{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T05:51:03Z","timestamp":1791525063244,"version":"4.3.4"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T00:00:00Z","timestamp":1791504000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1007\/s00285-026-02468-1","type":"journal-article","created":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T05:17:07Z","timestamp":1791523027000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A 2D fluid-based model for individual cell crawling"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6805-2120","authenticated-orcid":false,"given":"Christ\u00e8le","family":"Etchegaray","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicolas","family":"Meunier","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8746-5713","authenticated-orcid":false,"given":"Rapha\u00ebl","family":"Voituriez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,10,9]]},"reference":[{"key":"2468_CR1","doi-asserted-by":"crossref","unstructured":"Alamichel C, Meunier N (2025) Existence of traveling wave for a coupled incompressible Darcy\u2019s free boundary model with undercooling effect and surface tension. arXiv:2501.04576","DOI":"10.1051\/m2an\/2026013"},{"key":"2468_CR2","unstructured":"Alazard T, Magliocca M, Meunier N (2025) Mathematical study of a coupled incompressible Darcy s free boundary problem with surface tension. Journal de l\u2019\u00c9cole Polytechnique"},{"issue":"27","key":"2468_CR3","doi-asserted-by":"publisher","first-page":"10891","DOI":"10.1073\/pnas.1117814109","volume":"109","author":"E Anon","year":"2012","unstructured":"Anon E, Serra-Picamal X, Hersen P, Gauthier NC, Sheetz MP, Trepat X, Ladoux B (2012) Cell crawling mediates collective cell migration to close undamaged epithelial gaps. Proc Natl Acad Sci U S A 109(27):10891\u20136","journal-title":"Proc Natl Acad Sci U S A"},{"issue":"16","key":"2468_CR4","doi-asserted-by":"publisher","first-page":"5045","DOI":"10.1073\/pnas.1417257112","volume":"112","author":"E Barnhart","year":"2015","unstructured":"Barnhart E, Lee K-C, Allen GM, Theriot JA, Mogilner A (2015) Balance between cell-substrate adhesion and myosin contraction determines the frequency of motility initiation in fish keratocytes. Proc Natl Acad Sci U S A 112(16):5045\u201350","journal-title":"Proc Natl Acad Sci U S A"},{"issue":"2","key":"2468_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1140\/epjp\/i2011-11019-7","volume":"126","author":"M Ben Amar","year":"2011","unstructured":"Ben Amar M, Manyuhina O, Napoli G (2011) Cell motility: a viscous fingering analysis of active gels. Eur Phys J Plus 126(2):1\u201315","journal-title":"Eur Phys J Plus"},{"key":"2468_CR6","doi-asserted-by":"crossref","unstructured":"Berlyand L, Fuhrmann J, Rybalko V (2018) Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility. Commun Math Sci","DOI":"10.4310\/CMS.2018.v16.n3.a6"},{"issue":"1","key":"2468_CR7","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1007\/s00205-025-02153-5","volume":"250","author":"L Berlyand","year":"2026","unstructured":"Berlyand L, Safsten CA, Truskinovsky L (2026) Nonlinear stability in a free boundary model of active locomotion. Arch Ration Mech Anal 250(1):2","journal-title":"Arch Ration Mech Anal"},{"issue":"7","key":"2468_CR8","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.110.078102","volume":"110","author":"C Blanch-Mercader","year":"2013","unstructured":"Blanch-Mercader C, Casademunt J (2013) Spontaneous motility of actin lamellar fragments. Phys Rev Lett 110(7):078102","journal-title":"Phys Rev Lett"},{"issue":"2","key":"2468_CR9","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.ceb.2011.12.004","volume":"24","author":"JJ Bravo-Cordero","year":"2012","unstructured":"Bravo-Cordero