Topological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness

Topological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness | 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 Topological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness Simone Caramel This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8595475/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 Background This study investigates the deterministic constraints governing the structural integrity of proteomic networks. We propose a theoretical framework where protein stability is modeled through a hierarchy of universal fractal attractors, specifically focusing on the convergence of the Golden Ratio ( \(\:\varphi\:\approx\:1.618\) ) and the Feigenbaum constant ( \(\:\delta\:\approx\:4.669\) ). Methods We define two quantitative metrics: the Coherence Coefficient ( \(\:{C}_{b}\) ) and the Collective Phase-Coherence Coefficient ( \(\:{C}_{coll}\) ). These metrics measure the divergence between observed fractal dimensions ( \(\:{D}_{obs}\) ), derived from PDB atomic coordinates via mass-radius scaling laws, and theoretical geometric targets derived from the Feigenbaum-Fibonacci scale. The model was tested against a comprehensive dataset of 1,000 protein structures (curated from the RCSB PDB latest releases), categorized into four functional classes to assess the invariance of the topological parameters. Results The analysis identifies a critical regime of structural stability for \(\:0.89<{C}_{b}<0.95\) , consistent with the dynamics of self-organized criticality (SOC) at the "edge of chaos." A discontinuous phase transition is observed: values below \(\:{C}_{b}0.985\) (e.g., in amyloid aggregates and prions) indicate a shift toward pathological rigidity. A Topological Robustness Factor, defined by the product \(\:\varphi\:\cdot\:\delta\:\approx\:7.55\) , is proposed as a universal scaling constant for structural buffering across the dataset. Conclusions The results suggest that proteomic stability can be described as a state of geometric homeostasis maintained by non-linear oscillators. This framework provides a formal mathematical method to quantify structural robustness and predicts phase transitions in protein folding networks without relying on purely heuristic interpretations. Deterministic Chaos Topological Stability Phase-Locking Proteome Synchronization Nonlinear Dynamics Self-Organized Criticality Phase Transitions Kuramoto Model Full Text Additional Declarations No competing interests reported. Supplementary Files highlightsbc.docx 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-8595475","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":575933241,"identity":"da857ebb-93ab-40f5-9f15-451a187b6507","order_by":0,"name":"Simone Caramel","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA60lEQVRIiWNgGAWjYDACZmTOgwoGBgMQI4FoLQlnYFrw6kEGiW1QLfiskW/nTnxcUGEnb95+OvFB4rzD8ubszQcYHv7ArcXgMO9m4xlnkg3nnMndbJC47bDhzp5jCXgdZsDMu02at+0A4wyG3G0SQC2MG27kGODVIt8M0vLvgP0M/rfbfyTOOWxPUAvDYZCWhgOJMyRytzEkNhxOJKgF7BeeY8nJMyTebpZIOJaevOHMsYQDCWl4HNZ/duNjnho72xn8uRs/fKixtt1wvPngwx82eByGBprB5AHiNTAw1JGieBSMglEwCkYIAACHXVZ5wKqaFAAAAABJRU5ErkJggg==","orcid":"","institution":"","correspondingAuthor":true,"prefix":"","firstName":"Simone","middleName":"","lastName":"Caramel","suffix":""}],"badges":[],"createdAt":"2026-01-13 20:23:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8595475/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8595475/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":100746854,"identity":"7ae6c312-061e-4ada-9fc6-a10d14a46461","added_by":"auto","created_at":"2026-01-21 03:45:04","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1450014,"visible":true,"origin":"","legend":"","description":"","filename":"TopCoherencePhiDelta202601updatedwithfundingdeclaration.docx","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/2c7b5d3355dfce20af9ba706.docx"},{"id":100746888,"identity":"56507e9b-2150-45a6-b358-1c1fe324e502","added_by":"auto","created_at":"2026-01-21 03:45:19","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3555,"visible":true,"origin":"","legend":"","description":"","filename":"65aa2eb54b7f4f1c80d05a64003ab921.json","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/83f50aae4acf3e5fc2eb480c.json"},{"id":100746893,"identity":"eb94b66d-8736-4622-8677-4b7c2720eb69","added_by":"auto","created_at":"2026-01-21 03:45:21","extension":"docx","order_by":2,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":18869,"visible":true,"origin":"","legend":"","description":"","filename":"coverletterbcd1.docx","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/3120d9d29c689a33e4757936.docx"},{"id":100746866,"identity":"a5f4376a-4567-4f30-a1cc-9f24c0dd71aa","added_by":"auto","created_at":"2026-01-21 