Goldberg, Fuller, Caspar, Klug and Coxeter and a general approach to local symmetry-preserving operations

Cubic polyhedra with icosahedral symmetry where all faces are pentagons or hexagons have been studied in chemistry and biology as well as mathematics. In chemistry one of these is buckminsterfullerene, a pure carbon cage with maximal symmetry, whereas in biology they describe the structure of spherical viruses. Parameterized operations to construct all such polyhedra were first described by Goldberg in 1937 in a mathematical context and later by Caspar and Klug -- not knowing about Goldberg's work -- in 1962 in a biological context. In the meantime Buckminster Fuller also used subdivided icosahedral structures for the construction of his geodesic domes. In 1971 Coxeter published a survey article that refers to these constructions. Subsequently, the literature often refers to the Goldberg-Coxeter construction. This construction is actually that of Caspar and Klug. Moreover, there are essential differences between this (Caspar/Klug/Coxeter) approach and the approaches of Fuller and of Goldberg. We will sketch the different approaches and generalize Goldberg's approach to a systematic one encompassing all local symmetry-preserving operations on polyhedra

Decoration of the truncated tetrahedron - an Archimedean polyhedron - to produce a new class of convex equilateral polyhedra with tetrahedral symmetry

The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (T-d) fullerene cages. Cages in the first series have 28n(2) vertices (n >= 1). Cages in the second (leapfrog) series have 3 x 28n(2). We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (T-d) symmetry, these new polyhedra constitute a new class of "convex equilateral polyhedra with polyhedral symmetry". We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host's polyhedral symmetry

Cryo-EM structure of the mature dengue virus at 3.5-Å resolution.

Regulated by pH, membrane-anchored proteins E and M function during dengue virus maturation and membrane fusion. Our atomic model of the whole virion from cryo-electron microscopy at 3.5-Å resolution reveals that in the mature virus at neutral extracellular pH, the N-terminal 20-amino-acid segment of M (involving three pH-sensing histidines) latches and thereby prevents spring-loaded E fusion protein from prematurely exposing its fusion peptide. This M latch is fastened at an earlier stage, during maturation at acidic pH in the trans-Golgi network. At a later stage, to initiate infection in response to acidic pH in the late endosome, M releases the latch and exposes the fusion peptide. Thus, M serves as a multistep chaperone of E to control the conformational changes accompanying maturation and infection. These pH-sensitive interactions could serve as targets for drug discovery

Cryo-EM model of the bullet-shaped vesicular stomatitis virus.

Vesicular stomatitis virus (VSV) is a bullet-shaped rhabdovirus and a model system of negative-strand RNA viruses. Through direct visualization by means of cryo-electron microscopy, we show that each virion contains two nested, left-handed helices: an outer helix of matrix protein M and an inner helix of nucleoprotein N and RNA. M has a hub domain with four contact sites that link to neighboring M and N subunits, providing rigidity by clamping adjacent turns of the nucleocapsid. Side-by-side interactions between neighboring N subunits are critical for the nucleocapsid to form a bullet shape, and structure-based mutagenesis results support this description. Together, our data suggest a mechanism of VSV assembly in which the nucleocapsid spirals from the tip to become the helical trunk, both subsequently framed and rigidified by the M layer

Four Levels of Hierarchical Organization, Including Noncovalent Chainmail, Brace the Mature Tumor Herpesvirus Capsid against Pressurization

SummaryLike many double-stranded DNA viruses, tumor gammaherpesviruses Epstein-Barr virus and Kaposi’s sarcoma-associated herpesvirus withstand high internal pressure. Bacteriophage HK97 uses covalent chainmail for this purpose, but how this is achieved noncovalently in the much larger gammaherpesvirus capsid is unknown. Our cryoelectron microscopy structure of a gammaherpesvirus capsid reveals a hierarchy of four levels of organization: (1) Within a hexon capsomer, each monomer of the major capsid protein (MCP), 1,378 amino acids and six domains, interacts with its neighboring MCPs at four sites. (2) Neighboring capsomers are linked in pairs by MCP dimerization domains and in groups of three by heterotrimeric triplex proteins. (3) Small (∼280 amino acids) HK97-like domains in MCP monomers alternate with triplex heterotrimers to form a belt that encircles each capsomer. (4) One hundred sixty-two belts concatenate to form noncovalent chainmail. The triplex heterotrimer orchestrates all four levels and likely drives maturation to an angular capsid that can withstand pressurization

