{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T19:55:54Z","timestamp":1790625354016,"version":"4.1.0"},"reference-count":53,"publisher":"National Academy of Sciences","issue":"40","license":[{"start":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T00:00:00Z","timestamp":1790553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,10,6]]},"abstract":"<jats:p>\n                    Energy metabolism matches adenosine triphosphate (ATP) resynthesis to hydrolysis even during abrupt transitions such as exercise onset. In 1984, Alois Mader proposed a mechanistic model that captures this regulation through 33 coupled differential equations linking oxidative phosphorylation, glycolysis, and the phosphocreatine shuttle via adenosine diphosphate (ADP), phosphate, and pH feedback. Despite its potential, broader adoption has been limited by closed-source implementations and key publications available only in German. Here, we present MetaboliSim, an open-source software tool that simulates muscle energy metabolism in a virtual human parameterized by four physiological inputs: V\u0307O\n                    <jats:sub>2<\/jats:sub>\n                    max (oxidative capacity),\n                    <jats:italic toggle=\"yes\">v<\/jats:italic>\n                    Lamax (glycolytic capacity), body mass, and active muscle mass. Using MetaboliSim, we test Mader\u2019s equations against 10 experimentally established phenomena of exercise metabolism, including ATP homeostasis under fatigue, transient pH alkalinization, glycogen-dependent lactate thresholds, intensity-dependent fat oxidation, the V\u0307O\n                    <jats:sub>2<\/jats:sub>\n                    slow component, and work-rate-dependent exhaustion, without altering any equation or constant within the model. Mader\u2019s model qualitatively reproduces all 10 phenomena, generating similarly shaped response curves; absolute values depend on individual parameterization rather than being universally matched. The characteristic inverted-U of fat oxidation and the exercise intensity-dependent V\u0307O\n                    <jats:sub>2<\/jats:sub>\n                    kinetics, for example, both emerge from the same equation system. A Sobol sensitivity analysis (S01) reveals a calibration hierarchy: V\u0307O\n                    <jats:sub>2<\/jats:sub>\n                    max and body mass drive power predictions;\n                    <jats:italic toggle=\"yes\">v<\/jats:italic>\n                    Lamax and buffer capacity drive metabolite predictions. That a compact, 40-year-old equation system accounts for this breadth of metabolic phenomena suggests that a small number of regulatory feedback loops may suffice to explain a wide range of metabolic responses to exercise. MetaboliSim is freely available to enable independent testing, empirical calibration, and refinement.\n                  <\/jats:p>","DOI":"10.1073\/pnas.2525555123","type":"journal-article","created":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T19:02:56Z","timestamp":1790622176000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["The Mader model of muscle energy metabolism simulates key metabolic exercise phenomena"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0014-5878","authenticated-orcid":false,"given":"Jeffrey A.","family":"Rothschild","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01zvqw119","id-type":"ROR","asserted-by":"publisher"}],"name":"Sports Performance Research Institute New Zealand, Auckland University of Technology","place":["Auckland, New Zealand"]},{"name":"High Performance Sport New Zealand","place":["Auckland, New Zealand"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0954-3533","authenticated-orcid":false,"given":"Anna Katharina","family":"Dunst","sequence":"additional","affiliation":[{"name":"Institute for Applied Training Science","place":["Leipzig, Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8013-9576","authenticated-orcid":false,"given":"Jessie","family":"Axsom","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01z7r7q48","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Pathology and Laboratory Medicine, Children\u2019s Hospital of Philadelphia","place":["Philadelphia"]},{"id":[{"id":"https:\/\/ror.org\/00b30xv10","id-type":"ROR","asserted-by":"publisher"}],"name":"Department of Neurology, University of Pennsylvania","place":["Philadelphia"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5920-5842","authenticated-orcid":false,"given":"Henning","family":"Wackerhage","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02kkvpp62","id-type":"ROR","asserted-by":"publisher"}],"name":"School of Medicine and Health, Technical University of Munich","place":["Munich, Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9716-1697","authenticated-orcid":false,"given":"Hermann","family":"Heck","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04tsk2644","id-type":"ROR","asserted-by":"publisher"}],"name":"Professor Emeritus, Faculty of Sports Science, Ruhr University