03:45:10","extension":"docx","order_by":3,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":14522,"visible":true,"origin":"","legend":"","description":"","filename":"highlightsbc.docx","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/fd677ec3744748a46401ea89.docx"},{"id":100746874,"identity":"ff2c861f-42bc-4d0a-b191-e5eb524bed19","added_by":"auto","created_at":"2026-01-21 03:45:16","extension":"xml","order_by":4,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":117178,"visible":true,"origin":"","legend":"","description":"","filename":"65aa2eb54b7f4f1c80d05a64003ab9211enriched.xml","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/c36ec7e85cdee1d9e5967505.xml"},{"id":100746895,"identity":"005e2119-94af-40eb-912b-a4091a83df95","added_by":"auto","created_at":"2026-01-21 03:45:21","extension":"gif","order_by":5,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/a599a2b05bb08df5ce95329f.gif"},{"id":100746852,"identity":"e3644cee-5f1c-4f6f-8dcb-5e2446be07eb","added_by":"auto","created_at":"2026-01-21 03:45:03","extension":"gif","order_by":6,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/51768cb536dcd52203d952b0.gif"},{"id":100746869,"identity":"7493acc0-6a0c-4573-94b6-593ffa102a09","added_by":"auto","created_at":"2026-01-21 03:45:15","extension":"gif","order_by":7,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/0e082e39bbc4e933f684b857.gif"},{"id":100746894,"identity":"34398d98-97d3-422c-80bf-61c037337b96","added_by":"auto","created_at":"2026-01-21 03:45:21","extension":"gif","order_by":8,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/5f71856d85f541f54aa06716.gif"},{"id":100746876,"identity":"f48c7f96-d79a-4c8c-8585-debf921d64c4","added_by":"auto","created_at":"2026-01-21 03:45:17","extension":"gif","order_by":9,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/fc3d45fdfe339d64864fb6a3.gif"},{"id":100746870,"identity":"5bbfd636-9354-4da6-9f03-a94e34993725","added_by":"auto","created_at":"2026-01-21 03:45:15","extension":"jpeg","order_by":10,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":2994,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage14.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/3958af70df502dc9f2ccb0b3.jpeg"},{"id":100746897,"identity":"e0624259-e1b0-4c16-a435-cfecc3e2eb4b","added_by":"auto","created_at":"2026-01-21 03:45:22","extension":"gif","order_by":11,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/c3ef79adf29f255af5dcd0f3.gif"},{"id":100746873,"identity":"7f221d3d-3ef2-4e89-a7de-d84201943498","added_by":"auto","created_at":"2026-01-21 03:45:16","extension":"gif","order_by":12,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/57cb1a1439096437d259b111.gif"},{"id":100746901,"identity":"c6e10f0c-a0f2-40e0-b268-7a3b328acee4","added_by":"auto","created_at":"2026-01-21 03:45:23","extension":"gif","order_by":13,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/2bb6e7e4c2dfa00a0b0356c5.gif"},{"id":100796537,"identity":"4b138f53-39ad-4e91-9e1a-ec140f50a020","added_by":"auto","created_at":"2026-01-21 13:44:02","extension":"jpeg","order_by":14,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":400523,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage18.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/9f4ad9aa757ea4595ecaf699.jpeg"},{"id":100746856,"identity":"2eef2e90-f4db-4609-97c3-f2b4543c6bba","added_by":"auto","created_at":"2026-01-21 03:45:04","extension":"jpeg","order_by":15,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":872515,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage19.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/cacc0981d2468576564d67e4.jpeg"},{"id":100746908,"identity":"70bb3d5b-3af6-4bc4-a97e-e2a6750249ae","added_by":"auto","created_at":"2026-01-21 03:45:29","extension":"gif","order_by":16,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/26883d1bbe8c26fb33050106.gif"},{"id":100746878,"identity":"705165f7-a254-4091-89b6-c97ccc3dc792","added_by":"auto","created_at":"2026-01-21 03:45:17","extension":"jpeg","order_by":17,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":250627,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage20.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/5b2c7a0c191f0863f8a50a72.jpeg"},{"id":100796566,"identity":"c130d25f-4e14-42a9-9e1d-2f5ad941aad7","added_by":"auto","created_at":"2026-01-21 13:44:15","extension":"png","order_by":18,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":160660,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage21.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/359ee677016facce49165373.png"},{"id":100796649,"identity":"3b0433b9-3272-4cd5-bb8e-ccc452e5da08","added_by":"auto","created_at":"2026-01-21 13:44:47","extension":"png","order_by":19,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":146151,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage22.