JJ, Hodgson L, Condeelis J (2012) Directed cell invasion and migration during metastasis. Curr Opin Cell Biol 24(2):277\u201383","journal-title":"Curr Opin Cell Biol"},{"issue":"2","key":"2468_CR10","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/15\/2\/025022","volume":"15","author":"AC Callan-Jones","year":"2013","unstructured":"Callan-Jones AC, Voituriez R (2013) Active gel model of amoeboid cell motility. New J Phys 15(2):025022","journal-title":"New J Phys"},{"issue":"25","key":"2468_CR11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.100.258106","volume":"100","author":"AC Callan-Jones","year":"2008","unstructured":"Callan-Jones AC, Joanny J-F, Prost J (2008) Viscous-fingering-like instability of cell fragments. Phys Rev Lett 100(25):258106","journal-title":"Phys Rev Lett"},{"issue":"2","key":"2468_CR12","doi-asserted-by":"publisher","first-page":"594","DOI":"10.1137\/11083486X","volume":"72","author":"V Calvez","year":"2012","unstructured":"Calvez V, Hawkins RJ, Meunier N, Voituriez R (2012) Analysis of a nonlocal model for spontaneous cell polarization. SIAM J Appl Math 72(2):594\u2013622","journal-title":"SIAM J Appl Math"},{"key":"2468_CR13","doi-asserted-by":"crossref","unstructured":"Calvez V, Meunier N, Muller N, Voituriez R (2013) Numerical simulation of the dynamics of molecular markers involved in cell polarisation","DOI":"10.1007\/978-1-4614-7828-7_6"},{"issue":"4","key":"2468_CR14","doi-asserted-by":"publisher","first-page":"3843","DOI":"10.1137\/19M1267969","volume":"52","author":"A Cucchi","year":"2020","unstructured":"Cucchi A, Mellet A, Meunier N (2020) A Cahn\u2013Hilliard model for cell motility. SIAM J Math Anal 52(4):3843\u20133880. https:\/\/doi.org\/10.1137\/19M1267969","journal-title":"SIAM J Math Anal"},{"issue":"5","key":"2468_CR15","doi-asserted-by":"publisher","first-page":"2381","DOI":"10.3934\/dcds.2021194","volume":"42","author":"A Cucchi","year":"2022","unstructured":"Cucchi A, Mellet A, Meunier N (2022) Self polarization and traveling wave in a model for cell crawling migration. Discrete Contin Dyn Syst Ser A 42(5):2381. https:\/\/doi.org\/10.3934\/dcds.2021194","journal-title":"Discrete Contin Dyn Syst Ser A"},{"issue":"6","key":"2468_CR16","doi-asserted-by":"publisher","first-page":"2040","DOI":"10.1137\/16M1088715","volume":"77","author":"C Etchegaray","year":"2017","unstructured":"Etchegaray C, Meunier N, Voituriez R (2017) Analysis of a nonlocal and nonlinear Fokker\u2013Planck model for cell crawling migration. SIAM J Appl Math 77(6):2040\u20132065","journal-title":"SIAM J Appl Math"},{"key":"2468_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2025.113787","volume":"453","author":"C Garc\u00eda","year":"2026","unstructured":"Garc\u00eda C, Magliocca M, Meunier N (2026) Traveling motility of actin lamellar fragments under spontaneous symmetry breaking. J Differ Equ 453:113787","journal-title":"J Differ Equ"},{"key":"2468_CR18","doi-asserted-by":"crossref","unstructured":"Hawkins RJ, B\u00e9nichou O, Piel M, Voituriez R (2009) Rebuilding cytoskeleton roads: active-transport-induced polarization of cells. Phys Rev E Stat Nonlin Soft Matter Phys 80(4 Pt 1):040903","DOI":"10.1103\/PhysRevE.80.040903"},{"key":"2468_CR19","doi-asserted-by":"crossref","unstructured":"Jammalamadaka SR, SenGupta A (2001) Topics in circular statistics","DOI":"10.1142\/4031"},{"issue":"1","key":"2468_CR20","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1016\/j.physrep.2007.02.018","volume":"449","author":"F J\u00fclicher","year":"2007","unstructured":"J\u00fclicher F, Kruse K, Prost J, Joanny J-F (2007) Active behavior of the cytoskeleton. Phys Rep 449(1):3\u201328","journal-title":"Phys