Bochum","place":["Bochum, Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,9,28]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cell.2020.04.043"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1038\/s41586-023-06877-w"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/d41586-021-02480-z"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1038\/s42255-020-0251-4"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cmet.2024.09.010"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.1186\/s40798-026-00989-z"},{"key":"e_1_3_4_7_2","volume-title":"Eine Theorie zur Berechnung der Dynamik und des steady state von Phosphorylierungsszzustand und Stoffwechselaktivit\u00e4t der Muskelzelle als Folge des Energiebedarfs","author":"Mader A.","year":"1984","unstructured":"A. Mader, Eine Theorie zur Berechnung der Dynamik und des steady state von Phosphorylierungsszzustand und Stoffwechselaktivit\u00e4t der Muskelzelle als Folge des Energiebedarfs (Cologne, 1984)."},{"key":"e_1_3_4_8_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00421-002-0676-3"},{"key":"e_1_3_4_9_2","doi-asserted-by":"publisher","DOI":"10.1113\/JP281142"},{"key":"e_1_3_4_10_2","doi-asserted-by":"publisher","DOI":"10.3389\/fphys.2022.899670"},{"key":"e_1_3_4_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/s40279-025-02259-6"},{"key":"e_1_3_4_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9258(20)79895-9"},{"key":"e_1_3_4_13_2","first-page":"iv","article-title":"The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves","volume":"40","author":"Hill A. V.","year":"1910","unstructured":"A. V. Hill, The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J. Physiol. 40, iv\u2013vii (1910).","journal-title":"J. Physiol."},{"key":"e_1_3_4_14_2","volume-title":"Computersimulation. M\u00f6glichkeiten zur Theoriebildung und Ergebnisinterpretation","author":"Mader A.","year":"1994","unstructured":"A. Mader, H. Heck, \u201cEnergiestoffwechselregulation, Erweiterungen des theoretischen Konzepts und seiner Begr\u00fcndungen. Nachweis der praktischen N\u00fctzlichkeit der Simulation des Energiestoffwechsels\u201d in Computersimulation. M\u00f6glichkeiten zur Theoriebildung und Ergebnisinterpretation, A. Mader, H. Allmer, Eds. (Academia Verlag Sankt Augustin, 1994)."},{"key":"e_1_3_4_15_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.78.11.6714"},{"key":"e_1_3_4_16_2","first-page":"77","article-title":"Bioenergetics of intact human muscle. A 31P nuclear magnetic resonance study","volume":"1","author":"Taylor D. J.","year":"1983","unstructured":"D. J. Taylor, P. J. Bore, P. Styles, D. G. Gadian, G. K. 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Dtsch Z Sportmed 33, 304\u2013307 (1982).","journal-title":"Dtsch Z Sportmed"},{"key":"e_1_3_4_45_2","doi-asserted-by":"publisher","DOI":"10.1113\/expphysiol.2009.050500"},{"key":"e_1_3_4_46_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1748-1716.1985.tb07735.x"},{"key":"e_1_3_4_47_2","doi-asserted-by":"publisher","DOI":"10.1123\/ijspp.2025-0004"},{"key":"e_1_3_4_48_2","doi-asserted-by":"publisher","DOI":"10.1152\/japplphysiol.01063.2016"},{"key":"e_1_3_4_49_2","doi-asserted-by":"publisher","DOI":"10.1152\/physrev.00033.2024"},{"key":"e_1_3_4_50_2","unstructured":"K. Dunst V. Scharf C. Hesse A. Asteroth MetaboliSim: A Python implementation of the Mader model for dynamic and steady-state simulation of muscular energy metabolism. arXiv [Preprint] (2026). https:\/\/doi.org\/10.48550\/arXiv.2606.08366 (Accessed 10 September 2026)."},{"key":"e_1_3_4_51_2","first-page":"407","article-title":"Sensitivity estimates for nonlinear mathematical models","volume":"1","author":"Sobol M.","year":"1993","unstructured":"M. Sobol, Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407\u2013414 (1993).","journal-title":"Math. Model. Comput. Exp."},{"key":"e_1_3_4_52_2","unstructured":"K. Dunst V. Scharf C. Hesse A. Asteroth MetaboliSim: A Python implementation of the Mader model for dynamic and steady-state simulation of muscular energy metabolism (v1.0-pnas). Codeberg. https:\/\/codeberg.org\/3phos\/metabolisim. Deposited 10 September 2026."},{"key":"e_1_3_4_53_2","unstructured":"K. Dunst Data and analysis scripts for PNAS manuscript #2025-25555R (v1.0). Codeberg. https:\/\/codeberg.org\/KDunst\/PNAS-2025-25555R. Deposited 10 September 2026."}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.2525555123","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,28]],"date-time":"2026-09-28T19:04:35Z","timestamp":1790622275000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/10.1073\/pnas.2525555123"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,28]]},"references-count":53,"journal-issue":{"issue":"40","published-print":{"date-parts":[[2026,10,6]]}},"alternative-id":["10.1073\/pnas.2525555123"],"URL":"https:\/\/doi.org\/10.1073\/pnas.2525555123","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,28]]},"assertion":[{"value":"2025-09-15","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-07-01","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2026-09-28","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"e2525555123"}}