Twofold symmetry

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

Palindromic base pairs that are most frequent at the two ends are green, and the pseudo-twofold symmetry axis is Indicated by a red dot. 

The protein-DNA interactions have been analyzed in detail at high resolution in the complex between the 434 repressor fragment and the ORl containing 20mer DNA. A pseudo-twofold symmetry axis relates the halves of this complex. The symmetry is not exact since the nucleotide sequence of the DNA is slightly different in each half (see Table 8.2). However, the interactions between one protein subunit and one half of the DNA are very similar to those between the second subunit and the other half of the DNA since most of the bases that interact with the protein are identical in both halves. Details of the interaction are very similar to those in the complex with the palindromic synthetic 14mer of DNA shown in Figures 8.14 and 8.15. The base pairs at one end of the DNA, 1-14, 2-13, etc. are called base pairs 1, 2, etc. 

Many proteins that switch off or on gene expression in bacteria are dimeric molecules, and the DNA sequences that they specifically recognize are palindromic at their ends. The twofold symmetry of the protein is therefore matched by twofold symmetry at the ends of the recognition sequence. 

The structurally similar L and M subunits are related by a pseudo-twofold symmetry axis through the core, between the helices of the four-helix bundle motif. The photosynthetic pigments are bound to these subunits, most of them to the transmembrane helices, and they are also related by the same twofold symmetry axis (Figure 12.15). The pigments are arranged so that they form two possible pathways for electron transfer across the membrane, one on each side of the symmetry axis. 

Figure 12.15 Schematic arrangement of the photosynthetic pigments in the reaction center of Rhodopseudomonas viridis. The twofold symmetry axis that relates the L and the M subunits is aligned vertically in the plane of the paper. Electron transfer proceeds preferentially along the branch to the right. The periplasmic side of the membrane is near the top, and the cytoplasmic side is near the bottom of the structure. (From B. Furugren, courtesy of the Royal Swedish Academy of Science.)...

Small spherical viruses have a protein shell around their nucleic acid that is constructed according to icosahedral symmetry. Objects with icosahedral symmetry have 60 identical units related by fivefold, threefold, and twofold symmetry axes. Each such unit can accommodate one or severed polypeptide chains. Hence, virus shells are built up from multiples of 60 polypeptide chains. To preserve quasi-equivalent symmetry when packing subunits into the shell, only certain multiples (T = 1, 3, 4, 7...) are allowed. 

Figure 18.1 A crystal is built up from many billions of small identical units, or unit cells. These unit cells are packed against each other in three dimensions much as identical boxes are packed and stored in a warehouse. The unit cell may contain one or more than one molecule. Although the number of molecules per unit cell is always the same for all the unit cells of a single crystal, it may vary between different crystal forms of the same protein. The diagram shows in two dimensions several identical unit cells, each containing two objects packed against each other. The two objects within each unit cell are related by twofold symmetry to illustrate that each unit cell in a protein cr) stal can contain several molecules that are related by symmetry to each other. (The pattern is adapted from a Japanese stencil of unknown origin from the nineteenth century.)...

Note the twofold symmetry the sequence read 5 3 is the same in both strands. 

Table 11.5 lists many of the commonly used restriction endonucleases and their recognition sites. Because these sites all have twofold symmetry, only the sequence on one strand needs to be designated. 

An intensely colored by-product of the photolysis reaction of methyl-2-azidobenzoate has been identified as the first known derivative of 3,3 -diazaheptafulvalene 70 (94LA1165). Its molecular mass was established by elemental analysis and mass spectroscopy as that of a formal nitrene dimer, whereas and NMR studies demonstrated the twofold symmetry as well as the existence of a cross-conjugated 14 7r-electron system in 70. Involving l-azido-2,3-dimethoxy-5,6-dimethoxycarbonylbenzene in thermal decomposition reactions, the azaheptafulvalene 71 could be isolated and characterized spectroscopically and by means of X-ray diffraction. Tliis unusual fulvalene can be regarded as a vinylogous derivative of azafulvalenes (96JHC1333) (Scheme 28). 

At metaphase, mammalian chromosomes possess a twofold symmetry, with the identical duplicated sister chromatids connected at a centromere, the relative po-... 

Restriction endonuclease A deoxyribonuclease which cuts DNA at specific sequences which exhibit twofold symmetry about a point. Name derives from the fact that their presence in a bacterial cell prevents (restricts) the growth of many infecting bacteriophages. 

C13HuC1N307 2,2 -Anhydro-[5-chloro-l-(3,5-di-0-acetyl-/ -D-arabino-furanosyl)-6-oxocytosine] (ACAFCC)166 I4t Z — 8 D = 1.56 R = 0.066 for 818 intensities. The glycosyl disposition of the anhydronucleoside is constrained to the syn (— 68.8°) orientation. The conformation of the D-arabinofuranosyl group is a flattened 4E (232.6°, 18.0°), and the exocyclic, C-4 -C-5 bond torsion-angle is g auche+ (50.2°). The adjacent bases are connected by N-H O hydrogen-bonds between the N-4 atom of one molecule and the carbonyl oxygen atom (0-4) of another. The twofold-symmetry-related bases are stacked, with an interbase separation of 329 pm. 

A common pair of tailor s scissors (Fig. 11a) has the twofold symmetry (C2) we prefer. On opening the edges, the scissors gain bulkiness. One can say a pair of scissors is polar , too. One part serves for cutting, the other for handling. 

Another remarkable feature of most of the inclusion hosts of this chapter is that they possess at least approximately (i.e. nonerystallographic) twofold symmetry. 

When treating CF parameters in any of the two formalisms, non-specialists often overlook that the coefficients of the expansion of the CF potential (i.e. the values of CF parameters) depend on the choice of the coordinate system, so that conventions for assigning the correct reference framework are required. The conventional choice in which parameters are expressed requires the z-direction to be the principal symmetry axis, while the y-axis is chosen to coincide with a twofold symmetry axis (if present). Finally, the x-axis is perpendicular to both y- and z-axes, in such a way that the three axes form a right-handed coordinate system [31]. For symmetry in which no binary axis perpendicular to principal symmetry axis exists (e.g. C3h, Ctt), y is usually chosen so as to set one of the B kq (in Wybourne s approach) or Aq with q < 0 (in Stevens approach) to zero, thereby reducing the number of terms providing a non-zero imaginary contribution to the matrix elements of the ligand field Hamiltonian. Finally, for even lower symmetry (orthorhombic or monoclinic), the correct choice is such that the ratio of the Stevens parameter is restrained to X = /A (0, 1) and equivalently k =... 

Similar to the case without consideration of the GP effect, the nuclear probability densities ofAi and A 2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (1 0. However, the nuclear probability density for the lowest E state... 

It is also possible that a membrane might have an even lower symmetry than a chiral smectic-C liquid crystal in particular, it might lose the twofold rotational symmetry. This would occur if the molecular tilt defines one orientation in the membrane plane and the direction of one-dimensional chains defines another orientation. In that case, the free energy would take a form similar to Eq. (5) but with additional elastic constants favoring curvature. The argument for tubule formation presented above would still apply, but it would become more mathematically complex because of the extra elastic constants. As an approximation, we can suppose that there is one principal direction of elastic anisotropy, with some slight perturbations about the ideal twofold symmetry. In that approximation, we can use the results presented above, with 4) representing the orientation of the principal elastic anisotropy. 

Figure 8.6 Three-dimensional slice of C2 symmetrical SmC phase, showing tilt cone, polar axis (congruent with twofold symmetry axis), smectic layer planes, tilt plane, and polar plane.

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