{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T11:48:58Z","timestamp":1791978538599,"version":"4.4.0"},"reference-count":48,"publisher":"AIP Publishing","issue":"14","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["101097272"],"award-info":[{"award-number":["101097272"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-22-PEBB-0009"],"award-info":[{"award-number":["ANR-22-PEBB-0009"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,10,14]]},"abstract":"<jats:p>Path collective variables reduce a high-dimensional transition to a progress coordinate, s, and an orthogonal distance, z. Computing the free energy along s often requires restraining z, so that sampling embraces a tube centered on the reference path. However, a conventional fixed half-harmonic wall demands an a priori wall position that is both system dependent and s-dependent. Too tight a tube biases the path-projected free energy. Conversely, too loose a tube induces numerical instability and inter-channel leakage. We present an adaptive tube restraint, the wall position of which evolves on the fly to track a fixed contour \u0394F\u2217 of the orthogonal free energy F(z\u2223s), widening automatically in flat basins and narrowing at pinched saddles, using a user-selected transverse energy scale. We prove that, for a locally harmonic perpendicular well, a contour-following tube captures an s-independent fraction of the orthogonal partition function, so that in the hard-wall limit, its bias cancels out. In stark contrast, the bias of a fixed-width tube varies with s and distorts the free-energy landscape. We probe the algorithm on several model potentials, on N-acetyl-N\u2032-methylalanylamide isomerization, on the folding of the mini-protein chignolin, and on the folding-upon-binding of the ribose-binding protein. The method is implemented in the open-source Colvars library and is, therefore, usable within popular MD engines, such as NAMD, LAMMPS, GROMACS, and Tinker-HP.<\/jats:p>","DOI":"10.1063\/5.0351430","type":"journal-article","created":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T11:01:23Z","timestamp":1791975683000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A self-adaptive tube restraint for free-energy calculations along path collective variables"],"prefix":"10.1063","volume":"165","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6678-7801","authenticated-orcid":false,"given":"Radu A.","family":"Talmazan","sequence":"first","affiliation":[{"name":"Laboratoire de Physique et Chimie Th\u00e9oriques, Unit\u00e9 Mixte de Recherche No. 7019, Universit\u00e9 de Lorraine 1 , B.P. 70239, 54506 Vand\u0153uvre-l\u00e8s-Nancy Cedex,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3553-4718","authenticated-orcid":false,"given":"Cheng Giuseppe","family":"Chen","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique et Chimie Th\u00e9oriques, Unit\u00e9 Mixte de Recherche No. 7019, Universit\u00e9 de Lorraine 1 , B.P. 70239, 54506 Vand\u0153uvre-l\u00e8s-Nancy Cedex,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6914-7348","authenticated-orcid":false,"given":"Chenyu","family":"Tang","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique et Chimie Th\u00e9oriques, Unit\u00e9 Mixte de Recherche No. 7019, Universit\u00e9 de Lorraine 1 , B.P. 70239, 54506 Vand\u0153uvre-l\u00e8s-Nancy Cedex,","place":["France"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9122-1698","authenticated-orcid":false,"given":"Christophe","family":"Chipot","sequence":"additional","affiliation":[{"name":"Laboratoire de Physique et Chimie Th\u00e9oriques, Unit\u00e9 Mixte de Recherche No. 7019, Universit\u00e9 de Lorraine 1 , B.P. 70239, 54506 Vand\u0153uvre-l\u00e8s-Nancy Cedex,","place":["France"]},{"name":"Department of Biochemistry and Molecular Biology, University of Chicago 2 , Chicago, Illinois 60637,","place":["USA"]},{"name":"Department of Chemistry, University of Chicago 3 , Chicago, Illinois 60637,","place":["USA"]},{"name":"Theoretical and Computational Biophysics Group, Beckman Institute, and Department of Physics, University of Illinois at Urbana-Champaign 4 , Urbana, Illinois 61801,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,14]]},"reference":[{"key":"2026101407011957300_c1","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1002\/wcms.1157","volume":"4","year":"2014","journal-title":"Wiley Interdiscip. Rev.: Comput. Mol. Sci."},{"key":"2026101407011957300_c2","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.33011\/livecoms.4.1.1583","volume":"4","year":"2022","journal-title":"Living J. Comp. Mol. Sci."