Tensor structures helicity

The measured crystal optical activity, in general, can be either of molecular origin or due to the chiral helical arrangement of chiral or achiral molecules in the crystal, or both. The two factors are difficult to separate. Kobayashi defined a chirality factor r = (pc — ps)/pc = 1 — pslpc, where pc is the rotatory power per molecule of a randomly oriented crystal aggregate derived from the gyration tensors determined by HAUP, and ps that in solution [51]. It is a measure of the 4 crystal lattice structural contribution to the optical activity and represents the severity of the crystal lattice structural contribution to the optical activity, and represents the severity of the restriction of the freedom of molecular orientation by forming a crystal lattice. Quartz is a typical example of r = 1, as it does not contain chiral molecules or ions and its optical activity vanishes in random orientation (ps = 0). 

To illustrate the power of PISA wheels and dipolar waves to determine the structure of helical peptides and proteins in uniaxiaUy oriented lipid bilayers. Fig. 6a-c show SIMPSON/SIMMOL-simulated PISEMA spectra of an ideal 18-residue a-helix with a tilt angle of 10°-30° relative to Bq. In these simulations, we have tried to mimic experimental conditions by including a random distribution of the principal components of the chemical shift tensor and the dipolar coupling. The chemical shift distribution is 6 ppm for each principal element and has been established as follows we obtained — 85000 N isotropic chemical shifts reported to the BioMagResBank and selected only the — 31000 located in helical secondary stractures to have a data set independent on secondary chemical shifts. The standard deviation on the N chemical shifts for these resonances was — 6 ppm. With the lack of other statistically reliable experimental methods to establish such results for the individual principal elements of the N CSA tensor, we assumed the above variation of 6 ppm for all three principal elements. The variation of the H- N dipolar coupling was estimated by investigating the structures for a small number of a-helical membrane proteins for which the structures were established by liquid-state NMR spectroscopy. These showed standard deviations... 

Figure 8 depicts PISA wheels in which the helix tilt angle r varies from 15 to 90°. The magnitudes of the principal components of the 5-spin CSA tensor (533 = 64, 522 = 11, and 5n = 217 ppm) and the angles defining the relative orientations of the dipolar and chemical shift tensors (a = 0° and (1=17 ) were used in the simulations of PISA wheels presented in Fig. 8. The centers of the wheels as a function of helix tilt angle from both the peaks of the dipolar-coupling doublet are also shown as dashed lines in Fig. 8. The centers of the chemical shift and dipolar coupling tensors intersect at the isotropic values of the chemical shift (119.3 ppm) and the 4l " N dipolar coupling (0 Hz). The PISA wheel patterns can also be used to determine p-strand structures in lipid bilayers as the loop-like shapes of these p-strand resonances are very different from the wheel-like patterns of a-helices. It is clear from the...

The application of proton-driven CSA correlation spectroscopy to amino-acid specifically carboxylic-labeled spider silk [63] is shown in Fig. 4.11. Spider silk is known to consist of alanine- and glycine-rich domains [64, 65] and is known to be semicrystalline. The assignment of alanine to the (crystalline) /3-sheet domains [66] is clearly supported by the chemical-shift correlation spectrum of Fig. 4.11. Because the tensors in a j8-sheet structure are almost parallel, or antiparallel, with the tensors in spatial proximity, a diagonal spin-diffusion spectrum is expected for that structure and is indeed found. In contrast, the glycine spectrum shows considerable off-diagonal intensity. Simulations have shown that the spectrum is compatible with a local 3i-helical structure [63]. 

Both through-bond and pseudocontact contributions can be easily factorized into a series of products of two terms, each term depending either on the nucleus i (topologic and geometric location) or from the lanthanide j (electronic structure and crystal-field effects). For axial complexes, that is, possessing at least a three-fold axis as found in triple-stranded helicates, the molecular magnetic susceptibility tensor written in the principal magnetic axes system is symmetrical xx = mag-... 

A sufficient amount of oriented chiral molecules can be obtained in an induced cholesteric liquid crystal phase if the induced helical structure has been untwisted by an electric field. In the following description tensors are needed for the sake of simplicity (At least there are three tensors required the transition moment tensor (absorption tensor ,y), the rotational strength tensor (circular dichroism tensor A ,y), and the order tensor g,y33 (i,j= 1,2,3). If the molecules do not possess any symmetry, the principal axes of all of these tensors are differently oriented with respect to the molecular frame (the coordinate system in which only the three diagonal elements of a tensor are different from zero).) The only tensorial property, needed here explicitly, is the existence of three coordinates (components) of a tensor with respect to three specially chosen mutually perpendicular axes. This means that three information instead of one information about a molecule are needed instead of one CD value, namely Ae, three CD values, namely As, (i=l, 2, 3), have to be introduced. Ac is then one-third of a sum of the three so-called tensor coordinates of the CD tensor ... 

The three diagonal elements Ae P(v) (1=1, 2, 3) are proportional to products of electric dipole times electric quadrupole transition moments. They do not contribute to the isotropic CD because the sum over the three coordinates (v) (1 = 1, 2, 3) is zero. Asu, measured for oriented guest molecules in ordered liquid crystal phases, yield spectroscopic and structural information and, has been used, especially for the check of sector and helicity rules. First numerical quantum mechanical calculations of the CD tensor coordinates Asu have been published recently. 

In the helical structure this tensor, as well as the tensor of the dielectric anisotropy (ellipsoid) rotates upon the translation along the z-axis as shown in... 

Fig. 4.27. Then, the components of the director are n = (cosq z, smqz, 0). In the uniaxial approximation, there are only two principal components of the local dielectric tensor, Eh and Ej and two refraction indices, n and = n. As a rule n > n , and a uniaxial cholesteric is locally optically positive. For the overall helical structure, one can introduce average refraction indices, one along the helical...

To obtain the tensor of the cholesteric helical structure one should imagine that the local tensor rotates in the laboratory co-ordinate system, or, alternatively, to introduce a rotating co-ordinate system. In the latter case, one should make transformation... 

A cholesteric forms a helical structure and its optical properties are characterised by the tensor of dielectric permittivity rotating in space. We are already familiar with the form of the cholesteric tensor (see Section 4.7). It was Oseen [1] who suggested the first quantitative model of the helical cholesteric phase as a periodic medium with local anisotropy and very specific optical properties. First we shall discuss more carefully the Bragg reflection from the so-called cholesteric planes . 

Before further discussion of the cubic structured blue phase, let us consider a right-handed cholesteric phase with the helical axis in z direction. From Equation (13.32), we have the dielectric tensor... 

Physically, the rotational transformation is to untwist the twisted helical structure, so that in the rotating frame the medium now appears to be a simple birefringent material with a dielectric tensor Tj. = R T(2) R independent of z. This technique is analogous to the rotational-transformation technique used in magnetic resonance. After the transformation, Eq. (2) becomes... 

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