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/ddbddb444b3667692e26ce3d.png"},{"id":100746891,"identity":"1ca1cb2a-93d6-4906-8912-a6a63617d1ab","added_by":"auto","created_at":"2026-01-21 03:45:19","extension":"png","order_by":20,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":142090,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage23.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/adbac4c61e57084309caf2d6.png"},{"id":100746871,"identity":"c2781c23-fb62-4490-abad-e354b1ed2b1b","added_by":"auto","created_at":"2026-01-21 03:45:15","extension":"gif","order_by":21,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/3f76ec250940db8f3999bb81.gif"},{"id":100746864,"identity":"8073c7a2-47dc-44c3-a783-7dd323ae3dbd","added_by":"auto","created_at":"2026-01-21 03:45:07","extension":"gif","order_by":22,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/ac866308e89b9ccde6f6fad2.gif"},{"id":100746867,"identity":"5a956d58-4ca9-4fea-a768-8a01b9d1462b","added_by":"auto","created_at":"2026-01-21 03:45:12","extension":"gif","order_by":23,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/a5e082ac23f691f2cdb84e5f.gif"},{"id":100746865,"identity":"c4c64d52-d988-4589-85bc-782f17c0d936","added_by":"auto","created_at":"2026-01-21 03:45:07","extension":"gif","order_by":24,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/9df4ad6320dfdd12d537cd78.gif"},{"id":100746885,"identity":"1c5be4d4-53ac-40ad-9ec1-153f17e71134","added_by":"auto","created_at":"2026-01-21 03:45:18","extension":"gif","order_by":25,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/583642c3f459bcdcd2810b18.gif"},{"id":100746887,"identity":"ff226b06-9fd5-4e46-baa8-c645964b79e9","added_by":"auto","created_at":"2026-01-21 03:45:19","extension":"gif","order_by":26,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":43,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage1.gif","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/c3a1f4db302d2ae2c25f0b2a.gif"},{"id":100746900,"identity":"cf9b0eb3-ce31-4a31-8467-bf2993fe7c3b","added_by":"auto","created_at":"2026-01-21 03:45:22","extension":"jpeg","order_by":27,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":352462,"visible":true,"origin":"","legend":"","description":"","filename":"floatimage9.jpeg","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/6c86435bf75192cbc83aac83.jpeg"},{"id":100746861,"identity":"6422800f-c5bf-4b61-9ef0-0cf17caca638","added_by":"auto","created_at":"2026-01-21 03:45:06","extension":"png","order_by":28,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/5777fe736ad7d7761e25300c.png"},{"id":100746875,"identity":"a5d971dd-d9e6-422d-8775-3a57b3ea48a7","added_by":"auto","created_at":"2026-01-21 03:45:16","extension":"png","order_by":29,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/97a8d6fca368d5c7d0e37c4f.png"},{"id":100746883,"identity":"26f31916-3f62-4b29-9863-b9db56fc7c74","added_by":"auto","created_at":"2026-01-21 03:45:18","extension":"png","order_by":30,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/4eaa7472e374a6a0a0cddf77.png"},{"id":100746872,"identity":"2fac99ce-dfe9-41fd-8922-d857cda5bc84","added_by":"auto","created_at":"2026-01-21 03:45:16","extension":"png","order_by":31,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/a634bad814f8d2bc5c6d8321.png"},{"id":100746868,"identity":"a8afb314-f297-471e-8f48-a3035712a641","added_by":"auto","created_at":"2026-01-21 03:45:13","extension":"png","order_by":32,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/56fccd8fb876eccdbaef03d4.png"},{"id":100796558,"identity":"6dc5e259-dc86-4fd9-8d61-ab76c2b8128c","added_by":"auto","created_at":"2026-01-21 13:44:11","extension":"png","order_by":33,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":1247,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage14.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/81393d6214508458ca1399e1.png"},{"id":100804033,"identity":"85d5b65c-3169-44fe-8c20-3369fbefa991","added_by":"auto","created_at":"2026-01-21 14:35:13","extension":"png","order_by":34,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/0f67ba532a56b4f0abcc4833.png"},{"id":100746886,"identity":"5ce616cd-987b-46a4-8ae3-3ada3eef1f3a","added_by":"auto","created_at":"2026-01-21 03:45:19","extension":"png","order_by":35,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/327ca5208498cdd65d241b27.png"},{"id":100746859,"identity":"37c6d4a2-2284-49af-857a-a1e7be909307","added_by":"auto","created_at":"2026-01-21 03:45:05","extension":"png","order_by":36,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/379685bd1febe2e99667d795.png"},{"id":100746855,"identity":"5aa6fac9-6007-405e-9d45-7a2b17504e98","added_by":"auto","created_at":"2026-01-21 03:45:04","extension":"png","order_by":37,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":99361,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage18.