{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T20:59:10Z","timestamp":1765486750521,"version":"3.37.3"},"reference-count":38,"publisher":"Public Library of Science (PLoS)","issue":"3","license":[{"start":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T00:00:00Z","timestamp":1648080000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["MCB-1517671"],"award-info":[{"award-number":["MCB-1517671"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Graduate Assistance in Areas of National Need (GAANN) Fellowship","award":["P200A180093"],"award-info":[{"award-number":["P200A180093"]}]},{"name":"Brain & Behavior Institute, University of Maryland","award":["Brain and Behavior Initiative seed grant"],"award-info":[{"award-number":["Brain and Behavior Initiative seed grant"]}]},{"DOI":"10.13039\/100005825","name":"National Institute of Food and Agriculture","doi-asserted-by":"publisher","award":["2019-15 67015-29412"],"award-info":[{"award-number":["2019-15 67015-29412"]}],"id":[{"id":"10.13039\/100005825","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.ploscompbiol.org"],"crossmark-restriction":false},"short-container-title":["PLoS Comput Biol"],"abstract":"<jats:p>Stable isotope-assisted metabolic flux analysis (MFA) is a powerful method to estimate carbon flow and partitioning in metabolic networks. At its core, MFA is a parameter estimation problem wherein the fluxes and metabolite pool sizes are model parameters that are estimated, via optimization, to account for measurements of steady-state or isotopically-nonstationary isotope labeling patterns. As MFA problems advance in scale, they require efficient computational methods for fast and robust convergence. The structure of the MFA problem enables it to be cast as an equality-constrained nonlinear program (NLP), where the equality constraints are constructed from the MFA model equations, and the objective function is defined as the sum of squared residuals (SSR) between the model predictions and a set of labeling measurements. This NLP can be solved by using an algebraic modeling language (AML) that offers state-of-the-art optimization solvers for robust parameter estimation and superior scalability to large networks. When implemented in this manner, the optimization is performed with no distinction between state variables and model parameters. During each iteration of such an optimization, the system state is updated instead of being calculated explicitly from scratch, and this occurs concurrently with improvement in the model parameter estimates. This optimization approach starkly contrasts with traditional \u201cshooting\u201d methods where the state variables and model parameters are kept distinct and the system state is computed afresh during each iteration of a stepwise optimization. Our NLP formulation uses the MFA modeling framework of Wiechert et al. [1], which is amenable to incorporation of the model equations into an NLP. The NLP constraints consist of balances on either elementary metabolite units (EMUs) or cumomers. In this formulation, both the steady-state and isotopically-nonstationary MFA (inst-MFA) problems may be solved as an NLP. For the inst-MFA case, the ordinary differential equation (ODE) system describing the labeling dynamics is transcribed into a system of algebraic constraints for the NLP using collocation. This large-scale NLP may be solved efficiently using an NLP solver implemented on an AML. In our implementation, we used the reduced gradient solver CONOPT, implemented in the General Algebraic Modeling System (GAMS). The NLP framework is particularly advantageous for inst-MFA, scaling well to large networks with many free parameters, and having more robust convergence properties compared to the shooting methods that compute the system state and sensitivities at each iteration. Additionally, this NLP approach supports the use of tandem-MS data for both steady-state and inst-MFA when the cumomer framework is used. We assembled a software, eiFlux, written in Python and GAMS that uses the NLP approach and supports both steady-state and inst-MFA. We demonstrate the effectiveness of the NLP formulation on several examples, including a genome-scale inst-MFA model, to highlight the scalability and robustness of this approach. In addition to typical inst-MFA applications, we expect that this framework and our associated software, eiFlux, will be particularly useful for applying inst-MFA to complex MFA models, such as those developed for eukaryotes (e.g. algae) and co-cultures with multiple cell types.<\/jats:p>","DOI":"10.1371\/journal.pcbi.1009831","type":"journal-article","created":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T21:59:50Z","timestamp":1648159190000},"page":"e1009831","update-policy":"https:\/\/doi.org\/10.1371\/journal.pcbi.corrections_policy","source":"Crossref","is-referenced-by-count":11,"title":["Isotope-assisted metabolic flux analysis as an equality-constrained nonlinear program for improved scalability and robustness"],"prefix":"10.1371","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0943-9066","authenticated-orcid":true,"given":"Daniel J.","family":"Lugar","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6303-1328","authenticated-orcid":true,"given":"Ganesh","family":"Sriram","sequence":"additional","affiliation":[]}],"member":"340","published-online":{"date-parts":[[2022,3,24]]},"reference":[{"key":"pcbi.1009831.ref001","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1002\/(SICI)1097-0290(1999)66:2<69::AID-BIT1>3.0.CO;2-6","article-title":"Bidirectional