{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:50:14Z","timestamp":1760147414919,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T00:00:00Z","timestamp":1675036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we investigate some theoretical aspects related to the estimation approach proposed by Liebermeister and Klipp, 2006, in which general rate laws, derived from standardized enzymatic mechanisms, are exploited to kinetically describe the fluxes of a metabolic reaction network, and multiple metabolic steady-state measurements are exploited to estimate the unknown kinetic parameters. Further mathematical details are deeply investigated, and necessary conditions on the amount of information required to solve the identification problem are given. Moreover, theoretical results for the parameter identifiability are provided, and symmetrical and modular properties of the proposed approach are highlighted when the global identification problem is decoupled into smaller and simpler identification problems related to the single reactions of the network. Among the advantages of the proposed innovative approach are (i) non-restrictive conditions to guarantee the solvability of the parameter estimation problem, (ii) the unburden of the usual computational complexity for such identification problems, and (iii) the ease of obtaining the required number of measurements, which are actually steady-state data, experimentally easier to obtain with respect to the time-dependent ones. A simple example concludes the paper, highlighting the mentioned advantages of the method and the implementation of the related theoretical result.<\/jats:p>","DOI":"10.3390\/sym15020368","type":"journal-article","created":{"date-parts":[[2023,1,30]],"date-time":"2023-01-30T07:34:41Z","timestamp":1675064081000},"page":"368","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Parameter Identification in Metabolic Reaction Networks by Means of Multiple Steady-State Measurements"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2061-0239","authenticated-orcid":false,"given":"Giovanni","family":"Palombo","sequence":"first","affiliation":[{"name":"Institute for Systems Analysis and Computer Science \u2018A. Ruberti\u2019 (IASI-CNR), Via dei Taurini 19, 00185 Rome, Italy"},{"name":"SYSBIO\u2014Centre of Systems Biology, Piazza della Scienza 2, 20126 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6678-048X","authenticated-orcid":false,"given":"Alessandro","family":"Borri","sequence":"additional","affiliation":[{"name":"Institute for Systems Analysis and Computer Science \u2018A. Ruberti\u2019 (IASI-CNR), Via dei Taurini 19, 00185 Rome, Italy"},{"name":"Center of Excellence for Research DEWS, University of L\u2019Aquila, Via Vetoio 1, 67100 L\u2019Aquila, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0263-5697","authenticated-orcid":false,"given":"Federico","family":"Papa","sequence":"additional","affiliation":[{"name":"Institute for Systems Analysis and Computer Science \u2018A. Ruberti\u2019 (IASI-CNR), Via dei Taurini 19, 00185 Rome, Italy"},{"name":"SYSBIO\u2014Centre of Systems Biology, Piazza della Scienza 2, 20126 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marco","family":"Papi","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Campus Biomedico University of Rome, Via \u00c1lvaro del Portillo 21, 00128 Rome, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9371-2802","authenticated-orcid":false,"given":"Pasquale","family":"Palumbo","sequence":"additional","affiliation":[{"name":"SYSBIO\u2014Centre of Systems Biology, Piazza della Scienza 2, 20126 Milan, Italy"},{"name":"Department of Biotechnology and Biosciences, University of Milano-Bicocca, Piazza della Scienza 2, 20126 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1016\/j.biotechadv.2017.09.005","article-title":"Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks","volume":"35","author":"Saa","year":"2017","journal-title":"Biotechnol. Adv."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1038\/nature01149","article-title":"Escherichia coli K-12 undergoes adaptive evolution to achieve Silico predicted optimal growth","volume":"420","author":"Ibarra","year":"2002","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1126\/science.1216882","article-title":"Multidimensional optimality of microbial metabolism","volume":"336","author":"Schuetz","year":"2012","journal-title":"Science"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Li, C., Donizelli, M., Rodriguez, N., Dharuri, H., Endler, L., Chelliah, V., Li, L., He, E., Henry, A., and Stefan, M.I. (2010). BioModels Database: An enhanced, curated and annotated resource for published quantitative kinetic models. BMC Syst. Biol., 4.","DOI":"10.1186\/1752-0509-4-92"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1186\/1742-4682-3-41","article-title":"Bringing metabolic networks to life: Convenience rate law and thermodynamic constraints","volume":"3","author":"Liebermeister","year":"2006","journal-title":"Theor. Biol. Med. Model."