases [1][2][3]. Since metabolic fluxes are estimated by a computational analysis of the isotopic labeling data produced by a series of wet experiments [4][5][6][7], the development of an open software platform for 13 C-MFA is desired for further methodology improvement and wider applications for in vivo metabolic flux measurement.
In 13 C-MFA, after feeding of a 13 C-labeled 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 Embden-Meyerhof-Parnas (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 C-labeled 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 C-MFA 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, MDV sim is a simulated mass spectrum of metabolite . V and inp are the vectors of metabolic flux and isotopic labeling pattern of carbon source, respectively. A vector of metabolic flux V is fitted to the observed mass spectrum (MDV ) by a nonlinear optimization method: The optimized value V opt is the estimated metabolic flux distribution in the cells to minimize the covarianceweighted sum of squared difference. MDV 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 C-MFA such as 13CFLUX [8], 13CFLUX2 [9], C13 [10], Metran [11], FIA [12], influx s [13], and OpenFLUX [14].
In the case of conventional 13 C-MFA, 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 C-labeled 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][16][17]. For the isotopically nonstationary MFA (INST-13 C-MFA), 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 C-MFA [18,19]. Time course analysis by rapid sampling techniques has also been performed in INST-13 C-MFA to analyze the fast turnover of isotopic labeling enrichment in carbon central metabolism [20,21]. Despite these technical challenges, INST-13 C-MFA would be essential for the analysis of photoautotrophic organisms using BioMed Research International 3 CO 2 as a carbon source. Metabolic flux cannot be determined by conventional 13 C-MFA 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 quasi-steady metabolic state (e.g., primary cells or microorganisms in stationary phase). In order to analyze a time course dataset produced by INST-13 C-MFA, 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 C-MFA are also useful for facilitating the further development of INST-13 C-MFA [24].
Here, a novel open source software package for INST-13 C-MFA, OpenMebius (Open source software for Metabolic flux analysis), is reported. OpenMebius has been developed to perform INST-13 C-MFA and conventional 13 C-MFA using a user-defined 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.

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 C-MFA and INST-13 C-MFA using mass spectrometry data. Isotopic labeling enrichment of metabolites is described by a mass distribution vector (MDV) [25]: where MDV 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 MDV of the carbon skeleton, mass spectrum data are corrected for the presence of naturally occurring isotopes using the correction matrix [26].
In conventional 13 C-MFA, a metabolic model is an algebraic equation used to generate MDV sim from the vector of metabolic flux (V) and the isotopic labeling pattern of a carbon source ( inp ), as shown in (1).
Since the metabolic flux is determined in cells at metabolic steady state, V 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 user-defined configuration worksheet ( Figure 1(b)), taking into consideration the fluxes for biomass syntheses and product excretion. MDV sim 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 ( inp ) includes EMUs of the carbon source or the smaller size EMUs. The element ( , ) in row and column of matrices (V) and the element ( , ) of matrix (V) are described, respectively, as follows: In the case of INST-13 C-MFA, 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 , time-dependent changes in the isotopic labeling enrichment of metabolite are described by the differential equation as follows: where V in and V out 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 user-defined metabolic pathway and carbon transition network are provided on the project home page (http://www-shimizu.ist.osaka-u.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 MDV sim , = are standardized for each step to prevent divergence. Moreover, no specific libraries were used to implement the algorithm for solving differential equations.
fraction Biomass synthesis rate Production rate Step 1: initial estimates of metabolic fluxes are given following mass constraint Step 2: determination of metabolic fluxes to minimize the difference between simulated and measured isotopic labeling enrichment using the Levenberg-Marquardt method Step 1: initial estimates of metabolic fluxes are given at random following the constraints of the mass balance, biomass synthetic rate, and substrate consumption and production rate.
Step 2: metabolic fluxes are determined by minimizing the difference between simulated and measured isotopic labeling enrichment using the Levenberg-Marquardt method.

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 V is optimized to minimize the covariance-weighted sum of squared difference (SSD) using the Levenberg-Marquardt method [28] (Figure 2, Step 2): Here, MDV , = is the vector of experimental data at = . is the total number of measured metabolites for data fitting. is the total number of sampling points ( = 1 in the case of isotopically stationary), and MDV , = is the measurement covariance matrix with the measurement standard deviation located on the diagonal.

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 V opt, + and the objective function is reoptimized. Here, V opt, 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 Φ res, is the minimized SSD with one fixed flux, Φ res 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.

Implementation.
OpenMebius is a toolbox for conventional 13 C-MFA and INST-13 C-MFA using mass spectrometry data implemented in MATLAB on the Windows platform. Figure 3 shows a representative MATLAB code to perform INST-13 C-MFA on a simplified TCA cycle model mentioned below. A metabolic model is generated by the "ConstE-MUnetwork" function from user-defined 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 (Levenberg-Marquardt method). For a routine analysis, a batch execution of metabolic flux estimations is also supported. See Materials and Methods for detailed information.

Test Case of Isotopically Stationary MFA:
Simplified TCA Cycle Model. The performance of OpenMebius for conventional 13 C-MFA 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 (MDV ) of valine, lysine, aspartate, and   Figure 3: Representative Matlab code and structure of the metabolic model directory for isotopically nonstationary-13 C-metabolic flux analysis using OpenMebius. (a) A metabolic model "model" is generated from information described in "TCA cycle model" directory by "Modelconfig" and "ConstEMUnetwork" functions. After loading time course mass spectrometry data in "Artificial data" directory, isotopic labeling data of carbon sources and an initial metabolic flux distribution are prepared. A metabolic flux distribution ("Flux") is estimated by the "marquardt inst" function using nonlinear optimization (Levenberg-Marquardt method 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 C-MFA 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.

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 C-MFA 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 (V) and isotopic labeling patterns of substrate ( inp ) were identical to those of the previous section. The MDVs of oxaloacetate and succinate were sampled 17 times at 5-second intervals in silico, to which Gaussian noise (1%) was added to imitate actual measurements. Considering the simulated data as measured MDVs (MDV , = ), 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).
For a performance comparison between conventional 13 C-MFA and INST-13 C-MFA, the 95% confidence intervals of four representative reactions were determined by the grid search method (Figure 6 time points prepared above. In the case of conventional 13 C-MFA, a novel simulated dataset was prepared by the following procedure. From the MDV sim of oxaloacetate and succinate calculated using (1), 17 sets of simulated mass spectra (MDV ) 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 C-MFA were approximately 22% that of conventional 13 C-MFA (Figure 6(a)). The sharply curved parabolas were observed for INST-13 C-MFA, 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 C-MFA could be a reliable method to determine in vivo metabolic flux with narrow confidence intervals.

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 C-MFA from a user-defined metabolic network. Analysis using simulated data demonstrated not only the utility of OpenMebius for INST-13 C-MFA, but also its potential for use in metabolic flux analysis with reduced confidence intervals. OpenMebius provides an essential bioinformatics tool for INST-13 C-MFA to analyze metabolic flux in cells with slower metabolism (i.e., mammalian) [17] and cultivation with single carbon substrates (i.e., cyanobacteria) [15].