{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T22:56:04Z","timestamp":1790722564864,"version":"4.1.0"},"reference-count":86,"publisher":"American Physical Society (APS)","issue":"4","license":[{"start":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T00:00:00Z","timestamp":1759968000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004189","name":"Max-Planck-Gesellschaft","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004189","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2025,9,11]]},"abstract":"<jats:p>The flows of tissues of epithelial cells often involve T1 transitions. These neighbor exchanges are irreversible rearrangements crossing an energy barrier. Here, by an exact geometric construction, I determine this energy barrier for general, isolated T1 transitions dominated by line tensions. I show how deviations from regular cell packing reduce this energy barrier, but find that line tension fluctuations increase it on average. By another exact construction, I prove that the nonlinear tensions in vertex models of tissues also resist T1 transitions. My results thus form the basis for coarse-grained understanding of cell neighbor exchanges for continuum descriptions of epithelia.<\/jats:p>","DOI":"10.1103\/kdnb-bxdx","type":"journal-article","created":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T16:39:51Z","timestamp":1757608791000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":3,"title":["Geometry of T1 transitions in epithelia"],"prefix":"10.1103","volume":"112","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6663-0393","authenticated-orcid":true,"given":"Pierre A.","family":"Haas","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01bf9rw71","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute for the Physics of Complex Systems"},{"id":[{"id":"https:\/\/ror.org\/05b8d3w18","id-type":"ROR","asserted-by":"publisher"}],"name":"Max Planck Institute of Molecular Cell Biology and Genetics"},{"id":[{"id":"https:\/\/ror.org\/05hrn3e05","id-type":"ROR","asserted-by":"publisher"}],"name":"Center for Systems Biology Dresden"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2025,10,9]]},"reference":[{"key":"kdnb-bxdxCc1R1","doi-asserted-by":"publisher","DOI":"10.1038\/nrm3723","type":"journal-article"},{"key":"kdnb-bxdxCc2R1","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2019.0552","type":"journal-article"},{"key":"kdnb-bxdxCc3R1","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2000.0626","type":"journal-article"},{"key":"kdnb-bxdxCc4R1","doi-asserted-by":"publisher","DOI":"10.1016\/S1534-5807(02)00197-1","type":"journal-article"},{"key":"kdnb-bxdxCc5R1","doi-asserted-by":"publisher","DOI":"10.1242\/dev.120.4.827","type":"journal-article"},{"key":"kdnb-bxdxCc6R1","doi-asserted-by":"publisher","DOI":"10.1038\/nature02590","type":"journal-article"},{"key":"kdnb-bxdxCc7R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.mod.2016.12.001","type":"journal-article"},{"key":"kdnb-bxdxCc8R1","doi-asserted-by":"publisher","DOI":"10.7554\/eLife.95521","type":"journal-article"},{"key":"kdnb-bxdxCc9R1","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199662890.001.0001","type":"book","volume-title":"Foams: Structure and Dynamics","author":"I. 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Andreescu","year":"2009","type":"book"},{"key":"kdnb-bxdxCc36R1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187892","type":"journal-article"},{"key":"kdnb-bxdxCc37R1","doi-asserted-by":"publisher","DOI":"10.1137\/0116001","type":"journal-article"},{"key":"kdnb-bxdxCc38R1","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(94)90244-5","type":"journal-article"},{"key":"kdnb-bxdxCc38R2","unstructured":"show that the existence of the points F1, F2 and hence of the \u201cSteiner tree\u201d in Fig.   2 is equivalent with \u2220DAP2\u2264120\u2218, \u2220P2DA\u2264120\u2218, \u2220P1BC\u2264120\u2218, \u2220BCP1\u2264120\u2218, and \u03b8\u2264120\u2218. In this work, I shall assume tacitly that the first four conditions hold. This is of course a strong mathematical assumption, but not perhaps a strong physical assumption on the geometry of cell vertices surrounding a T1 transition edge.","type":"other"},{"key":"kdnb-bxdxCc39R1","volume-title":"Introduction to Geometry","author":"H.\u00a0S.\u00a0M. Coxeter","year":"1969","type":"book"},{"key":"kdnb-bxdxCc40R1","volume-title":"Advanced Euclidean Geometry","author":"R.\u00a0A. Johnson","year":"1960","type":"book"},{"key":"kdnb-bxdxCc41R1","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2002.11919871","type":"journal-article"},{"key":"kdnb-bxdxCc42R1","volume-title":"Hadamard's Plane Geometry: A Reader's Companion","author":"M. Saul","year":"2019","type":"book"},{"key":"kdnb-bxdxCc43R1","unstructured":"See Supplemental Material at http:\/\/link.aps.org\/supplemental\/10.1103\/kdnb-bxdx, which includes Refs.   [12, 13, 44, 45, 46, 47, 48, 49, 50], for details of the calculations and numerical results for a mean-field vertex model of an isolated T1 transition.","type":"other"},{"key":"kdnb-bxdxCc44R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cub.2007.11.049","type":"journal-article"},{"key":"kdnb-bxdxCc45R1","doi-asserted-by":"publisher","DOI":"10.1140\/epje\/i2010-10677-0","type":"journal-article"},{"key":"kdnb-bxdxCc46R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.bpj.2013.11.4498","type":"journal-article"},{"key":"kdnb-bxdxCc47R1","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2015.0519","type":"journal-article"},{"key":"kdnb-bxdxCc48R1","doi-asserted-by":"publisher","DOI":"10.1098\/rstb.2015.0520","type":"journal-article"},{"key":"kdnb-bxdxCc49R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.129.048102","type":"journal-article"},{"key":"kdnb-bxdxCc50R1","unstructured":"Matlab (The MathWorks, Inc.) code for minimisation of the mean-field vertex model energy of an isolated T1 transition derived in the Supplemental Material   [43] is available online at https:\/\/zenodo.org\/records\/16738082.","type":"other"},{"key":"kdnb-bxdxCc51R1","volume-title":"Contributions to Location Analysis","author":"R.