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/c080ef055efb033737a4d506.png"},{"id":100746881,"identity":"c180c008-c2ed-48d7-a30a-cb12364f74c0","added_by":"auto","created_at":"2026-01-21 03:45:18","extension":"png","order_by":38,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":214338,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage19.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/027fc6644ca343780206d724.png"},{"id":100746884,"identity":"73684f89-436b-44f5-a967-38ec665ed89b","added_by":"auto","created_at":"2026-01-21 03:45:18","extension":"png","order_by":39,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/e3446251ed2bfd4f80b1ec42.png"},{"id":100746879,"identity":"16f0f48f-625d-4718-bc37-8904887d9cd2","added_by":"auto","created_at":"2026-01-21 03:45:17","extension":"png","order_by":40,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":39255,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage20.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/b6138218f10a583b378aa8c5.png"},{"id":100746896,"identity":"76fc7eff-2fef-48e0-8ded-92c3c98119a3","added_by":"auto","created_at":"2026-01-21 03:45:22","extension":"png","order_by":41,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":75208,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage21.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/c09b9f15c702aa48072b3706.png"},{"id":100746877,"identity":"e7acbb75-0eb9-4cc0-8c6f-7080d37f1592","added_by":"auto","created_at":"2026-01-21 03:45:17","extension":"png","order_by":42,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":47206,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage22.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/98cf59f22ef4074a4aea4aa4.png"},{"id":100746910,"identity":"9e3ae832-4cd3-4e35-8c6e-ed9607473c35","added_by":"auto","created_at":"2026-01-21 03:45:30","extension":"png","order_by":43,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":64787,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage23.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/8c5e9ce7ddcd0753edcbb131.png"},{"id":100746892,"identity":"0c875964-dd78-4973-babd-fda4c3c0f699","added_by":"auto","created_at":"2026-01-21 03:45:20","extension":"png","order_by":44,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/b46607208c98255a5c51e720.png"},{"id":100746889,"identity":"e3f22850-593a-4e3b-a7be-fdf4cb5cb506","added_by":"auto","created_at":"2026-01-21 03:45:19","extension":"png","order_by":45,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/9b783f5efe0cb3b88eb3855d.png"},{"id":100746853,"identity":"13f536b8-e8a5-43cd-85b5-35868b8d98fc","added_by":"auto","created_at":"2026-01-21 03:45:03","extension":"png","order_by":46,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/37d13834e60923fcf18a5aed.png"},{"id":100746880,"identity":"0e6be1c0-dbb6-47f0-a479-118548f949c9","added_by":"auto","created_at":"2026-01-21 03:45:17","extension":"png","order_by":47,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/b30876a3138e8a2609773025.png"},{"id":100746911,"identity":"d3349838-dbbe-4044-b72f-19bcc889fe0f","added_by":"auto","created_at":"2026-01-21 03:45:30","extension":"png","order_by":48,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/671124c0fb38541a7bd3525f.png"},{"id":100746857,"identity":"79554210-f90f-4ff6-bc0d-14210e746a63","added_by":"auto","created_at":"2026-01-21 03:45:05","extension":"png","order_by":49,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":321,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage1.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/5781b856610e6f43bf6b64d0.png"},{"id":100746882,"identity":"8d615dc3-8b42-43c9-83be-5481b804ea62","added_by":"auto","created_at":"2026-01-21 03:45:18","extension":"png","order_by":50,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":79101,"visible":true,"origin":"","legend":"","description":"","filename":"Onlinefloatimage9.png","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/cccb6065ec00444a158d96e3.png"},{"id":100746899,"identity":"9a275012-9c2c-4607-9b72-89a93caa2715","added_by":"auto","created_at":"2026-01-21 03:45:22","extension":"xml","order_by":51,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":113199,"visible":true,"origin":"","legend":"","description":"","filename":"65aa2eb54b7f4f1c80d05a64003ab9211structuring.xml","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/a2c00bc824dc4869e113d5fc.xml"},{"id":100746860,"identity":"fa18645f-cb08-4753-ad52-e7392d850d43","added_by":"auto","created_at":"2026-01-21 03:45:05","extension":"html","order_by":52,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":147851,"visible":true,"origin":"","legend":"","description":"","filename":"earlyproof.html","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/93848d196c378223ef74396a.html"},{"id":101751101,"identity":"327391c5-5ff9-4fbd-8bdf-994ecce54e9d","added_by":"auto","created_at":"2026-02-03 10:13:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1258265,"visible":true,"origin":"","legend":"","description":"","filename":"TopCoherencePhiDelta202601updatedwithfundingdeclaration.