Using MR-Chordless Circuits for Efficient Enumeration of Autocatalytic Cores in Large Chemical Reaction Networks

Using MR-Chordless Circuits for Efficient Enumeration of Autocatalytic Cores 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 Using MR-Chordless Circuits for Efficient Enumeration of Autocatalytic Cores in Large Chemical Reaction Networks Richard Golnik, Thomas Gatter, Nicola Vassena, Peter Florian Stadler This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8659930/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 11 You are reading this latest preprint version Abstract Autocatalysis is an important property of chemical reaction networks (CRNs) that is particularly prevalent in metabolic networks. A set of well-defined autocatalytic cores prominently features minimal subsystems that determine the autocatalytic capabilities. Recently, a graph-theoretic characterization has become available that enabled the enumeration of moderate size autocatalytic cores in real-life metabolic networks. Here, we improve on this approach in two ways: (1) We elaborate on algorithms for the enumeration of elementary circuits restricted to so-called MR-chordless circuits relevant for autocatalysis detection. (2) We interleave our new algorithm with tests for autocatalysis to further limit the number of circuits that need to be stored for the construction of autocatalytic cores more complex than elementary MR-chordless circuits. Combined, these innovations achieve a performance gain of several orders of magnitude and make it possible to exhaustively enumerate all autocatalytic cores in real-life metabolic reaction networks comprising several hundreds of metabolites and reactions. Importantly, we find that reaction networks with irreversible reactions contain complex autocatalytic cores comprising more than a single "cycle with an ear". Such structures exceed the established classification of autocatalytic cores for fully reversible networks into five types. Scientific contribution We developed a graph-theoretic algorithm to enumerate autocatalytic cores that is efficient enough to handle genome-scale metabolic models of various species exhaustively. To this end, we devised a novel method to enumerate elementary circuits in bipartite digraphs free of one particular type of chord, which outperforms efficient standard enumeration algorithms of elementary circuits by up to 4 orders of magnitude. We show that for CRNs comprising both reducible and irreducible reactions more complex autocatalytic cores exist than envisioned by the established classification schemes for reversible CRNs. Full Text Additional Declarations No competing interests reported. Supplementary Files SupplementaryInformation.pdf Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 04 Apr, 2026 Reviews received at journal 26 Mar, 2026 Reviews received at journal 25 Mar, 2026 Reviewers agreed at journal 23 Feb, 2026 Reviewers agreed at journal 21 Feb, 2026 Reviewers agreed at journal 17 Feb, 2026 Reviewers agreed at journal 08 Feb, 2026 Reviewers invited by journal 04 Feb, 2026 Editor assigned by journal 02 Feb, 2026 Submission checks completed at journal 28 Jan, 2026 First submitted to journal 27 Jan, 2026 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. 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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-8659930","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":584574912,"identity":"24f6647e-fb1d-4637-8e42-7f785d010a66","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":584574914,"identity":"3f0d16f6-fa79-4f56-b3b9-6a187d0c642a","order_by":1,"name":"Thomas Gatter","email":"","orcid":"","institution":"Leipzig University","correspondingAuthor":false,"prefix":"","firstName":"Thomas","middleName":"","lastName":"Gatter","suffix":""},{"id":584574916,"identity":"e37e9fc9-6430-43d4-8645-00197b10ea22","order_by":2,"name":"Nicola Vassena","email":"","orcid":"","institution":"Leipzig University","correspondingAuthor":false,"prefix":"","firstName":"Nicola","middleName":"","lastName":"Vassena","suffix":""},{"id":584574918,"identity":"56bdc516-b3c1-40f5-9314-6d3870ecef32","order_by":3,"name":"Peter Florian Stadler","email":"","orcid":"","institution":"Leipzig University","correspondingAuthor":false,"prefix":"","firstName":"Peter","middleName":"Florian","lastName":"Stadler","suffix":""}],"badges":[],"createdAt":"2026-01-21 12:39:39","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8659930/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8659930/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":101880804,"identity":"fd2a00c8-5950-4b89-b0e1-608bad760511","added_by":"auto","created_at":"2026-02-04 15:06:34","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3137608,"visible":true,"origin":"","legend":"","description":"","filename":"JCheminfAutogatito.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8659930/v1_covered_3b239946-05fc-4b6b-bab5-fb3d9111de22.pdf"},{"id":101783099,"identity":"92088828-8744-4b25-b3a1-ff83d7444894","added_by":"auto","created_at":"2026-02-03 15:18:51","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":125362,"visible":true,"origin":"","legend":"","description":"","filename":"SupplementaryInformation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8659930/v1/3d33cbe27c224b15814972bd.