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Search Results (5)

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Keywords = Cayley graph, protein cage

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23 pages, 925 KiB  
Article
Bi-Symmetric Polyhedral Cages with Nearly Maximally Connected Faces and Small Holes
by Bernard Piette
Symmetry 2025, 17(6), 940; https://doi.org/10.3390/sym17060940 - 12 Jun 2025
Viewed by 450
Abstract
Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where [...] Read more.
Polyhedral cages (p-cages) provide a good description of the geometry of some families of artificial protein cages. In this paper we identify p-cages made out of two families of equivalent polygonal faces/protein rings, where each face has at least four neighbours and where the holes are contributed by at most four faces. We start the construction from a planar graph made out of two families of equivalent nodes. We construct the dual of the solid corresponding to that graph, and we tile its faces with regular or nearly regular polygons. We define an energy function describing the amount of irregularity of the p-cages, which we then minimise using a simulated annealing algorithm. We analyse over 600,000 possible geometries but restrict ourselves to p-cages made out of faces with deformations not exceeding 10%. We then present graphically some of the most promising geometries for protein nanocages. Full article
(This article belongs to the Special Issue Chemistry: Symmetry/Asymmetry—Feature Papers and Reviews)
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20 pages, 731 KiB  
Article
Bi-Symmetric Polyhedral Cages with Maximally Connected Faces and Small Holes
by Bernard Piette and Árpad Lukács
Symmetry 2025, 17(1), 101; https://doi.org/10.3390/sym17010101 - 10 Jan 2025
Cited by 2 | Viewed by 765
Abstract
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 [...] Read more.
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%. Full article
(This article belongs to the Section Mathematics)
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22 pages, 693 KiB  
Article
Biequivalent Planar Graphs
by Bernard Piette
Axioms 2024, 13(7), 437; https://doi.org/10.3390/axioms13070437 - 28 Jun 2024
Cited by 3 | Viewed by 994
Abstract
We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an [...] Read more.
We define biequivalent planar graphs, which are a generalisation of the uniform polyhedron graphs, as planar graphs made out of two families of equivalent nodes. Such graphs are required to identify polyhedral cages with geometries suitable for artificial protein cages. We use an algebraic method, which is followed by an algorithmic method, to determine all such graphs with up to 300 nodes each with valencies ranging between three and six. We also present a graphic representation of every graph found. Full article
(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics)
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29 pages, 756 KiB  
Article
Near-Miss Bi-Homogenous Symmetric Polyhedral Cages
by Bernard Piette and Árpad Lukács
Symmetry 2023, 15(9), 1804; https://doi.org/10.3390/sym15091804 - 21 Sep 2023
Cited by 4 | Viewed by 1711
Abstract
Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of [...] Read more.
Following the discovery of an artificial protein cage with a paradoxical geometry, we extend the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages, for which all the faces are equivalent, and define bi-homogeneous symmetric polyhedral cages made of two different types of faces, where all the faces of a given type are equivalent. We parametrise the possible connectivity configurations for such cages, analytically derive p-cages that are regular, and numerically compute near-symmetric p-cages made of polygons with 6 to 18 edges and with deformation not exceeding 10%. Full article
(This article belongs to the Special Issue Symmetry and Asymmetry in Nature-Inspired, Bio-Based Materials)
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28 pages, 825 KiB  
Article
Near-Miss Symmetric Polyhedral Cages
by Bernard M. A. G. Piette and Árpad Lukács
Symmetry 2023, 15(3), 717; https://doi.org/10.3390/sym15030717 - 13 Mar 2023
Cited by 6 | Viewed by 3113
Abstract
Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons [...] Read more.
Following the experimental discovery of several nearly symmetric protein cages, we define the concept of homogeneous symmetric congruent equivalent near-miss polyhedral cages made out of P-gons. We use group theory to parameterize the possible configurations and we minimize the irregularity of the P-gons numerically to construct all such polyhedral cages for P=6 to P=20 with deformation of up to 10%. Full article
(This article belongs to the Section Mathematics)
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