Geometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35

Geometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35 | 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 Geometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35 Dimitrios Christodoulou This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8457155/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 Protein folding is traditionally described as stochastic motion across a rugged free-energy landscape, where trajectories diverge widely even under identical ini- tial conditions. This paradigm explains ensemble behaviour but provides limited deterministic structure at the level of individual folding pathways. Here we examine an alternative formulation for a class of ultrafast monomeric proteins, treating folding as a deterministic geometric contraction on an invariant- constrained admissible manifold. Instead of modelling thermal diffusion or optimisation of an atomistic potential, the analysis imposes four classes of structural invariants—steric feasibility, torsional admissibility, hydrophobic monotonicity, and topological regularity—which restrict conformational space sufficiently to stabilise a unique curvature-minimising pathway. Using the villin headpiece HP35 as a model system, we demonstrate that these invariants yield a reproducible, 42-step macro-trajectory that is robust to perturbations of initial coordinates and variations in numerical resolution. Backbone curvature, torsional deviation, and topological irregularity decrease monotonically, while hydrophobic solvent-accessible surface area contracts without backtracking, producing a strictly directional collapse. The deterministic trajectory naturally separates into four mechanistic phases—early helix nucleation, hydrophobic core ordering, mesoscale tertiary refinement, and sub-angstrom geometric exhaustion—each dominated by different invariant classes. Across repeated deterministic reconstructions, RMSD convergence to the native state follows an exponential decay profile with a stable contraction factor, and aromatic distances evolve in a fixed sequence consistent with experimental observations. Although the model contains no explicit energetic potentials, implicit solvent terms, or stochastic components, the resulting trajectory aligns with known ultrafast- folding signatures, including early helix onset, ordered hydrophobic burial, and minimal intermediate heterogeneity. Perturbation and mutational analyses further indicate that the deterministic macro-path is sequence-specific and governed by the geometry of the invariant manifold rather than by statistical sampling. Taken together, these findings indicate that for the villin headpiece HP35, folding dynamics can be consistently described within a constraint-based geometric framework, providing a complementary perspective to traditional energy-landscape descriptions. All reproducibility checkpoints and validation hashes are reported in the Supplementary Information. Structural Biology Computational Biology Biophysics Mathematical and Theoretical Biology Computational Mathematics Mathematical Physics HP35 Structural Biology Biophysics Protein Folding Folding Pathways Deterministic Folding Geometric Constraints Conformational Dynamics. Figures Figure 1 Figure 2 Figure 3 Figure 4 Full Text Additional Declarations The authors declare no competing interests. Supplementary Files HP35SI42Steps.pdf Supplementary Information: Deterministic Geometric Folding of the Villin Headpiece HP35 (42-Step Trajectory) 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-8457155","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":565958767,"identity":"daf78c28-d8d3-4eb3-9ac9-d636619b9845","order_by":0,"name":"Dimitrios Christodoulou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFklEQVRIiWNgGAWjYBCDBH4G5gYGBgMgk52BjYGhgAgtkg2MIC1APcwgLQZEaDE4ANLCQIQW8/b2hw9+7rHLMz5+sPFzRcEfeXln5mMPfhgwyPOLHcCqRebMGWPDnmfJxWZnEpslzxgYGG48zJZu2GPAYDhzdgJWLRISOWwSPAcOJG67wdgg2WBgwLixmcdMgscA6NTbOLTIP3/+8w9Qy+YZjM0/gVrsQVok/+DTIsFgxgyyZYMEYxvIlsT5zDxm0nht4ckxlpY5kJw440xim2WDgXHyBma2dGMZAwncfmE//vDjmwN2if3thw/fbPgjZzu/vfnYwzcVNvL80ti1YAKDAxCziFQOAvINJCgeBaNgFIyCEQEASIdb8e0aQxoAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0009-3572-7158","institution":"Clarita Global Research Systems","correspondingAuthor":true,"prefix":"","firstName":"Dimitrios","middleName":"","lastName":"Christodoulou","suffix":""}],"badges":[],"createdAt":"2025-12-26 18:25:12","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8457155/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8457155/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":99797346,"identity":"b4ac19bb-b72d-48a1-8866-9a2535fd8e50","added_by":"auto","created_at":"2026-01-08 13:45:37","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":67958,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eMonotonic geometric contraction along the invariant-constrained folding manifold for HP35.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eManifold curvature as a function of folding step for the villin headpiece HP35, showing a progressive and monotonic reduction across the deterministic folding trajectory, consistent with contraction on an invariant-constrained geometric manifold.\u003c/p\u003e","description":"","filename":"curvature.png","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1/441b6b3c1294897ab882fbb5.png"},{"id":99729932,"identity":"cfbb06f1-118d-4a50-9181-4698a7dffa49","added_by":"auto","created_at":"2026-01-07 17:30:06","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":73916,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eHydrophobic SASA contraction during folding of HP35.\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eSmall stepwise fluctuations reflect local side-chain rearrangements rather than stochastic sampling, while the overall monotonic decrease indicates progressive hydrophobic core formation along the deterministic folding trajectory.\u003c/p\u003e","description":"","filename":"sasa.png","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1/32d1d9f86aa080ba1e0e4bcd.png"},{"id":99729933,"identity":"bdc7be5b-8e69-47f0-a860-975a0237b148","added_by":"auto","created_at":"2026-01-07 17:30:06","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":107006,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eAblation RMSD traces under partial constraint removal.