Rep"},{"issue":"2","key":"2468_CR21","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1088\/1478-3975\/3\/2\/005","volume":"3","author":"K Kruse","year":"2006","unstructured":"Kruse K, Joanny JF, J\u00fclicher F, Prost J (2006) Contractility and retrograde flow in lamellipodium motion. Phys Biol 3(2):130\u20137","journal-title":"Phys Biol"},{"issue":"3","key":"2468_CR22","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/S0092-8674(00)81280-5","volume":"84","author":"DA Lauffenburger","year":"1996","unstructured":"Lauffenburger DA, Horwitz AF (1996) Cell migration: a physically integrated molecular process. Cell 84(3):359\u201369","journal-title":"Cell"},{"key":"2468_CR23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.101.022404","volume":"101","author":"I Lavi","year":"2020","unstructured":"Lavi I, Meunier N, Voituriez R, Casademunt J (2020) Motility and morphodynamics of confined cells. Phys Rev E 101:022404. https:\/\/doi.org\/10.1103\/PhysRevE.101.022404","journal-title":"Phys Rev E"},{"issue":"6","key":"2468_CR24","doi-asserted-by":"publisher","first-page":"1589","DOI":"10.4310\/CMS.2022.v20.n6.a6","volume":"20","author":"T Lepoutre","year":"2022","unstructured":"Lepoutre T, Meunier N (2022) Analysis of a model of cell crawling migration. Commun Math Sci 20(6):1589\u20131611. https:\/\/doi.org\/10.4310\/CMS.2022.v20.n6.a6","journal-title":"Commun Math Sci"},{"issue":"2","key":"2468_CR25","doi-asserted-by":"publisher","first-page":"152","DOI":"10.1016\/j.matpur.2013.05.006","volume":"101","author":"T Lepoutre","year":"2014","unstructured":"Lepoutre T, Meunier N, Muller N (2014) Cell polarisation model: the 1D case. Journal des Mathematiques Pures et Appliquees 101(2):152\u2013171","journal-title":"Journal des Mathematiques Pures et Appliquees"},{"issue":"17","key":"2468_CR26","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1016\/j.cub.2012.07.052","volume":"22","author":"P Maiuri","year":"2012","unstructured":"Maiuri P, Terriac E, Paul-Gilloteaux P, Vignaud T, McNally K, Onuffer J, Thorn K, Nguyen PA, Georgoulia N, Soong D et al (2012) The first world cell race. Curr Biol 22(17):673\u2013675","journal-title":"Curr Biol"},{"issue":"2","key":"2468_CR27","doi-asserted-by":"publisher","first-page":"374","DOI":"10.1016\/j.cell.2015.01.056","volume":"161","author":"P Maiuri","year":"2015","unstructured":"Maiuri P, Rupprecht J-F, Wieser S, Ruprecht V, B\u00e9nichou O, Carpi N, Coppey M, De Beco S, Gov N, Heisenberg C-P, Lage Crespo C, Lautenschlaeger F, Le Berre M, Lennon-Dumenil A-M, Raab M, Thiam H-R, Piel M, Sixt M, Voituriez R (2015) Actin flows mediate a universal coupling between cell speed and cell persistence. Cell 161(2):374\u201386","journal-title":"Cell"},{"key":"2468_CR28","doi-asserted-by":"crossref","unstructured":"Meunier N, Souplet P (2025) Convergence, concentration and critical mass phenomena in a model of cell motion with boundary signal production","DOI":"10.2422\/2036-2145.202305_012"},{"key":"2468_CR29","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1146\/annurev-fluid-010816-060238","volume":"50","author":"A Mogilner","year":"2018","unstructured":"Mogilner A, Manhart A (2018) Intracellular fluid mechanics: coupling cytoplasmic flow with active cytoskeletal gel. Annu Rev Fluid Mech 50:347\u2013370","journal-title":"Annu Rev Fluid Mech"},{"key":"2468_CR30","unstructured":"Muller N (2013) \u00c9tudes math\u00e9matiques et num\u00e9riques de probl\u00e8mes non-lin\u00e9aires et non-locaux issus de la biologie. PhD thesis, Universit\u00e9 Paris Descartes"},{"issue":"4","key":"2468_CR31","doi-asserted-by":"publisher","first-page":"1004795","DOI":"10.1371\/journal.pcbi.1004795","volume":"12","author":"N Muller","year":"2016","unstructured":"Muller N, Piel M, Calvez V, Voituriez R, Gon\u00e7alves-S\u00e1 J, Guo C-L, Jiang X, Murray