Fivefold axis

Figure Bl.8.7. A convergent beam diffraction pattern of the fivefold axis of a quasicrystal, as in figure Bl.8.6. The diffraction rings show that the synnnetry is fivefold, not tenfold. (Courtesy of L Bendersky.)...

The asymmetric unit contains one copy each of the subunits VPl, VP2, VP3, and VP4. VP4 is buried inside the shell and does not reach the surface. The arrangement of VPl, VP2, and VP3 on the surface of the capsid is shown in Figure 16.12a. These three different polypeptide chains build up the virus shell in a way that is analogous to that of the three different conformations A, C, and B of the same polypeptide chain in tomato bushy stunt virus. The viral coat assembles from 12 compact aggregates, or pen tamers, which contain five of each of the coat proteins. The contours of the outward-facing surfaces of the subunits give to each pentamer the shape of a molecular mountain the VPl subunits, which correspond to the A subunits in T = 3 plant viruses, cluster at the peak of the mountain VP2 and VP3 alternate around the foot and VP4 provides the foundation. The amino termini of the five VP3 subunits of the pentamer intertwine around the fivefold axis in the interior of the virion to form a p stmcture that stabilizes the pentamer and in addition interacts with VP4. 

Figure 16.1S Subunits VPl In picomaviruses antibody cluster around the fivefold axis (a) so that their binding sites tips form a protrusion on the viral surface.

The symmetry axis with maximum strength for each function of set I is 69° 1.35 from the fivefold axis of the pentagonal antiprism, corresponding to the ratio 0.38341 of half height to radius of the antiprism, with the corresponding values for set II of 41° 47.65 and 1.1186. 

Whereas only hh0 can diffract with the beam along a fivefold axis of the icosatwin because of the tilt of the cubic crystals, any plane in the zone 110 of the decatwin can diffract because the seed orients the cubic crystals with this zone axis parallel to the fivefold axis of the twin. It is seen from Fig. 1 (b) that the meridional spots of decagonal FeAl are the same as for icosahedral MnAl6, representing orders of hh0, but that there are many more equatorial spots, eleven instead of three. Indices are assigned in Table II some spots involve double... 

TABLE 1. The meridional (P-P) diffraction maxima of the MnAU icosatwin fivefold-axis ED photograph (Ref. 3). Values of intensity, / radius, R observed and calculated interplanar distances, and dale and indices. Scale Rd -75.6 mm A, ao —23.36 A. 

In 1985 Dr. Michael Rossmann and his colleagues determined for the first time the three-dimensional structure of a human rhinovirus [15], Their studies, performed with human rhinovirus type 14 (HRV-14), revealed the structure as an eicosahedron consisting of four proteins designated VP1, VP2, VP3, and VP4 forming a protomeric unit, combined to form a fivefold axis of symmetry (Fig. 2). The surface of the capsid... 

Fig. 9 XRD of quenched samples of Sc MgjCu/ia r The v = 46.0 and 48.5 products are the 1/1 AC, that for y = 48.0 is the i-QC, and y = 47.0 results in a mixture of the QC and Sc2CuGa3. The inset and the six integers show the ED pattern of the i-QC along a fivefold axis and the sixdimensional indices. (Reproduced with permission from [79]. Copyright 2005 American Chemical Society)...

Now the molecular orbitals in equation (3) also have interesting transformation properties. Consider the effect of a rotation through 2w/5 radians about the fivefold axis of symmetry, perpendicular to the plane of the ring. For example, in place of 0+i take that function which is obtained from it by replacing atom 5 (and thus 0b) by atom 4 (and thus 04), 4 by 3, and so on, namely... 

If the tiling in Figure 2.8 is rotated 36° about the fivefold axis and translated the right amount, there is coincidence of the vertices with those of the original tiling... 

Electron diffraction pattern of an AIMn quasicrystal along the fivefold axis (left) and a computed Fourier pattern of a three-dimensional Penrose tiling (right). From C. Janot, Quasicrystals, A Primer, 2nd ed. (London Oxford Univ. Press, 1994), p. 3, figure 1.24. 

Figure 7.40 Structure of the spherical lanthanide p-sulfonatocalix[4]arene assembly (a) partial space filling view along the pseudo-fivefold axis. Pyridine N oxide and one calixarene are shown in stick mode. S03 groups line the surface of the sphere, aryl rings define the hydrophobic shell and the polar core comprises 30 water molecules and two Na+ ions, (b) cut away view showing an [Na(H20)6] 2 cluster within the core. (Reprinted with permission from AAAS from [52]).

Also, quasi-crystals of apparent macroscopic fivefold symmetry do exist, despite the toological restriction that an object with a fivefold axis cannot tile two-dimensional space these funny quasi-crystals do have voids in them, which "fill" the remainder of the space needed. 

Figure 12. (a) Distances d between atoms situated on a fivefold axis, belonging to layers n and n - 1 (n = 0 refers to central atom), calculated for seven relaxed MIC models with complete layers. The quantity varies inversely as the mean model diameter, (b) Distances d, between nearest-neighbor atoms situated in the neighborhood of the center of the faces belonging to successive layers n. 

Figure 11.4. Three subunits of the Lymnaea AChBP viewed perpendicular to the fivefold axis of symmetry and from the outside of the pentamer. The inner and outer sheets of the j8-sandwich are blue and red, respectively, whereas the putative ligand HEPES is purple. The approximate positions of the a carbons of residues discussed in the text are marked with arrows on the foremost subunit. The approximate positions of the cell membrane and of the M2-M3 loop are shown diagrammatically. The inner jS-sheet (blue) is thought to rotate after agonist binding and to interact with the M2-M3 loop (as indicated by the asterisk). See color insert.

Figure 11.4. Three subunits of the Lymnaea ACbBP viewed perpendicular to the fivefold axis of symmetry and from the ofitsihe di the pfelfratner. The inner an ...

Cyclophilin A is a simple example pentamer and decamer share the same isometric form lattice. Moreover, the mutual orientation of the pentamers in the decameric configuration is fixed through a linear scaling transformation leaving the form lattice invariant. The geometry of the entire biomacromolecular system and of its components is expressible in term of the single parameter tq, which is the shortest distance of the monomeric chain from the fivefold axis. 

Pentagonal clusters. For each coat protein one has six decamers with symmetry 52 and 12 pentamers with symmetry 5 about the fivefold axis. The enclosing... 

Each plane orthogonal to a threefold or fivefold axis contains exclusively the points of a single quasilattice, q, a or b, but each plane orthogonal to a twofold axis might contain points of all three quasilattices, as shown in Table 12-2. 

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