reaction steps in metabolic networks: III. Explicit solution and analysis of isotopomer labeling systems","volume":"66","author":"W Wiechert","year":"1999","journal-title":"Biotechnol Bioeng"},{"key":"pcbi.1009831.ref002","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1016\/j.ymben.2020.11.002","article-title":"A guide to metabolic flux analysis in metabolic engineering: Methods, tools and applications","volume":"63","author":"MR Antoniewicz","year":"2021","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref003","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.copbio.2018.02.013","article-title":"Isotopically nonstationary metabolic flux analysis (INST-MFA): putting theory into practice","volume":"54","author":"YE Cheah","year":"2018","journal-title":"Curr Opin Biotechnol"},{"key":"pcbi.1009831.ref004","doi-asserted-by":"crossref","first-page":"979","DOI":"10.1016\/j.copbio.2013.03.024","article-title":"Isotopically non-stationary metabolic flux analysis: complex yet highly informative","volume":"24","author":"W Wiechert","year":"2013","journal-title":"Curr Opin Biotechnol"},{"key":"pcbi.1009831.ref005","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.ymben.2006.09.001","article-title":"Elementary metabolite units (EMU): A novel framework for modeling isotopic distributions","volume":"9","author":"MR Antoniewicz","year":"2007","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref006","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1016\/j.ymben.2004.02.003","article-title":"Improvements in metabolic flux analysis using carbon bond labeling experiments: bondomer balancing and Boolean function mapping","volume":"6","author":"G Sriram","year":"2004","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref007","doi-asserted-by":"crossref","first-page":"731","DOI":"10.1002\/bit.10429","article-title":"Cumulative bondomers: A new concept in flux analysis from 2D [13C, 1H] COSY NMR data","volume":"80","author":"WA van Winden","year":"2002","journal-title":"Biotechnol Bioeng"},{"key":"pcbi.1009831.ref008","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1752-0509-5-129","article-title":"Fluxomers: a new approach for 13C metabolic flux analysis","volume":"5","author":"O Srour","year":"2011","journal-title":"BMC Syst Biol"},{"key":"pcbi.1009831.ref009","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1002\/bit.21632","article-title":"An elementary metabolite unit (EMU) based method of isotopically nonstationary flux analysis","volume":"99","author":"JD Young","year":"2008","journal-title":"Biotechnol Bioeng"},{"key":"pcbi.1009831.ref010","doi-asserted-by":"crossref","first-page":"e1006828","DOI":"10.1371\/journal.pcbi.1006828","article-title":"Scalable nonlinear programming framework for parameter estimation in dynamic biological system models","volume":"15","author":"S Shin","year":"2019","journal-title":"PLOS Comput Biol"},{"key":"pcbi.1009831.ref011","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"S Kirkpatrick","year":"1983","journal-title":"Science"},{"key":"pcbi.1009831.ref012","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.biosystems.2005.06.016","article-title":"A hybrid approach for efficient and robust parameter estimation in biochemical pathways","volume":"83","author":"M Rodriguez-Fernandez","year":"2006","journal-title":"Biosystems"},{"key":"pcbi.1009831.ref013","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1049\/iet-syb:20060067","article-title":"Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting","volume":"1","author":"M Peifer","year":"2007","journal-title":"IET Syst Biol"},{"key":"pcbi.1009831.ref014","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898718577","volume-title":"Practical Methods for Optimal Control and Estimation Using Nonlinear Programming","author":"JT Betts","year":"2010","edition":"2"},{"key":"pcbi.1009831.ref015","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/BF02591747","article-title":"CONOPT: A GRG code for large sparse dynamic nonlinear optimization problems","volume":"31","author":"A. Drud","year":"1985","journal-title":"Math Program."},{"key":"pcbi.1009831.ref016","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1287\/ijoc.6.2.207","article-title":"CONOPT\u2014A Large-Scale GRG Code","volume":"6","author":"A. Drud","year":"1994","journal-title":"ORSA J Comput"},{"key":"pcbi.1009831.ref017","unstructured":"General Algebraic Modeling System (GAMS). 2751 Prosperity Ave, Suite 210, Fairfax VA 22031: GAMS Development Corporation. Fairfax VA, USA; 2021."},{"key":"pcbi.1009831.ref018","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.ymben.2010.11.006","article-title":"Tandem mass spectrometry: A novel approach for metabolic flux analysis","volume":"13","author":"J Choi","year":"2011","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref019","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1002\/(SICI)1097-0290(1999)66:2<86::AID-BIT2>3.0.CO;2-A","article-title":"Bidirectional reaction steps in metabolic networks: IV. Optimal design of isotopomer labeling experiments","volume":"66","author":"M M\u00f6llney","year":"1999","journal-title":"Biotechnol Bioeng"},{"volume-title":"Optimization","year":"1969","author":"J Abadie","key":"pcbi.1009831.ref020"},{"key":"pcbi.1009831.ref021","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1016\/j.ymben.2006.05.006","article-title":"Computational tools for isotopically