{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:40:47Z","timestamp":1759920047683,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T00:00:00Z","timestamp":1736467200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Leverhulme Trust Research Project","award":["RPG-2020-306"],"award-info":[{"award-number":["RPG-2020-306"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of one family have five neighbors and P1 edges, while those of the other family have six neighbors and P2 edges. We restrict ourselves to polyhedral cages where the holes are adjacent to four faces at most. We characterize all p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%.<\/jats:p>","DOI":"10.3390\/sym17010101","type":"journal-article","created":{"date-parts":[[2025,1,10]],"date-time":"2025-01-10T08:03:05Z","timestamp":1736496185000},"page":"101","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9777-603X","authenticated-orcid":false,"given":"Bernard","family":"Piette","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5737-1393","authenticated-orcid":false,"given":"\u00c1rpad","family":"Luk\u00e1cs","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Durham University, Durham DH1 3LE, UK"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2056","DOI":"10.1021\/nl3002155","article-title":"Gold nanoparticle-induced formation of artificial protein capsids","volume":"12","author":"Malay","year":"2012","journal-title":"Nano Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1038\/s41586-019-1185-4","article-title":"An ultra-stable gold-coordinated protein cage displaying reversible assembly","volume":"569","author":"Malay","year":"2019","journal-title":"Nature"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3187","DOI":"10.1021\/acs.nanolett.1c04222","article-title":"Artificial Protein Cage with Unusual Geometry and Regularly Embedded Gold Nanoparticles","volume":"22","author":"Majsterkiewicz","year":"2022","journal-title":"Nano Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"20210679","DOI":"10.1098\/rspa.2021.0679","article-title":"Characterization of near-miss connectivity-invariant homogeneous convex polyhedral cages","volume":"478","author":"Piette","year":"2022","journal-title":"Proc. R. Soc. A."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Piette, B.M.A.G., and Luk\u00e1cs, A. (2023). Near-Miss Symmetric Polyhedral Cage. Symmetry, 15.","DOI":"10.3390\/sym15091804"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Piette, B.M.A.G., and Luk\u00e1cs, A. (2023). Near-Miss Bi-Homogenous Symmetric Polyhedral Cage. Symmetry, 15.","DOI":"10.3390\/sym15091804"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Piette, B.M.A.G. (2024). Biequivalent Planar Graphs. Axioms, 13.","DOI":"10.3390\/axioms13070437"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"13293","DOI":"10.1039\/C9RA10983H","article-title":"Enzyme encapsulation by protein cages","volume":"10","author":"Chakraborti","year":"2020","journal-title":"RSC Adv."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"e2104964119","DOI":"10.1073\/pnas.2104964119","article-title":"A prototype protein nanocage minimized from carboxysomes with gated oxygen permeability","volume":"119","author":"Gao","year":"2022","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"13701","DOI":"10.1021\/acs.chemrev.1c00308","article-title":"Protein Assembly by Design","volume":"12","author":"Zhu","year":"2021","journal-title":"Chem. Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1038\/s41467-021-21184-6","article-title":"Self-assembly and regulation of protein cages from pre-organised coiled-coil modules","volume":"12","author":"Lapenta","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"13480","DOI":"10.1021\/acs.chemrev.0c00672","article-title":"Design and Applications of Water-Soluble Coordination Cages","volume":"120","author":"Percastegui","year":"2020","journal-title":"Chem. Rev."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"172","DOI":"10.1038\/s41586-019-1928-2","article-title":"Constructing protein polyhedra via orthogonal chemical interactions","volume":"578","author":"Golub","year":"2020","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liang, Y., Furukawa, H., Sakamoto, K., Inaba, H., and Matsuura, K. (2022). Anticancer Activity of Reconstituted Ribonuclease S-Decorated Artificial Viral Capsid. ChemBioChem, 23.","DOI":"10.1002\/cbic.202200220"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2018","DOI":"10.1021\/acs.bioconjchem.2c00030","article-title":"Engineering Self-Assembling Protein Nanoparticles for Therapeutic Delivery","volume":"33","author":"Olshefsky","year":"2022","journal-title":"Bioconjug. Chem."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"e2103971","DOI":"10.1002\/advs.202103971","article-title":"Ultrasmall Luminescent Metal Nanoparticles: Surface Engineering Strategies for Biological Targeting and Imaging","volume":"9","author":"Luo","year":"2022","journal-title":"Adv. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"4146","DOI":"10.1021\/acs.biomac.1c00630","article-title":"Artificial Protein Cage Delivers Active Protein Cargos to the Cell Interior","volume":"22","author":"Naskalska","year":"2021","journal-title":"Biomacromolecules"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"5410","DOI":"10.1038\/s41467-020-19112-1","article-title":"Two-tier supramolecular encapsulation of small molecules in a protein cage","volume":"11","author":"Edwardson","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"eabj9424","DOI":"10.1126\/sciadv.abj9424","article-title":"Chemically induced protein cage assembly with programmable opening and cargo release","volume":"8","author":"Stupka","year":"2022","journal-title":"Sci. Adv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4414","DOI":"10.1038\/s41467-019-12367-3","article-title":"Structural puzzles in virology solved with an overarching icosahedral design principle","volume":"10","author":"Twarock","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1021\/acsnanoscienceau.2c00019","article-title":"Shape-morphing of an artificial protein cage with unusual geometry induced by a single amino acid change","volume":"2","author":"Sharma","year":"2022","journal-title":"ACS Nanosci. Au"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"10393","DOI":"10.1021\/acs.chemrev.1c00763","article-title":"Beyond Platonic: How to Build Metal-Organic Polyhedra Capable of Binding Low-Symmetry, Information-Rich Molecular Cargoes","volume":"122","author":"McTernan","year":"2022","journal-title":"Chem. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.mib.2021.03.003","article-title":"Structure-based design of novel polyhedral protein nanomaterials","volume":"61","author":"Khmelinskaia","year":"2021","journal-title":"Curr. Opin. Microbiol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"31817","DOI":"10.1073\/pnas.2015183117","article-title":"A complete rule set for designing symmetry combination materials from protein molecules","volume":"117","author":"Laniado","year":"2020","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_25","first-page":"436","article-title":"An Artificial Protein Cage Made from a 12-Membered Ring","volume":"12","author":"Stupka","year":"2024","journal-title":"J. Mater. Chem."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1126\/science.220.4598.671","article-title":"Optimization by Simulated Annealing","volume":"220","author":"Kirkpatrick","year":"1983","journal-title":"Science"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/101\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T10:26:34Z","timestamp":1759919194000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/1\/101"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,10]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["sym17010101"],"URL":"https:\/\/doi.org\/10.3390\/sym17010101","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,1,10]]}}}