Cyclic symmetry

Fig. 1. Model of a ligand gated ion channel (LGIC) where (a) is the structure of a generic LGIC subunit showing the two cysteine (Cys) residues common to all LGIC subunits, and (b) shows the arrangement of five such subunits as a pentamer having psuedo-cyclic symmetry delineating a gated, fluid-filled...

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)... 

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. 

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. 

As one can see in (1.14a), the variables (x, y, z) are cycled in the three derivatives, each appearing once in the numerator, once in the denominator, and once as the constant variable. The cyclic symmetry makes it easy (and advisable) to commit this identity to memory, even if it can be easily rederived from (1.10) for use as needed. 

A ring formed using exclusively heterologous interactions possesses cyclic symmetry. The trimer in Eq. 7-25 has a threefold axis Each subunit can be superimposed on the next by rotation through 360°/ 3. The oligomer is said to have C3 symmetry. Many real proteins, including all of those with 3,5, or another uneven number of identical protomers, appear to be formed of subunits arranged with cyclic symmetry. 

The southern bean mosaic virus has an eight-stranded antiparallel (3-barrel structure closely similar to that of the major domain of the bushy stunt viruses but lacking the second hinged domain. The problem of quasi-equivalence is resolved by the presence of an N-terminal extension that binds onto a subunit across the quasi-six-fold axis to give a set of three subunits (labeled C in Fig. 7-19) that associate with true three-fold symmetry and another set (B) with a slightly different conformation fitting between them.68 92 The subunits A, which have a third conformation, fit together around the five-fold axis in true cyclic symmetry. 

Fig. S-I2 Diagrammatic representation of hemoglobin showing its cyclic symmetry.

Alcohol dehydrogenase is a dimeric protein with C2, and pyruvate carboxylase is a tetrameric protein with C4 symmetries respectively. The cyclic symmetry allows the polar... 

The magnitude of this vector is equal to the area of the polygon. Because of cyclic symmetry, we can also write... 

The Platt model just described may also be used for systems that do not have full cyclic symmetry, for example, naphthalene (C,oHg). The < )+ and ( ) orbitals of Equation 3.52 are no longer degenerate, but recognizable as cosine and sine functions. The Lj, Lb and B, functions may still be constructed and are energy ordered in the same way. 

The pressure vessel and reflector structures have a cyclic symmetry every 30 degrees. There would be limited energy exchange in the hoop direction. There is currently no information on energy exchange through the top and bottom cuts of the pressure vessel. 

Fig. 2.38. The arrangements of protomers in cyclic symmetry (adapted from Klotz et a/., 1975).

In cyclic symmetry, there is a -fold rotational axis, n being the number of subunits. Figure 2.38 shows the arrangements of protomers for dimer, trimer, tetramer, pentamer, and hexamer. The dimer has the Cj symmetry (each protomer coincides to the other by a rotation of 360°/2), the trimer has the C3 symmetry (rotation of 360°/3) and is required for the protomer transposition. The cyclic symmetry is referred to as C in the Shoenflies notation, n being the number of protomers. In the International Hermann-Mauguin notation it is referred to as n. 

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