A, Meunier N (2016) A predictive model for yeast cell polarization in pheromone gradients. PLoS Comput Biol 12(4):1004795","journal-title":"PLoS Comput Biol"},{"issue":"11","key":"2468_CR32","doi-asserted-by":"publisher","first-page":"1005862","DOI":"10.1371\/journal.pcbi.1005862","volume":"13","author":"M Nickaeen","year":"2017","unstructured":"Nickaeen M, Novak IL, Pulford S, Rumack A, Brandon J, Slepchenko BM, Mogilner A (2017) A free-boundary model of a motile cell explains turning behavior. PLoS Comput Biol 13(11):1005862","journal-title":"PLoS Comput Biol"},{"issue":"8","key":"2468_CR33","doi-asserted-by":"publisher","first-page":"4553","DOI":"10.3934\/dcds.2016.36.4553","volume":"36","author":"D Oelz","year":"2016","unstructured":"Oelz D, Mogilner A (2016) A drift-diffusion model for molecular motor transport in anisotropic filament bundles. Discrete Contin Dynam Syst 36(8):4553\u20134567. https:\/\/doi.org\/10.3934\/dcds.2016.36.4553","journal-title":"Discrete Contin Dynam Syst"},{"issue":"2","key":"2468_CR34","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1038\/nphys3224","volume":"11","author":"J Prost","year":"2015","unstructured":"Prost J, J\u00fclicher F, Joanny J (2015) Active gel physics. Nat Phys 11(2):111\u2013117","journal-title":"Nat Phys"},{"key":"2468_CR35","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.111.108102","volume":"111","author":"P Recho","year":"2013","unstructured":"Recho P, Putelat T, Truskinovsky L (2013) Contraction-driven cell motility. Phys Rev Lett 111:108102. https:\/\/doi.org\/10.1103\/PhysRevLett.111.108102","journal-title":"Phys Rev Lett"},{"key":"2468_CR36","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1016\/j.jmps.2015.08.006","volume":"84","author":"P Recho","year":"2015","unstructured":"Recho P, Putelat T, Truskinovsky L (2015) Mechanics of motility initiation and motility arrest in crawling cells. J Mech Phys Solids 84:469\u2013505. https:\/\/doi.org\/10.1016\/j.jmps.2015.08.006","journal-title":"J Mech Phys Solids"},{"key":"2468_CR37","doi-asserted-by":"publisher","first-page":"1799","DOI":"10.1090\/tran\/8824","volume":"376","author":"V Rybalko","year":"2023","unstructured":"Rybalko V, Berlyand L (2023) Emergence of traveling waves and their stability in a free boundary model of cell motility. Trans Am Math Soc 376:1799\u20131844","journal-title":"Trans Am Math Soc"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02468-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-026-02468-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-026-02468-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T05:17:08Z","timestamp":1791523028000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-026-02468-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,9]]},"references-count":37,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2026,11]]}},"alternative-id":["2468"],"URL":"https:\/\/doi.org\/10.1007\/s00285-026-02468-1","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,10,9]]},"assertion":[{"value":"3 June 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 July 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 September 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 October 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors have no conflict of interest to declare that are relevant to the content of this article.","order":1,"name":"Ethics","label":"Conflict of interest","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","label":"Ethics approval and consent to participate","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"All authors gave their consent for publication.","order":3,"name":"Ethics","label":"Consent for publication","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"54"}}