Atom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks

Atom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Atom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks Richard Golnik, Peter Florian Stadler, Thomas Gatter This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5888287/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Isotope tracing experiments are an important source of information in particular for the analysis of complex metabolic networks. Their evaluation, however, is a notoriously difficult and time-consuming task. Here, we derive a simple and complete solution for an important special case, namely the specific labeling of a single atom in a single feed compound. Starting from the atom-to-atom map of a reaction we first derive a transition matrix that exactly describes the propagation of the label from reactants to products, taking into account the relevant symmetries. These are subsequently combined into a global atom transition network. Assuming a steady-state flux F through the chemical reaction network we derive a system of affine differential equations describing how the reaction network is flooded by labeled atoms from an external reservoir. This leads to a unique asymptotically stable steady state distribution of labels that can be computed by solving a non-singular system of linear equations. Linear combinations of these single-atom labeling patterns also solve the problem of computing the enrichment of multiple, simultaneous labels, albeit without providing information on correlations between distinct labels. Isotope tracing Positional Enrichment Atom transition networks Symmetries in atom-atom maps Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5888287","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":406213766,"identity":"5d307b2a-0ff4-4236-82a6-4e415ec93e36","order_by":0,"name":"Richard Golnik","email":"data:image/png;base64,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","orcid":"","institution":"Leipzig University","correspondingAuthor":true,"prefix":"","firstName":"Richard","middleName":"","lastName":"Golnik","suffix":""},{"id":406213767,"identity":"b0074ce7-7617-4d0a-bc0c-3bba4f4c73eb","order_by":1,"name":"Peter Florian Stadler","email":"","orcid":"","institution":"Leipzig University","correspondingAuthor":false,"prefix":"","firstName":"Peter","middleName":"Florian","lastName":"Stadler","suffix":""},{"id":406213768,"identity":"09a01c6f-a3c3-4837-8bd5-70e93532bed9","order_by":2,"name":"Thomas Gatter","email":"","orcid":"","institution":"Leipzig University","correspondingAuthor":false,"prefix":"","firstName":"Thomas","middleName":"","lastName":"Gatter","suffix":""}],"badges":[],"createdAt":"2025-01-23 12:23:17","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5888287/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5888287/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":84923608,"identity":"c85bd632-ef87-4746-9f07-645d7115d366","added_by":"auto","created_at":"2025-06-18 20:53:27","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1090568,"visible":true,"origin":"","legend":"","description":"","filename":"IceTM.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5888287/v1_covered_0ada535d-a75c-4d99-9c51-c58a2e2b40d8.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eAtom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Isotope tracing, Positional Enrichment, Atom transition networks, Symmetries in atom-atom maps","lastPublishedDoi":"10.21203/rs.3.rs-5888287/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5888287/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eIsotope tracing experiments are an important source of information in\u0026nbsp; particular for the analysis of complex metabolic networks. Their\u0026nbsp; evaluation, however, is a notoriously difficult and time-consuming task.\u0026nbsp; Here, we derive a simple and complete solution for an important special\u0026nbsp; case, namely the specific labeling of a single atom in a single feed\u0026nbsp; compound. Starting from the atom-to-atom map of a reaction we first\u0026nbsp; derive a transition matrix that exactly describes the propagation of the\u0026nbsp; label from reactants to products, taking into account the relevant\u0026nbsp; symmetries. These are subsequently combined into a global atom transition\u0026nbsp; network. Assuming a steady-state flux \u003cem\u003eF\u003c/em\u003e through the chemical\u0026nbsp; reaction network we derive a system of affine differential equations\u0026nbsp; describing how the reaction network is flooded by labeled atoms from an\u0026nbsp; external reservoir. This leads to a unique asymptotically stable steady\u0026nbsp; state distribution of labels that can be computed by solving a\u0026nbsp; non-singular system of linear equations. Linear combinations of these\u0026nbsp; single-atom labeling patterns also solve the problem of computing the\u0026nbsp; enrichment of multiple, simultaneous labels, albeit without providing\u0026nbsp; information on correlations between distinct labels. \u0026nbsp;\u003c/p\u003e","manuscriptTitle":"Atom Transition Networks and Isotope Labeling Patterns in Large Chemical Reaction Networks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-01-28 04:07:28","doi":"10.21203/rs.3.rs-5888287/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3637e004-b24a-4c39-a206-94320d01f4c2","owner":[],"postedDate":"January 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-06-18T20:53:13+00:00","versionOfRecord":[],"versionCreatedAt":"2025-01-28 04:07:28","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5888287","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5888287","identity":"rs-5888287","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}