},{"key":"2026101407011957300_c3","doi-asserted-by":"publisher","first-page":"12562","DOI":"10.1073\/pnas.202427399","volume":"99","year":"2002","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026101407011957300_c4","doi-asserted-by":"publisher","first-page":"126601","DOI":"10.1088\/0034-4885\/71\/12\/126601","volume":"71","year":"2008","journal-title":"Rep. Prog. Phys."},{"key":"2026101407011957300_c5","doi-asserted-by":"publisher","first-page":"020603","DOI":"10.1103\/physrevlett.100.020603","volume":"100","year":"2008","journal-title":"Phys. Rev. Lett."},{"key":"2026101407011957300_c6","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1146\/annurev-physchem-040215-112229","volume":"67","year":"2016","journal-title":"Annu. Rev. Phys. Chem."},{"key":"2026101407011957300_c7","doi-asserted-by":"publisher","first-page":"090601","DOI":"10.1103\/physrevlett.113.090601","volume":"113","year":"2014","journal-title":"Phys. Rev. Lett."},{"key":"2026101407011957300_c8","doi-asserted-by":"publisher","first-page":"2731","DOI":"10.1021\/acs.jpclett.0c00497","volume":"11","year":"2020","journal-title":"J.\u00a0Phys. Chem. Lett."},{"key":"2026101407011957300_c9","doi-asserted-by":"publisher","first-page":"9169","DOI":"10.1063\/1.1410978","volume":"115","year":"2001","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c10","doi-asserted-by":"publisher","first-page":"144120","DOI":"10.1063\/1.2829861","volume":"128","year":"2008","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c11","doi-asserted-by":"publisher","first-page":"1129","DOI":"10.1021\/jp506633n","volume":"119","year":"2015","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c12","doi-asserted-by":"publisher","first-page":"3676","DOI":"10.1021\/acs.jpcb.6b10055","volume":"121","year":"2017","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c13","doi-asserted-by":"publisher","first-page":"3506","DOI":"10.1021\/acs.jctc.6b00447","volume":"12","year":"2016","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c14","doi-asserted-by":"publisher","first-page":"4738","DOI":"10.1021\/acs.jpclett.8b01994","volume":"9","year":"2018","journal-title":"J.\u00a0Phys. Chem. Lett."},{"key":"2026101407011957300_c15","doi-asserted-by":"publisher","first-page":"6732","DOI":"10.1073\/pnas.0408098102","volume":"102","year":"2005","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026101407011957300_c16","doi-asserted-by":"publisher","first-page":"3345","DOI":"10.1080\/00268976.2013.813594","volume":"111","year":"2013","journal-title":"Mol. Phys."},{"key":"2026101407011957300_c17","doi-asserted-by":"publisher","first-page":"3651","DOI":"10.1021\/acs.jcim.2c00770","volume":"62","year":"2022","journal-title":"J.\u00a0Chem. Inf. Model."},{"key":"2026101407011957300_c18","doi-asserted-by":"publisher","first-page":"1165","DOI":"10.1021\/ct3001377","volume":"8","year":"2012","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c19","doi-asserted-by":"publisher","first-page":"5867","DOI":"10.1021\/acs.jctc.6c00492","volume":"22","year":"2026","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c20","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1021\/acs.jcim.5c02906","volume":"66","year":"2026","journal-title":"J.\u00a0Chem. Inf. Model."},{"key":"2026101407011957300_c21","doi-asserted-by":"publisher","first-page":"052301","DOI":"10.1103\/physrevb.66.052301","volume":"66","year":"2002","journal-title":"Phys. Rev. B"},{"key":"2026101407011957300_c22","doi-asserted-by":"publisher","first-page":"6688","DOI":"10.1021\/jp0455430","volume":"109","year":"2005","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c23","doi-asserted-by":"publisher","first-page":"024106","DOI":"10.1063\/1.2212942","volume":"125","year":"2006","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c24","doi-asserted-by":"publisher","first-page":"194103","DOI":"10.1063\/1.3130083","volume":"130","year":"2009","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c25","doi-asserted-by":"publisher","first-page":"5214","DOI":"10.1021\/acs.jpcb.3c02028","volume":"127","year":"2023","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c26","doi-asserted-by":"publisher","first-page":"6358","DOI":"10.1073\/pnas.1303186110","volume":"110","year":"2013","journal-title":"Proc. Natl. Acad. Sci. U. S. A."},{"key":"2026101407011957300_c27","doi-asserted-by":"publisher","first-page":"2837","DOI":"10.1038\/s41596-020-0342-4","volume":"15","year":"2020","journal-title":"Nat. Protoc."