instationary 13C labeling experiments under metabolic steady state conditions","volume":"8","author":"K N\u00f6h","year":"2006","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref022","first-page":"145","article-title":"From stationary to instationary metabolic flux analysis","volume":"92","author":"W Wiechert","year":"2005","journal-title":"Adv Biochem Eng Biotechnol"},{"key":"pcbi.1009831.ref023","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1109\/TCBB.2016.2544299","article-title":"Metabolic Flux Analysis in Isotope Labeling Experiments Using the Adjoint Approach","volume":"14","author":"S Mottelet","year":"2017","journal-title":"IEEE\/ACM Trans Comput Biol Bioinform"},{"key":"pcbi.1009831.ref024","doi-asserted-by":"crossref","first-page":"1333","DOI":"10.1093\/bioinformatics\/btu015","article-title":"INCA: a computational platform for isotopically non-stationary metabolic flux analysis","volume":"30","author":"JD Young","year":"2014","journal-title":"Bioinformatics"},{"key":"pcbi.1009831.ref025","doi-asserted-by":"crossref","first-page":"e627014","DOI":"10.1155\/2014\/627014","article-title":"OpenMebius: An Open Source Software for Isotopically Nonstationary 13C-Based Metabolic Flux Analysis","volume":"2014","author":"S Kajihata","year":"2014","journal-title":"BioMed Res Int"},{"key":"pcbi.1009831.ref026","doi-asserted-by":"crossref","first-page":"e1005331","DOI":"10.1371\/journal.pcbi.1005331","article-title":"Scalable Parameter Estimation for Genome-Scale Biochemical Reaction Networks","volume":"13","author":"F Fr\u00f6hlich","year":"2017","journal-title":"PLOS Comput Biol"},{"key":"pcbi.1009831.ref027","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1021\/ie00050a015","article-title":"Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equation systems","volume":"30","author":"IB Tjoa","year":"1991","journal-title":"Ind Eng Chem Res"},{"key":"pcbi.1009831.ref028","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0377-0427(99)00134-X","article-title":"Stiff differential equations solved by Radau methods","volume":"111","author":"E Hairer","year":"1999","journal-title":"J Comput Appl Math"},{"volume-title":"19th AIAA Computational Fluid Dynamics","year":"2009","author":"HT Huynh","key":"pcbi.1009831.ref029"},{"key":"pcbi.1009831.ref030","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.copbio.2012.10.011","article-title":"Tandem mass spectrometry for measuring stable-isotope labeling","volume":"24","author":"MR Antoniewicz","year":"2013","journal-title":"Curr Opin Biotechnol"},{"key":"pcbi.1009831.ref031","doi-asserted-by":"crossref","first-page":"4628","DOI":"10.1021\/ac300611n","article-title":"Measuring Complete Isotopomer Distribution of Aspartate Using Gas Chromatography\/Tandem Mass Spectrometry","volume":"84","author":"J Choi","year":"2012","journal-title":"Anal Chem"},{"key":"pcbi.1009831.ref032","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1002\/bit.24344","article-title":"Collisional fragmentation of central carbon metabolites in LC-MS\/MS increases precision of 13C metabolic flux analysis","volume":"109","author":"M R\u00fchl","year":"2012","journal-title":"Biotechnol Bioeng"},{"key":"pcbi.1009831.ref033","volume-title":"Numerical Recipes 3rd Edition: The Art of Scientific Computing","author":"WH Press","year":"2007","edition":"3"},{"key":"pcbi.1009831.ref034","doi-asserted-by":"crossref","first-page":"1496","DOI":"10.1039\/C3MB70348G","article-title":"Flux and reflux: metabolite reflux in plant suspension cells and its implications for isotope-assisted metabolic flux analysis","volume":"10","author":"S Nargund","year":"2014","journal-title":"Mol Biosyst"},{"key":"pcbi.1009831.ref035","doi-asserted-by":"crossref","first-page":"e1004321","DOI":"10.1371\/journal.pcbi.1004321","article-title":"Escher: a web application for building, sharing, and embedding data-rich visualizations of biological pathways","volume":"11","author":"ZA King","year":"2015","journal-title":"PLOS Comput Biol"},{"volume-title":"Plant Metabolism: Methods and Protocols","year":"2014","author":"JD Young","key":"pcbi.1009831.ref036"},{"key":"pcbi.1009831.ref037","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1016\/j.ymben.2018.03.008","article-title":"Elucidation of photoautotrophic carbon flux topology in Synechocystis PCC 6803 using genome-scale carbon mapping models","volume":"47","author":"S Gopalakrishnan","year":"2018","journal-title":"Metab Eng"},{"key":"pcbi.1009831.ref038","doi-asserted-by":"crossref","first-page":"656","DOI":"10.1016\/j.ymben.2011.08.002","article-title":"Mapping photoautotrophic metabolism with isotopically nonstationary 13C flux analysis","volume":"13","author":"JD Young","year":"2011","journal-title":"Metab Eng"}],"container-title":["PLOS Computational Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009831","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,3,24]],"date-time":"2022-03-24T22:00:50Z","timestamp":1648159250000},"score":1,"resource":{"primary":{"URL":"https:\/\/dx.plos.org\/10.1371\/journal.pcbi.1009831"}},"subtitle":[],"editor":[{"given":"Costas D.","family":"Maranas","sequence":"first","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2022,3,24]]},"references-count":38,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3,24]]}},"URL":"https:\/\/doi.org\/10.1371\/journal.pcbi.1009831","relation":{},"ISSN":["1553-7358"],"issn-type":[{"type":"electronic","value":"1553-7358"}],"subject":[],"published":{"date-parts":[[2022,3,24]]}}}