Comprehensive Pan-Genomic Characterization of Adrenocortical Carcinoma

SummaryWe describe a comprehensive genomic characterization of adrenocortical carcinoma (ACC). Using this dataset, we expand the catalogue of known ACC driver genes to include PRKAR1A, RPL22, TERF2, CCNE1, and NF1. Genome wide DNA copy-number analysis revealed frequent occurrence of massive DNA loss followed by whole-genome doubling (WGD), which was associated with aggressive clinical course, suggesting WGD is a hallmark of disease progression. Corroborating this hypothesis were increased TERT expression, decreased telomere length, and activation of cell-cycle programs. Integrated subtype analysis identified three ACC subtypes with distinct clinical outcome and molecular alterations which could be captured by a 68-CpG probe DNA-methylation signature, proposing a strategy for clinical stratification of patients based on molecular markers

Some New Symmetric Equilateral Embeddings of Platonic and Archimedean Polyhedra

The icosahedron and the dodecahedron have the same graph structures as their algebraic conjugates, the great dodecahedron and the great stellated dodecahedron. All four polyhedra are equilateral and have planar faces—thus “EP”—and display icosahedral symmetry. However, the latter two (star polyhedra) are non-convex and “pathological” because of intersecting faces. Approaching the problem analytically, we sought alternate EP-embeddings for Platonic and Archimedean solids. We prove that the number of equations—E edge length equations (enforcing equilaterality) and 2 E − 3 F face (torsion) equations (enforcing planarity)—and of variables ( 3 V − 6 ) are equal. Therefore, solutions of the equations up to equivalence generally leave no degrees of freedom. As a result, in general there is a finite (but very large) number of solutions. Unfortunately, even with state-of-the-art computer algebra, the resulting systems of equations are generally too complicated to completely solve within reasonable time. We therefore added an additional constraint, symmetry, specifically requiring solutions to display (at least) tetrahedral symmetry. We found 77 non-classical embeddings, seven without intersecting faces—two, four and one, respectively, for the (graphs of the) dodecahedron, the icosidodecahedron and the rhombicosidodecahedron

A first step towards general local operations preserving orientation-preserving symmetries but not necessarily orientation-reversing automorphisms from Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations

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A Keplerian Ag90 nest of Platonic and Archimedean polyhedra in different symmetry groups

Nested polyhedra are compelling but incredibly complex synthetic targets in cluster chemistry. Here, the authors synthesize a Ag90 nanocluster comprising three concentric polyhedra with apparently incompatible octahedral (Oh) and icosahedral (Ih) symmetry, a mathematical oddity that is solved by the shells’ symmetric arrangement around rotational 2- and 3-fold axes

A Keplerian Ag90 nest of Platonic and Archimedean polyhedra in different symmetry groups.

Polyhedra are ubiquitous in chemistry, biology, mathematics and other disciplines. Coordination-driven self-assembly has created molecules mimicking Platonic, Archimedean and even Goldberg polyhedra, however, nesting multiple polyhedra in one cluster is challenging, not only for synthesis but also for determining the alignment of the polyhedra. Here, we synthesize a nested Ag90 nanocluster under solvothermal condition. This pseudo-Th symmetric Ag90 ball contains three concentric Ag polyhedra with apparently incompatible symmetry. Specifically, the inner (Ag6) and middle (Ag24) shells are octahedral (Oh), an octahedron (a Platonic solid with six 3.3.3.3 vertices) and a truncated octahedron (an Archimedean solid with twenty-four 4.6.6 vertices), whereas the outer (Ag60) shell is icosahedral (Ih), a rhombicosidodecahedron (an Archimedean solid with sixty 3.4.5.4 vertices). The Ag90 nanocluster solves the apparent incompatibility with the most symmetric arrangement of 2- and 3-fold rotational axes, similar to the arrangement in the model called Kepler's Kosmos, devised by the mathematician John Conway