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1_covered_a8b67c9b-c5bd-46a8-a4d8-cc174b2522fa.pdf"},{"id":100746851,"identity":"e37aeb6a-3e35-4af8-a373-3cc23b304ab5","added_by":"auto","created_at":"2026-01-21 03:45:03","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":14522,"visible":true,"origin":"","legend":"","description":"","filename":"highlightsbc.docx","url":"https://assets-eu.researchsquare.com/files/rs-8595475/v1/1547dca3953dbd741b97543a.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eTopological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"Deterministic Chaos, Topological Stability, Phase-Locking, Proteome Synchronization, Nonlinear Dynamics, Self-Organized Criticality, Phase Transitions, Kuramoto Model","lastPublishedDoi":"10.21203/rs.3.rs-8595475/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8595475/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003ch2\u003eBackground\u003c/h2\u003e \u003cp\u003eThis study investigates the deterministic constraints governing the structural integrity of proteomic networks. We propose a theoretical framework where protein stability is modeled through a hierarchy of universal fractal attractors, specifically focusing on the convergence of the Golden Ratio (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varphi\\:\\approx\\:1.618\\)\u003c/span\u003e\u003c/span\u003e) and the Feigenbaum constant (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\delta\\:\\approx\\:4.669\\)\u003c/span\u003e\u003c/span\u003e).\u003c/p\u003e\u003ch2\u003eMethods\u003c/h2\u003e \u003cp\u003eWe define two quantitative metrics: the Coherence Coefficient (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{b}\\)\u003c/span\u003e\u003c/span\u003e) and the Collective Phase-Coherence Coefficient (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{coll}\\)\u003c/span\u003e\u003c/span\u003e). These metrics measure the divergence between observed fractal dimensions (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{D}_{obs}\\)\u003c/span\u003e\u003c/span\u003e), derived from PDB atomic coordinates via mass-radius scaling laws, and theoretical geometric targets derived from the Feigenbaum-Fibonacci scale. The model was tested against a comprehensive dataset of 1,000 protein structures (curated from the RCSB PDB latest releases), categorized into four functional classes to assess the invariance of the topological parameters.\u003c/p\u003e\u003ch2\u003eResults\u003c/h2\u003e \u003cp\u003eThe analysis identifies a critical regime of structural stability for \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:0.89\u0026lt;{C}_{b}\u0026lt;0.95\\)\u003c/span\u003e\u003c/span\u003e, consistent with the dynamics of self-organized criticality (SOC) at the \"edge of chaos.\" A discontinuous phase transition is observed: values below \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{b}\u0026lt;0.89\\)\u003c/span\u003e\u003c/span\u003e correlate with increased entropic disorder (Topological Gap), while values exceeding \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{C}_{b}\u0026gt;0.985\\)\u003c/span\u003e\u003c/span\u003e (e.g., in amyloid aggregates and prions) indicate a shift toward pathological rigidity. A Topological Robustness Factor, defined by the product \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\varphi\\:\\cdot\\:\\delta\\:\\approx\\:7.55\\)\u003c/span\u003e\u003c/span\u003e, is proposed as a universal scaling constant for structural buffering across the dataset.\u003c/p\u003e\u003ch2\u003eConclusions\u003c/h2\u003e \u003cp\u003eThe results suggest that proteomic stability can be described as a state of geometric homeostasis maintained by non-linear oscillators. This framework provides a formal mathematical method to quantify structural robustness and predicts phase transitions in protein folding networks without relying on purely heuristic interpretations.\u003c/p\u003e","manuscriptTitle":"Topological Phase Transitions and Phase-Locking Dynamics in Proteomic Networks: A Deterministic Model of Structural Robustness","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-21 03:42:50","doi":"10.21203/rs.3.rs-8595475/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":"3783ced7-d73f-4cd3-a683-b17f39c11e6b","owner":[],"postedDate":"January 21st, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-01-23T08:11:10+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-21 03:42:50","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8595475","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8595475","identity":"rs-8595475","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}