[{"status":"public","abstract":[{"text":"Modern experimental methods in programmable self-assembly make it possible to precisely design particle concentrations, shapes and interactions. However, more physical insight is needed before we can take full advantage of this vast design space to assemble nanostructures with complex form and function. Here we show how a substantial part of this design space can be quickly and comprehensively understood by identifying a class of thermodynamic constraints that act on it. These thermodynamic constraints form a high-dimensional convex polyhedron that determines which nanostructures can be assembled at high equilibrium yield and reveals limitations that govern the coexistence of structures. We validate our predictions through detailed, quantitative assembly experiments of nanoscale particles synthesized using DNA origami. Our results uncover physical relationships underpinning many-component programmable self-assembly in equilibrium and form the basis for robust inverse design, applicable to various systems from biological protein complexes to synthetic nanomachines.","lang":"eng"}],"article_processing_charge":"Yes (via OA deal)","publication":"Nature Physics","ddc":["570","540"],"language":[{"iso":"eng"}],"related_material":{"link":[{"url":"https://ista.ac.at/en/news/behind-natures-blueprints/","description":"News on ISTA website","relation":"press_release"}]},"year":"2026","doi":"10.1038/s41567-025-03120-3","month":"01","has_accepted_license":"1","publication_status":"epub_ahead","type":"journal_article","article_type":"original","publication_identifier":{"eissn":["1745-2481"],"issn":["1745-2473"]},"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","image":"/images/cc_by.png"},"day":"08","corr_author":"1","author":[{"first_name":"Maximilian","id":"5eb8629e-15b2-11ec-abd3-e6f3e5e01f32","full_name":"Hübl, Maximilian","last_name":"Hübl"},{"first_name":"Thomas E.","full_name":"Videbæk, Thomas E.","last_name":"Videbæk"},{"first_name":"Daichi","full_name":"Hayakawa, Daichi","last_name":"Hayakawa"},{"first_name":"W. Benjamin","last_name":"Rogers","full_name":"Rogers, W. Benjamin"},{"last_name":"Goodrich","full_name":"Goodrich, Carl Peter","id":"EB352CD2-F68A-11E9-89C5-A432E6697425","first_name":"Carl Peter","orcid":"0000-0002-1307-5074"}],"title":"A polyhedral structure controls programmable self-assembly","oa_version":"Published Version","acknowledgement":"We thank B. Isaac and A. Tiano for their technical support with the electron microscopy and S. Waitukaitis for helpful comments on the manuscript. The TEM images were prepared and imaged at the Brandeis Electron Microscopy facility. This work was supported by the Gesellschaft für Forschungsförderung Niederösterreich under project FTI23-G-011 (M.C.H. and C.P.G.), the Brandeis University Materials Research Science and Engineering Center (MRSEC) under grant number NSF DMR-2011846 (T.E.V., D.H. and W.B.R.) and the Smith Family Foundation (W.B.R.). Open access funding provided by Institute of Science and Technology (IST Austria).","publisher":"Springer Nature","date_published":"2026-01-08T00:00:00Z","_id":"21006","OA_type":"hybrid","quality_controlled":"1","PlanS_conform":"1","date_created":"2026-01-20T10:02:19Z","user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","oa":1,"citation":{"chicago":"Hübl, Maximilian, Thomas E. Videbæk, Daichi Hayakawa, W. Benjamin Rogers, and Carl Peter Goodrich. “A Polyhedral Structure Controls Programmable Self-Assembly.” <i>Nature Physics</i>. Springer Nature, 2026. <a href=\"https://doi.org/10.1038/s41567-025-03120-3\">https://doi.org/10.1038/s41567-025-03120-3</a>.","mla":"Hübl, Maximilian, et al. “A Polyhedral Structure Controls Programmable Self-Assembly.” <i>Nature Physics</i>, Springer Nature, 2026, doi:<a href=\"https://doi.org/10.1038/s41567-025-03120-3\">10.1038/s41567-025-03120-3</a>.","ieee":"M. Hübl, T. E. Videbæk, D. Hayakawa, W. B. Rogers, and C. P. Goodrich, “A polyhedral structure controls programmable self-assembly,” <i>Nature Physics</i>. Springer Nature, 2026.","ista":"Hübl M, Videbæk TE, Hayakawa D, Rogers WB, Goodrich CP. 2026. A polyhedral structure controls programmable self-assembly. Nature Physics.","ama":"Hübl M, Videbæk TE, Hayakawa D, Rogers WB, Goodrich CP. A polyhedral structure controls programmable self-assembly. <i>Nature Physics</i>. 2026. doi:<a href=\"https://doi.org/10.1038/s41567-025-03120-3\">10.1038/s41567-025-03120-3</a>","short":"M. Hübl, T.E. Videbæk, D. Hayakawa, W.B. Rogers, C.P. Goodrich, Nature Physics (2026).","apa":"Hübl, M., Videbæk, T. E., Hayakawa, D., Rogers, W. B., &#38; Goodrich, C. P. (2026). A polyhedral structure controls programmable self-assembly. <i>Nature Physics</i>. Springer Nature. <a href=\"https://doi.org/10.1038/s41567-025-03120-3\">https://doi.org/10.1038/s41567-025-03120-3</a>"},"main_file_link":[{"url":"https://doi.org/10.1038/s41567-025-03120-3","open_access":"1"}],"OA_place":"publisher","scopus_import":"1","date_updated":"2026-04-28T11:56:45Z","department":[{"_id":"CaGo"},{"_id":"GradSch"}],"project":[{"grant_number":"FTI23-G-011","_id":"8dd93da8-16d5-11f0-9cad-d2c70200d9a5","name":"Dynamically reconfigurable self-assembly with triangular DNA-origami bricks"}]}]