Helices, translational symmetry

During the past half a century, fundamental scientific discoveries have been aided by the symmetry concept. They have played a role in the continuing quest for establishing the system of fundamental particles [7], It is an area where symmetry breaking has played as important a role as symmetry. The most important biological discovery since Darwin s theory of evolution was the double helical structure of the matter of heredity, DNA, by Francis Crick and James D. Watson (Figure 1-2) [8], In addition to the translational symmetry of helices (see, Chapter 8), the molecular structure of deoxyribonucleic acid as a whole has C2 rotational symmetry in accordance with the complementary nature of its two antiparallel strands [9], The discovery of the double helix was as much a chemical discovery as it was important for biology, and lately, for the biomedical sciences. 

It should be remarked here that the Bloch-form of the one-electron orbitals [equation (2)] automatically implies translational symmetry. In the case of onedimensional polymers this symmetry operation can be combined with a simultaneous rotation around the polymer axis (helix operation). It can be shown that if the AO s xtt 0 are properly transformed in the translated and rotated elementary cells,8 7 the above described formalism can still be applied. 

Two important types of polymer chain are the planar zigzag and the helix, but whatever form the chains take in crystallites, they must be straight on a crystallite-size scale and the straight chains must pack side by side parallel to each other in the crystal. The ideas of standard bond lengths, angles and orientations around bonds discussed in chapter 3 can be used to predict likely possible low-energy model chain conformations that exhibit translational symmetry (see example 4.3). As discussed in the... 

It is of interest to note that one may change the translation lattice of Fig. 5.3 by replacing the translation lattice vector c with the molecular helix lattice, keeping the translation symmetries a and This would lead to a match of the molecular helix symmetry with the crystal symmetry and even for irrational helices, a crystal stracture symmetry would be recognized. In fact, a whole set of new lattices can be generated replacing all three translation symmetry operations by helix symmetry operations [5]. Since a 1 1/1 hehx has a translational symmetry, this new space lattice description with helices would contain the traditional crystallography as a special case. 

Finally, it is noteworthy that this derivations differs from those given by McCubbinS > and Ukrainski,S"> who start in the usual way with simple translational symmetry and analyze a posteriori the effects of other symmetry operations, such as the helix operation. 

An electric dipole operator, of importance in electronic (visible and uv) and in vibrational spectroscopy (infrared) has the same symmetry properties as Ta. Magnetic dipoles, of importance in rotational (microwave), nmr (radio frequency) and epr (microwave) spectroscopies, have an operator with symmetry properties of Ra. Raman (visible) spectra relate to polarizability and the operator has the same symmetry properties as terms such as x2, xy, etc. In the study of optically active species, that cause helical movement of charge density, the important symmetry property of a helix to note, is that it corresponds to simultaneous translation and rotation. Optically active molecules must therefore have a symmetry such that Ta and Ra (a = x, y, z) transform as the same i.r. It only occurs for molecules with an alternating or improper rotation axis, Sn. 

A convenient method for defining helical symmetry and calculating the distribution of Intensity In a fiber pattern was devised by Cochran, Crick and Vand (CCV) (4). As Indicated in Figure 1, the molecular conformation is treated as a regular series of diffraction units uniformly spaced along a helix of pitch, P, with axial separation, s. In PTFE, one helix defines the carbon positions two helices define fluorine positions. If there is a meaningful translational Identity, c, It follows that P/s will be the ratio of small whole numbers ... 

Classically, a molecule is optically active when in an electronic transition there is a helical movement of charge density. A characteristic of a helix is that it corresponds to a simultaneous translation and rotation and so (...) optically active molecules are those in which a transition is simultaneously both electric dipole (charge translation) and magnetic dipole (charge rotation) allowed. As an alternative general statement, one can say that optically active molecules do not have any Sn axis, where n can assume any value (n = 1 corresponds to a mirror plane and n = 2 to a centre of symmetry)."... 

It was easy to show that we can formulate the method also in the case of a combined symmetry operation (for instance helix operation = translation + rotation) instead of simple translation ( ). In this case k is defined on the combined symmetry operation and from going from one cell to the next one, one has (1) to put the nuclei in the positions required by the symmetry operation and (2) one has to rotate accordingly also the basis set. 

Consider an ideally infinite helical chain whose crystallographic repeat contains N chemical repeat units, each with P atoms. A screw symmetry operation transforms one chemical unit into the next, with a being the rotation about the helix axis and d the translation along the axis. Let r" denote the ith internal displacement coordinate associated with the nth chemical repeat unit. The potential energy, by analogy with Eq. (3), is given by... 

Helical symmetry The polymeric proteins of filamentous viruses and the cytoskel-ton possess helical symmetry, in which subunits are related by a translation, as well as a rotational component. Actin, myosin, tubulin and various other fibrous proteins all interact with helical symmetry, which is often called screw symmetry. Screw symmetry, which relates the positions of adjacent subunits, combines a translation along the helix axis with the rotation. Actin forms a two-stranded helix of globular actin subunits. However, important variations in the helix parameters occur (Egehnan et al, 1982). The rise per subunit is relatively constant, but the twist or relative rotation around the helix axis is highly variable. This polymorphic tendency is probably important for the smooth functioning of muscle contraction, which involves considerable force generation. 

Here a is the radius. Is this helix left- or right-handed Write down the parameterization of its enantiomer. The symmetry of a helix is based on a screw axis, which corresponds to a translation in t. It is composed of a translation along the z-direction with a concomitant rotation in the xy-plane. Now decorate the helix with atoms at points tk/a = 2nk/m, where k and m are integers. Determine the screw symmetry of this molecular helix. If n/m is irrational, the helix is noncommensurate. Will it stiU have a symmetry in this case ... 

Note that a uniform sign change of t would leave the right-handed helix unchanged. For the discrete helix, the screw symmetry consists of a translation in the z-direction over a distance InaIm in combination with a rotation around the z-axis over an angle Innlm.lfmis, irrational, the helix will not be periodic, and the screw symmetry is lost. 

We can apply the formalism developed in the preceding section also in the case of a combined symmetry operation. To show this let us consider a helix in which we pass from one unit to the next by a translation t and simultaneous rotation a. We can then introduce the helix operator... 

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