},{"key":"2026101407011957300_c28","doi-asserted-by":"publisher","first-page":"2898","DOI":"10.1021\/jacsau.5c00460","volume":"5","year":"2025","journal-title":"JACS Au"},{"key":"2026101407011957300_c29","doi-asserted-by":"publisher","first-page":"020601","DOI":"10.1103\/physrevlett.109.020601","volume":"109","year":"2012","journal-title":"Phys. Rev. Lett."},{"key":"2026101407011957300_c30","doi-asserted-by":"publisher","first-page":"072320","DOI":"10.1063\/1.5027392","volume":"149","year":"2018","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c31","doi-asserted-by":"publisher","first-page":"9202","DOI":"10.1021\/acs.jctc.3c00938","volume":"19","year":"2023","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c32","doi-asserted-by":"publisher","first-page":"144104","DOI":"10.1063\/5.0288060","volume":"163","year":"2025","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c33","doi-asserted-by":"publisher","first-page":"174109","DOI":"10.1063\/5.0202156","volume":"160","year":"2024","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c34","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1021\/acs.jctc.8b00563","volume":"15","year":"2019","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c35","doi-asserted-by":"publisher","first-page":"11108","DOI":"10.1021\/acs.jpcb.4c05604","volume":"128","year":"2024","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c36","doi-asserted-by":"publisher","first-page":"054103","DOI":"10.1063\/1.2432340","volume":"126","year":"2007","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c37","volume-title":"NIST Handbook of Mathematical Functions","author":"Olver","year":"2010"},{"key":"2026101407011957300_c38","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1021\/jp504920s","volume":"119","year":"2015","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c39","doi-asserted-by":"publisher","first-page":"1615","DOI":"10.1002\/jcc.21305","volume":"30","year":"2009","journal-title":"J.\u00a0Comput. Chem."},{"key":"2026101407011957300_c40","doi-asserted-by":"publisher","first-page":"3696","DOI":"10.1021\/acs.jctc.5b00255","volume":"11","year":"2015","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c41","doi-asserted-by":"publisher","first-page":"044130","DOI":"10.1063\/5.0014475","volume":"153","year":"2020","journal-title":"J.\u00a0Chem. Phys."},{"key":"2026101407011957300_c42","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1038\/s43588-025-00828-3","volume":"5","year":"2025","journal-title":"Nat. Comput. Sci."},{"key":"2026101407011957300_c43","doi-asserted-by":"publisher","first-page":"5199","DOI":"10.1021\/acs.jctc.9b00761","volume":"15","year":"2019","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c44","doi-asserted-by":"publisher","first-page":"3886","DOI":"10.1021\/acs.jctc.1c00103","volume":"17","year":"2021","journal-title":"J.\u00a0Chem. Theory Comput."},{"key":"2026101407011957300_c45","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1038\/s43588-024-00645-0","volume":"4","year":"2024","journal-title":"Nat. Comput. Sci."},{"key":"2026101407011957300_c46","doi-asserted-by":"publisher","first-page":"30206","DOI":"10.1016\/s0021-9258(18)43798-2","volume":"269","year":"1994","journal-title":"J.\u00a0Biol. Chem."},{"key":"2026101407011957300_c47","doi-asserted-by":"publisher","first-page":"2898","DOI":"10.1021\/acs.jpcb.0c11600","volume":"125","year":"2021","journal-title":"J.\u00a0Phys. Chem. B"},{"key":"2026101407011957300_c48","doi-asserted-by":"publisher","first-page":"6245","DOI":"10.1038\/s41467-026-74165-y","volume":"17","year":"2026","journal-title":"Nat. Commun."}],"container-title":["The Journal of Chemical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0351430\/21170660\/144117_1_5.0351430.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0351430\/21170660\/144117_1_5.0351430.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T11:01:24Z","timestamp":1791975684000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jcp\/article\/165\/14\/144117\/3406520\/A-self-adaptive-tube-restraint-for-free-energy"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,14]]},"references-count":48,"journal-issue":{"issue":"14","published-print":{"date-parts":[[2026,10,14]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0351430","relation":{},"ISSN":["0021-9606","1089-7690"],"issn-type":[{"value":"0021-9606","type":"print"},{"value":"1089-7690","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,10,14]]},"published":{"date-parts":[[2026,10,14]]},"article-number":"144117"}}