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1186\/1742-4682-3-42","article-title":"Bringing metabolic networks to life: Integration of kinetic, metabolic, and proteomic data","volume":"3","author":"Liebermeister","year":"2006","journal-title":"Theor. Biol. Med. Model."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1038\/msb4100155","article-title":"A genome-scale metabolic reconstruction for Escherichia coli K-12 MG1655 that accounts for 1260 ORFs and thermodynamic information","volume":"3","author":"Feist","year":"2007","journal-title":"Mol. Syst. Biol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1155","DOI":"10.1038\/nbt1492","article-title":"A consensus yeast metabolic network reconstruction obtained from a community approach to systems biology","volume":"26","author":"Herrgard","year":"2008","journal-title":"Nat. Biotechnol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"659","DOI":"10.1038\/nbt1401","article-title":"The growing scope of applications of genome-scale metabolic reconstructions using Escherichia coli","volume":"26","author":"Feist","year":"2008","journal-title":"Nat. Biotechnol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5606","DOI":"10.1529\/biophysj.108.135442","article-title":"Ensemble modeling of metabolic networks","volume":"95","author":"Tran","year":"2008","journal-title":"Biophys. J."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"16298","DOI":"10.1021\/jp108764b","article-title":"Parameter Balancing in Kinetic Models of Cell Metabolism","volume":"114","author":"Lubitz","year":"2010","journal-title":"J. Phys. Chem. B"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Stanford, N.J., Lubitz, T., Smallbone, K., Klipp, E., Mendes, P., and Liebermeister, W. (2013). Systematic Construction of Kinetic Models from Genome-Scale Metabolic Networks. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0079195"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1007\/s10295-015-1585-x","article-title":"Methods and advances in metabolic flux analysis: A mini-review","volume":"42","author":"Antoniewicz","year":"2015","journal-title":"J. Ind. Microbiol. Biotechnol."},{"key":"ref_14","unstructured":"Klipp, E., Liebermeister, W., Wierling, C., Kowald, A., Lehrach, H., and Herwig, R. (2009). Systems Biology: A Textbook, Wiley."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Noor, E., Flamholz, A., Bar-Even, A., Davidi, D., Milo, R., and Liebermeister, W. (2016). The protein cost of metabolic fluxes: Prediction from enzymatic rate laws and cost minimization. PLoS Comput. Biol., 12.","DOI":"10.1371\/journal.pcbi.1005167"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1038\/nbt.1614","article-title":"What is flux balance analysis?","volume":"28","author":"Orth","year":"2010","journal-title":"Nat. Biotechnol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1006\/mben.2001.0187","article-title":"13C metabolic flux analysis","volume":"3","author":"Wiechert","year":"2001","journal-title":"Metab. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1186\/1475-2859-8-25","article-title":"OpenFLUX: Efficient modelling software for 13C-based metabolic flux analysis","volume":"8","author":"Quek","year":"2009","journal-title":"Microb. Cell Factories"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1093\/bioinformatics\/bts646","article-title":"13CFLUX2\u2014High-performance software suite for 13C-metabolic flux analysis","volume":"29","author":"Weitzel","year":"2013","journal-title":"Bioinformatics"},{"key":"ref_20","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":"Young","year":"2014","journal-title":"Bioinformatics"},{"key":"ref_21","unstructured":"Allendoerfer, C.B. (1974). Calculus of Several Variables and Differentiable Manifolds, Macmillan."},{"key":"ref_22","unstructured":"Ross, J., and Nystr\u00f6m, D.W. (2018). Differentiability of the argmin function and a minimum principle for semiconcave subsolutions. arXiv."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/368\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:19:52Z","timestamp":1760120392000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/368"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,30]]},"references-count":22,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020368"],"URL":"https:\/\/doi.org\/10.3390\/sym15020368","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,1,30]]}}}