Spherical icosahedral symmetry

The protein shells of spherical viruses have icosahedral symmetry... 

The asymmetric unit of an icosahedron can contain one or several polypeptide chains. The protein shell of a spherical virus with icosahedral symmetry... 

Small spherical viruses have a protein shell around their nucleic acid that is constructed according to icosahedral symmetry. Objects with icosahedral symmetry have 60 identical units related by fivefold, threefold, and twofold symmetry axes. Each such unit can accommodate one or severed polypeptide chains. Hence, virus shells are built up from multiples of 60 polypeptide chains. To preserve quasi-equivalent symmetry when packing subunits into the shell, only certain multiples (T = 1, 3, 4, 7...) are allowed. 

Virus symmetry The nucleocapsids of viruses are constructed in highly symmetrical ways. Symmetry refers to the way in which the protein morphological units are arranged in the virus shell. When a symmetrical structure is rotated around an axis, the same form is seen again after a certain number of degrees of rotation. Two kinds of symmetry are recognized in viruses which correspond to the two primary shapes, rod and spherical. Rod-shaped viruses have helical symmetry and spherical viruses have icosahedral symmetry. 

The nature of the electronic states for fullerene molecules depends sensitively on the number of 7r-electrons in the fullerene. The number of 7r-electrons on the Cgo molecule is 60 (i.e., one w electron per carbon atom), which is exactly the correct number to fully occupy the highest occupied molecular orbital (HOMO) level with hu icosahedral symmetry. In relating the levels of an icosahedral molecule to those of a free electron on a thin spherical shell (full rotational symmetry), 50 electrons fully occupy the angular momentum states of the shell through l = 4, and the remaining 10 electrons are available... 

At the present time, there is no accepted chelating agent which can be used against common influenza viruses in humans. A virus has a core of either DNA or RNA and a protective coat of many identical protein units. All viruses are either rods or spheres, that is the protein coats are cylindrical shells having helical symmetry or spherical shells having icosahedral symmetry. Viruses reproduce inside living cells, where each viral nucleic acid directs the synthesis of about 1000 fresh viruses. These are then released and the host cell may die. 

Applying ab initio quantum-chemical methods and density functional theory in the local density approximation, different (BH) spherical clusters for n — 12,20,32,42 and 92 have been investigated. Most of the clusters show nearly icosahedral symmetry. The hydrogen atoms are bonded to the spherical surface as prickles. The relative stability of the spheres measured as the binding energy per molecule has been analyzed. All the clusters studied are very stable, and the spherical (BH)32 cluster Seems to be the most stable structure. The effect of the hydrogen atoms is to increase the stability of the bare boron clusters. 

Fig. 5. Comparison of HSV-1 procapsid and capsid structures. Outside (A and B) and inside (C and D) views of reconstructions of the HSV-1 procapsid (A and C) and capsid (B and D). All views are shovm looking along a 3-fold icosahedral axis. Although it retains the same icosahedral symmetry, as shovm by the hexagonal arrangement of the capsomers apparent in the internal view (C), the procapsid subunits are much less clearly defined than those in the capsid. In addition, the contacts between individual subunits are much more tenuous and the continuous floor, formed by extensions from the bases of the capsomers, is not present. Note also the contrast in shape between the spherical procapsid and the polyhedral capsid. Scale bar 500 A. [Image supplied by Benes Trus.]...

In the case of the icosahedral point groups, Ih and I, Table 3.10, the analysis is more complicated and there is a need to identify the combinations of the spherical harmonics, which will generate higher dimensional irreducibile subspaces. For example, at level 3, there are 7 harmonics, but the irreducible subspaces in icosahedral symmetry are four-fold [Gu] and three-fold [T2u]. It is found that three of the original functions can be carried over to provide basis functions in icosahedral, symmetry but that four distinct linear combinations of... 

The most impressive assemblies are the spherical viruses with icosahedral symmetry. Sixty copies of a fundamental unit are arranged on the surface of a shell. These structures have 532 symmetry meaning that there are twofold, threefold, and fivefold axes of symmetry. Icosahedral symmetry is the most efficient arrangement for a closed shell and as such is analogous to the geodesic dome. 

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

T ux,Tiuy,Tiu transform as the x, y, z)-axes themselves or, for illustration purposes, a set iPx,Py, Pz) of /t-orbitals. Actually, as the Tiu orbitals form a basis for the spherical harmonics with angular momentum L = 1, this basis can be used whatever orientation is chosen. However, the same cannot be said for the HOMO orbitals as the L = 5 harmonics decompose as Tiu 0 72m 0 77 in icosahedral symmetry. 

Consultation of polyhedron models revealed the structure of 1 to conform to a snub cube, one of the 13 Archimedean solids, in which the vertices of the square faces correspond to the comers of 2 and the centroids of the eight triangles that adjoin three squares correspond to the eight water molecules. Indeed, to us, the ability of six resorcin[4]arenes to self-assemble to form 1 was reminiscent of spherical viruses in which identical copies of proteins self-assemble, by way of noncovalent forces, to form viral capsids having icosahedral symmetry and a shell-like enclosure. In fact, owing to the fit displayed by its components, 1 exhibits a topology that agrees with the theory of vims shell stmcture which states that... 

In general, carbon onions are spherical, which implies a considerable number of defects in every shell as otherwise they would have to be Goldberg fullerenes with a faceted icosahedral symmetry. 

If we intend to construct a giant species similar in size and shape to spherical viruses with icosahedral symmetry - in the sense of an Archimedean synthesis - we have to find a reaction system in which the aforementioned pentagonal units which are present as disposition insolution can get linked, and be placed at the 12 corners of an icosahedron (see below). 

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