Symmetry asymmetric unit

Any symmetric object is built up from smaller pieces that are identical and that are related to each other by symmetry. An icosahedron can therefore be divided into a number of smaller identical pieces called symmetry-related units. Protein subunits are asymmetric objects hence, a symmetry axis cannot pass through them. The minimum number of protein subunits that can form a virus shell with icosahedral symmetry is therefore equal to... 

The symmetry properties of an icosahedron are not restricted to the surface but extend through the whole volume. An asymmetric unit is therefore a part of this volume it is a wedge from the surface to the center of the icosahedron. Sixty such wedges completely fill the volume of the icosahedron. 

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

Figure 16.4 The division of the surface of an icosahedron into asymmetric units, (a) One triangular face is divided into three asymmetric units into which an object is placed. These are related by the threefold symmetry axis.

Figure 16.S Schematic illustration of the way the 60 protein subunits are arranged around the shell of safellite tobacco necrosis virus. Each subunit is shown as an asymmetric A. The view is along one of the threefold axes, as in Figure 16.3a. (a) Three subunifs are positioned on one triangular tile of an Icosahedron, in a similar way to that shown in 16.4a. The red lines represent a different way to divide the surface of the icosahedron into 60 asymmetric units. This representation will be used in the following diagrams because it is easier to see the symmetry relations when there are more than 60 subunits in the shells, (b) All subunits are shown on the surface of the virus, seen in the same orientation as 16.4a. The shell has been subdivided into 60 asymmetric units by the red lines. When the corners are joined to the center of the virus, the particle is divided into 60 triangular wedges, each comprising an asymmetric unit of the virus. In satellite tobacco necrosis virus each such unit contains one polypeptide chain...

In the T = 4 structure there are 240 subunits (4 x 60) in four different environments, A, B, C, and D, in the asymmetric unit. The A subunits interact around the fivefold axes, and the D subunits around the threefold axes (Figure 16.7). The B and C subunits are arranged so that two copies of each interact around the twofold axes in addition to two D subunits. For a T = 4 structure the twofold axes thus form pseudosixfold axes. The A, B, and C subunits interact around pseudothreefold axes clustered around the fivefold axes. There are 60 such pseudothreefold axes. The T = 4 structure therefore has a total of 80 threefold axes 20 with strict icosahedral symmetry and 60 with pseudosymmetry. 

Commonly, only the atomic coordinates for the atoms in one asymmetric unit are listed. Atoms that can be generated from these by symmetry operations are not listed. Which symmetry operations are to be applied is revealed by stating the space group (cf Section 3.3). When the lattice parameters, the space group, and the atomic coordinates are known, all structural details can be deduced. In particular, all interatomic distances and angles can be calculated. 

C2 Z = 4 Dx = 1.41 R = 0.102 for 4,115 intensities. The structure is a 3 2 complex of proflavine and CpG. The asymmetrical unit contains one CpG molecule, 1.5 proflavine molecules, 0.5 sulfate ion, and 11 5 water molecules. Two CpG molecules form an antiparallel, Watson-Crick, miniature duplex, with a proflavine intercalated between the base pairs through the wide groove. The double helix has exact (crystallographic), two-fold symmetry, and the crystallographic, two-fold axis passes through the C-9-N-10 vector of the intercalated proflavine. A second and a third molecule of proflavine are stacked on top of the C - G pairs ... 

Seven crystal systems as described in Table 3.2 occur in the 32 point groups that can be assigned to protein crystals. For crystals with symmetry higher than triclinic, particles within the cell are repeated as a consequence of symmetry operations. The number of asymmetric units within the unit cell is related but not necessarily equal to the number of molecules in a unit cell, depending on how the molecules are related by symmetry operations. From the symmetry in the X-ray diffraction pattern and the systematic absence of specific reflections in the pattern, it is possible to deduce the space group to which the crystal belongs. 

To the extent that a crystal is a perfectly ordered structure, the specificity of a reaction therein is determined by the crystallographic symmetry. A crystal is built up by repeated translations, in three dimensions, of the contents of the unit cell. However, the space group usually contains elements additional to the pure translations, such as a center of inversion, rotation axis, and mirror plane. These elements can interrelate molecules within the unit cell. The smallest structural unit that can develop the whole crystal on repeated applications of all operations of the space group is called the asymmetric unit. This unit can consist of a fraction of a molecule, sometimes fractions of two or more molecules, a single whole molecule, or more than one molecule. If, for example, a molecule lies on a crystallographic center of inversion, the asymmetric unit will contain half... 

The complexity that may arise even for a relatively simple molecule is illustrated by iminodiacetic acid, H02CCH2NHCH2C02H. This acid has three crystal forms. The bond angles and lengths are essentially the same in all the forms, but the torsional angles differ, providing three different molecular conformations (19). In addition, one of the forms has two symmetry-independent molecules, of slightly different conformations, in its asymmetric unit. 

An example of the second effect is provided by mono-sec-butyl phthalamide (17a) (56). In the crystal the two enantiomers of this molecule are miscible in all proportions. The racemate crystallizes in space group PI (two general positions in the unit cell) with four molecules per unit cell. Thus there are two molecules in the asymmetric unit. The sec-butyl moieties adopt the anti conformation (the two geometries are shown schematically in 17b) and exhibit conformational disorder to different extents at the two symmetry-independent sites. 

An interesting molecule of the type just discussed is 8-dimethylamino-1 -naphthoic acid, in whose crystal there are two symmetry-independent molecules in the asymmetric unit. One of these shows distortions in agreement with the above generalizations, whereas there are qualitatively different distortions in the second molecule. The explanation is that the crystal is actually a 1 1 molecular compound of the amino acid, which shows the N 0=0 interaction, and the corresponding zwitterion, which does not. 

There are a number of possible explanations for the formation of more than one photodimer. First, due care is not always taken to ensure that the solid sample that is irradiated is crystallographically pure. Indeed, it is not at all simple to establish that all the crystals of the sample that will be exposed to light are of the same structure as the single crystal that was used for analysis of structure. A further possible cause is that there are two or more symmetry-independent molecules in the asymmetric unit then each will have a different environment and can, in principle, have contacts with neighbors that are suited to formation of different, topochemical, photodimers. This is illustrated by 61, which contrasts with monomers 62 to 65, which pack with only one molecule per asymmetric unit. Similarly, in monomers containing more than one olefinic bond there may be two or more intermolecular contacts that can lead to different, topochemical, dimers. Finally, any disorder in the crystal, for example due to defective structure or molecular-orientational disorder, can lead to formation of nontopochemical products in addition to the topochemical ones formed in the ordered phase. This would be true, too, in those cases where there is reaction in the liquid phase formed, for example, by local melting. 

Fig. 9.5 n = 2 shells, (a) symmetry, (b) D2d symmetry (tennis ball) (c) subunit of D2d shell depicting the four asymmetric units. 

Td, possesses 32 symmetry, and requires a minimum of 12 asymmetric units the cube and octahedron, which belong to the point group Oh, possess 432 symmetry, and require a minimum of 24 asymmetric units and the dodecahedron and icosahedron, which belong to the point group Ih, possess 532 symmetry, and require a minimum of 60 asymmetric units. The number of asymmetric units required to generate each shell doubles if mirror planes are present in these structures. 

Although the number of atoms in a crystal is extremely high, we can imagine the crystal as generated by a spatial reproduction of the asymmetric unit by means of symmetry operations. The calculation can thus be restricted to a particular portion of space, defined as the Brillouin zone (Brillouin, 1953). 

---