Symmetry dihedral

FIGURE 6.44 Several possible symmetric arrays of identical protein snbnnits, inclnding (a) cyclic symmetry, (b) dihedral symmetry, and (c) cubic symmetry, inclnding examples of tetrahedral (T), octahedral (O), and icosahedral (I) symmetry. (Irving GAs)... 

Hosts based upon n= 2 subunits possess dihedral symmetry and their structures may be considered to be based upon tennis balls, rugby balls, prisms, and antiprisms (see ref. [11]). 

There are several forms of rotational symmetry. The simplest is cyclic symmetry, involving rotation about a single axis (Fig. 4—24a). If subunits can be superimposed by rotation about a single axis, the protein has a symmetry defined by convention as Gn (C for cyclic, n for the number of subunits related by the axis). The axis itself is described as an w-fold rotational axis. The a/3 protomers of hemoglobin (Fig. 4-23) are related by C2 symmetry. A somewhat more complicated rotational symmetry is dihedral symmetry, in which a twofold rotational axis intersects an w-fold axis at right angles. The symmetry is defined as DTO (Fig. 4—24b). A protein with dihedral symmetry has 2n protomers. 

Proteins with cyclic or dihedral symmetry are particularly common. More complex rotational symmetries are possible, but only a few are regularly encountered. One example is icosahedral symmetry. An icosahedron is a regular 12-cornered polyhedron having 20 equilateral triangular faces (Fig. 4-24c). Each face can... 

FIGURE 4-24 Rotational symmetry in proteins, (a) In cyclic symmetry, subunits are related by rotation about a single n-fold axis, where n is the number of subunits so related. The axes are shown as black lines the numbers are values of n. Only two of many possible Cn arrangements are shown, (b) In dihedral symmetry, all subunits can be related by rotation about one or both of two axes, one of which is twofold. D2 symmetry is most common, (c) Icosahedral symmetry. Relating all 20 triangular faces of an icosahedron requires rotation about one or more of three separate rotational axes twofold, threefold, and fivefold. An end-on view of each of these axes is shown at the right. 

Nevertheless, a pair of interactions between the left-hand subunit in the top ring and the subunit in the lower ring at the right does exist, even if it is only electrostatic and at a distance. An example of a tetra-meric enzyme with perfect dihedral symmetry of the type shown in Fig. 

Since its introduction into clinical use in about 1979 the immunosuppresant cyclosporin has been responsible for a revolution in human organ transplantation.3 The exact mechanism of action in suppressing T-lymphocyte-mediated autoimmune responses is still not completely clear, but cyclosporin, a cyclic lipophilic peptide from a fungus, was found to bind to specific proteins that were named cydophilins.d Human cyclophilin A is a 165-residue protein which associates, in the crystal form, as a decamer with five-fold rotational and dihedral symmetry.6 This protein is also found in almost all... 

If the a and /3 chains are considered to be identical, then hemoglobin has dihedral symmetry with two rotational axes, and with the four subunits arranged at the apices of the tetrahedron. 

Catacondensed Benzenoids with Dihedral Symmetry and Centrosymmetry... 

A catacondensed benzenoid with dihedral symmetry, viz. D2h is either a branched system or an (unbranched) linear acene. A centrosymmetrical (C2h) catacondensed benzenoid is either branched or unbranched. The D2h systems under consideration have been enumerated by the efficient algorithm invoking SCS s (cf. Sect. 6.4) [80], Table 20, in combination with Table 17, shows the known numbers for the branched catacondensed Dlh and C2h benzenoids. The numbers of unbranched catacondensed benzenoids with C2h symmetry are found under the designation d in Tables 14 and 15 for h < 20 and 21 < h < 30, respectively. 

Table 20. Numbers of branched catacondensed benzenoids with dihedral symmetry and centrosymmetry ...

Benzenoids of both dihedral symmetry (D2h) and centrosymmetry (C2k) are divided into two kinds (i) the first kind, where the systems have a central hexagon (ii) the second kind, where they have an edge in the centre, a central edge. 

Table 37. Numbers of classified benzenoids with dihedral symmetry (D2k)...

There is a difference between biphenyl- and 1,3-butadiene systems with respect to symmetry Atropisomeric biphenyl compounds, because of their dihedral symmetry, need a specific substitution pattern to be chiral. In contrast, nonplanar, helical butadienes belong to the point groups C2 or Cj and are chiral without bearing specific substituents. 

The simple formal analysis outlined above reveals several interesting aspects of the Ajg - Tig CD predicted for chiral Co(III) complexes of trigonal dihedral symmetry. First,... 

Dihedral symmetry planes will not be considered further. Molecules in the group Td do not have a unique principal axis but have several axes of the same order. The cd planes each contain one of these axes but neither contain the other axes nor are at right-angles to them. 

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