\u00a0L. 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Vickers","year":"2015","journal-title":"Forum Geom."},{"key":"kdnb-bxdxCc53R4","unstructured":"This theorem shows that F1 and F2 are \u201cSteiner\u201d points of triangles AF2D and BF1C, respectively, in the same way as they are Fermat points in the homogeneous case.","type":"other"},{"key":"kdnb-bxdxCc54R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.devcel.2017.09.018","type":"journal-article"},{"key":"kdnb-bxdxCc55R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-019-0618-1","type":"journal-article"},{"key":"kdnb-bxdxCc56R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41567-021-01215-1","type":"journal-article"},{"key":"kdnb-bxdxCc57R1","doi-asserted-by":"publisher","DOI":"10.1039\/D0SM01559H","type":"journal-article"},{"key":"kdnb-bxdxCc58R1","doi-asserted-by":"publisher","DOI":"10.1103\/9ktk-6rqc","type":"journal-article"},{"key":"kdnb-bxdxCc59R1","doi-asserted-by":"publisher","DOI":"10.1080\/10724117.2000.11975111","type":"journal-article"},{"key":"kdnb-bxdxCc60R1","unstructured":"This fact is surely folklore, but I could not find an exact reference. To prove it, I denote by rA,rB,rC,rD the position vectors of A,B,C,D and compute the gradient of E(r)=\u2225r\u2212rA\u22252+\u2225r\u2212rB\u22252+\u2225r\u2212rC\u22252+\u2225r\u2212rD\u22252, \u2207E=24r\u2212rA\u2212rB\u2212rC\u2212rD, which has a single stationary point at r=(rA+rB+rC+rD)\/4, i.e., at the barycenter, and this is clearly a minimum.","type":"other"},{"key":"kdnb-bxdxCc61R1","doi-asserted-by":"publisher","DOI":"10.1039\/C9SM01068H","type":"journal-article"},{"key":"kdnb-bxdxCc62R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.bpj.2015.05.019","type":"journal-article"},{"key":"kdnb-bxdxCc63R1","doi-asserted-by":"publisher","DOI":"10.1093\/imammb\/dqx008","type":"journal-article"},{"key":"kdnb-bxdxCc64R1","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1010135","type":"journal-article"},{"key":"kdnb-bxdxCc65R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.105.064611","type":"journal-article"},{"key":"kdnb-bxdxCc66R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.5.013143","type":"journal-article"},{"key":"kdnb-bxdxCc67R1","doi-asserted-by":"publisher","DOI":"10.1039\/D2SM01580C","type":"journal-article"},{"key":"kdnb-bxdxCc68R1","doi-asserted-by":"publisher","DOI":"10.1039\/D3SM00874F","type":"journal-article"},{"key":"kdnb-bxdxCc69R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/ad3099","type":"journal-article"},{"key":"kdnb-bxdxCc70R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.108.L042602","type":"journal-article"},{"key":"kdnb-bxdxCc71R1","doi-asserted-by":"publisher","DOI":"10.1039\/D3SM00254C","type":"journal-article"},{"key":"kdnb-bxdxCc72R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.108.044118","type":"journal-article"},{"key":"kdnb-bxdxCc73R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.7.013039","type":"journal-article"},{"key":"kdnb-bxdxCc74R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.92.052713","type":"journal-article"},{"key":"kdnb-bxdxCc75R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.99.022411","type":"journal-article"},{"key":"kdnb-bxdxCc76R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.130.198401","type":"journal-article"},{"key":"kdnb-bxdxCc78R1","doi-asserted-by":"publisher","DOI":"10.1103\/PRXLife.3.023002","type":"journal-article"},{"key":"kdnb-bxdxCc79R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jtbi.2003.10.001","type":"journal-article"},{"key":"kdnb-bxdxCc80R1","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/ac23f1","type":"journal-article"},{"key":"kdnb-bxdxCc81R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.cdev.2021.203718","type":"journal-article"},{"key":"kdnb-bxdxCc82R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevResearch.4.043148","type":"journal-article"},{"key":"kdnb-bxdxCc83R1","doi-asserted-by":"publisher","DOI":"10.1038\/s41556-023-01332-4","type":"journal-article"}],"container-title":["Physical Review E"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.aps.org\/article\/10.1103\/kdnb-bxdx","content-type":"unspecified","content-version":"vor","intended-application":"syndication"},{"URL":"http:\/\/harvest.aps.org\/v2\/journals\/articles\/10.1103\/kdnb-bxdx\/fulltext","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T22:00:15Z","timestamp":1790719215000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.aps.org\/doi\/10.1103\/kdnb-bxdx"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,9]]},"references-count":86,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,10]]}},"URL":"https:\/\/doi.org\/10.1103\/kdnb-bxdx","relation":{"has-preprint":[{"id-type":"doi","id":"10.48550\/arXiv.2504.16765","asserted-by":"subject"}]},"ISSN":["2470-0045","2470-0053"],"issn-type":[{"value":"2470-0045","type":"print"},{"value":"2470-0053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,9]]},"assertion":[{"value":"2025-04-23","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"Published by the American Physical Society. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2025","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92025 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"L042401"}}