\u003c/strong\u003e\u003cbr\u003e\nRoot-mean-square deviation (RMSD) trajectories comparing the full invariant-constrained folding framework with ablated variants in which individual geometric constraints are removed. The full model exhibits rapid and monotonic convergence toward the native structure, whereas ablated configurations show delayed convergence, increased fluctuations, or divergence. This demonstrates that deterministic folding arises from the coupled action of the complete geometric constraint set rather than from any single component.\u003c/p\u003e","description":"","filename":"S4.png","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1/bd761247d606b2387bcc4769.png"},{"id":99729935,"identity":"a25159d5-eb20-412e-bd49-86486217ddc1","added_by":"auto","created_at":"2026-01-07 17:30:06","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":55411,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eRMSD convergence along the deterministic folding trajectory.\u003c/strong\u003e\u003cbr\u003e\nMonotonic reduction of RMSD to the experimental native structure across the 42-step folding trajectory for HP35, indicating stable convergence without reliance on stochastic sampling or energetic optimization.\u003c/p\u003e","description":"","filename":"rmsd.png","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1/551512f26db047d94eaa00f0.png"},{"id":99805028,"identity":"4bdbf90e-9d83-4616-964b-6cfac47e38ac","added_by":"auto","created_at":"2026-01-08 14:15:28","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":698106,"visible":true,"origin":"","legend":"","description":"","filename":"H35Main42Steps.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1_covered_644bd730-decc-45a2-9324-f512facdbbd2.pdf"},{"id":99729936,"identity":"6a844c11-e9f8-434e-bbff-1e6e94011365","added_by":"auto","created_at":"2026-01-07 17:30:06","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":308831,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cstrong\u003eSupplementary Information: \u003c/strong\u003eDeterministic Geometric Folding of the Villin Headpiece HP35 (42-Step Trajectory)\u003c/p\u003e","description":"","filename":"HP35SI42Steps.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8457155/v1/26c39deec4d86aeac1316cb7.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eGeometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Clarita Global Research Systems ","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":"HP35, Structural Biology, Biophysics, Protein Folding, Folding Pathways, Deterministic Folding, Geometric Constraints, Conformational Dynamics.","lastPublishedDoi":"10.21203/rs.3.rs-8457155/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8457155/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eProtein folding is traditionally described as stochastic motion across a rugged\u003c/p\u003e\n\u003cp\u003efree-energy landscape, where trajectories diverge widely even under identical ini-\u003c/p\u003e\n\u003cp\u003etial conditions. This paradigm explains ensemble behaviour but provides limited deterministic structure at the level of individual folding pathways.\u003c/p\u003e\n\u003cp\u003eHere we examine an alternative formulation for a class of ultrafast monomeric proteins, treating folding as a deterministic geometric contraction on an invariant- constrained admissible manifold. Instead of modelling thermal diffusion or optimisation of an atomistic potential, the analysis imposes four classes of structural invariants—steric feasibility, torsional admissibility, hydrophobic monotonicity, and topological regularity—which restrict conformational space sufficiently to stabilise a unique curvature-minimising pathway.\u003c/p\u003e\n\u003cp\u003eUsing the villin headpiece HP35 as a model system, we demonstrate that these\u003c/p\u003e\n\u003cp\u003einvariants yield a reproducible, 42-step macro-trajectory that is robust to perturbations of initial coordinates and variations in numerical resolution. Backbone curvature, torsional deviation, and topological irregularity decrease monotonically, while hydrophobic solvent-accessible surface area contracts without backtracking, producing a strictly directional collapse. The deterministic trajectory naturally separates into four mechanistic phases—early helix nucleation, hydrophobic core ordering,\u003c/p\u003e\n\u003cp\u003emesoscale tertiary refinement, and sub-angstrom geometric exhaustion—each dominated by different invariant classes. Across repeated deterministic reconstructions, RMSD convergence to the native state follows an exponential decay profile with a stable contraction factor, and aromatic distances evolve in a fixed sequence consistent with experimental observations.\u003c/p\u003e\n\u003cp\u003eAlthough the model contains no explicit energetic potentials, implicit solvent\u003c/p\u003e\n\u003cp\u003eterms, or stochastic components, the resulting trajectory aligns with known ultrafast- folding signatures, including early helix onset, ordered hydrophobic burial, and minimal intermediate heterogeneity. Perturbation and mutational analyses further indicate that the deterministic macro-path is sequence-specific and governed by the geometry of the invariant manifold rather than by statistical sampling. Taken together, these findings indicate that for the villin headpiece HP35, folding dynamics can be consistently described within a constraint-based geometric framework, providing a complementary perspective to traditional energy-landscape descriptions.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eAll reproducibility checkpoints and validation hashes are reported in the Supplementary Information.\u003c/p\u003e","manuscriptTitle":"Geometric Contraction on an Invariant-Constrained Manifold: A 42-Step Folding Trajectory for the Villin Headpiece HP35","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-07 17:30:02","doi":"10.21203/rs.3.rs-8457155/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":"f59d2d67-49d0-4b67-909d-b91516e02092","owner":[],"postedDate":"January 7th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":60765255,"name":"Structural Biology"},{"id":60765256,"name":"Computational Biology"},{"id":60765257,"name":"Biophysics"},{"id":60765258,"name":"Mathematical and Theoretical Biology"},{"id":60765259,"name":"Computational Mathematics"},{"id":60765260,"name":"Mathematical Physics"}],"tags":[],"updatedAt":"2026-01-07T17:30:02+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-07 17:30:02","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8457155","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8457155","identity":"rs-8457155","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}