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Using MR-Chordless Circuits for Efficient Enumeration of Autocatalytic Cores in Large Chemical Reaction Networks","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"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":"journal-of-cheminformatics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"chin","sideBox":"Learn more about [Journal of Cheminformatics](https://jcheminf.biomedcentral.com/)","snPcode":"13321","submissionUrl":"https://submission.nature.com/new-submission/13321/3","title":"Journal of Cheminformatics","twitterHandle":"@jcheminf","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-8659930/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8659930/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAutocatalysis is an important property of chemical reaction\u0026nbsp; networks (CRNs) that is particularly prevalent in metabolic networks. A set of\u0026nbsp; well-defined autocatalytic cores prominently features minimal\u0026nbsp; subsystems that determine the autocatalytic capabilities. Recently, a\u0026nbsp; graph-theoretic characterization has become available that enabled the\u0026nbsp; enumeration of moderate size autocatalytic cores in real-life metabolic\u0026nbsp; networks. Here, we improve on this approach in two ways: (1) We elaborate\u0026nbsp; on algorithms for the enumeration of elementary circuits restricted to\u0026nbsp; so-called MR-chordless circuits relevant for autocatalysis\u0026nbsp; detection. (2) We interleave our new algorithm with tests for\u0026nbsp; autocatalysis to further limit the number of circuits that need to be\u0026nbsp; stored for the construction of autocatalytic cores more complex than\u0026nbsp; elementary MR-chordless circuits. Combined, these innovations achieve a\u0026nbsp; performance gain of several orders of magnitude and make it possible to\u0026nbsp; exhaustively enumerate all autocatalytic cores in real-life\u0026nbsp; metabolic reaction networks comprising several hundreds of metabolites\u0026nbsp; and reactions. Importantly, we find that reaction networks with\u0026nbsp; irreversible reactions contain complex autocatalytic cores comprising\u0026nbsp; more than a single \"cycle with an ear\". Such structures exceed the\u0026nbsp; established classification of autocatalytic cores for fully reversible\u0026nbsp; networks into five types.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eScientific contribution\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eWe developed a graph-theoretic algorithm to enumerate autocatalytic\u0026nbsp; cores that is efficient enough to handle genome-scale metabolic models\u0026nbsp; of various species exhaustively. To this end, we devised a novel method\u0026nbsp; to enumerate elementary circuits in bipartite digraphs free of one\u0026nbsp; particular type of chord, which outperforms efficient standard\u0026nbsp; enumeration algorithms of elementary circuits by up to 4 orders of\u0026nbsp; magnitude. We show that for CRNs comprising both\u0026nbsp;\u0026nbsp; reducible and irreducible reactions more complex autocatalytic cores\u0026nbsp;\u0026nbsp; exist than envisioned by the established classification schemes for reversible CRNs.\u003c/p\u003e","manuscriptTitle":"Using MR-Chordless Circuits for Efficient Enumeration of Autocatalytic Cores in Large Chemical Reaction Networks","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-03 15:18:46","doi":"10.21203/rs.3.rs-8659930/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-04T11:18:58+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-26T15:28:28+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-25T18:31:20+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"249359333588070622876227192953519871534","date":"2026-02-23T10:03:35+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"102833444150374363077773942193188364974","date":"2026-02-21T23:11:02+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"290107376065728193688261152747956202834","date":"2026-02-17T10:46:31+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"170106282582660491345439635217506413748","date":"2026-02-08T09:03:45+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-02-04T19:56:08+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-02T19:14:35+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-28T10:56:50+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Cheminformatics","date":"2026-01-27T10:47:22+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-cheminformatics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"chin","sideBox":"Learn more about [Journal of Cheminformatics](https://jcheminf.biomedcentral.com/)","snPcode":"13321","submissionUrl":"https://submission.nature.com/new-submission/13321/3","title":"Journal of Cheminformatics","twitterHandle":"@jcheminf","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"BMC/SO AJ","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"3637e004-b24a-4c39-a206-94320d01f4c2","owner":[],"postedDate":"February 3rd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[],"tags":[],"updatedAt":"2026-05-19T09:40:17+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-03 15:18:46","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8659930","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8659930","identity":"rs-8659930","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}