Tensor alignment

Figure 1 The principal sources of structural data are the NOEs, which give information on the spatial proximity d of protons coupling constants, which give information on dihedral angles < i and residual dipolar couplings, which give information on the relative orientation 0 of a bond vector with respect to the molecule (to the magnetic anisotropy tensor or an alignment tensor). Protons are shown as spheres. The dashed line indicates a coordinate system rigidly attached to the molecule.

S. Blenk, H. Ehrentraut, W. Muschik. Statistical foundation of macroscopic balances for liquid crystals in alignment tensor formulation. Physica A 77 119-138, 1991. 

In the principal axis frame of the alignment tensor, A, the dipolar coupling between two nuclei, P and Q, as a function of the polar coordinates, 6 and tp, is given by... 

Fig. 8.1 Orientation of two dipolar coupling vectors in a protein segment. The vectors connect the amide Hn and 15N atoms. In this case the interaction vector coincides with the chemical bond. The axis system of the alignment tensor is designated as A, Aw Aa. The angles ( n, y>n, and 02, define the orientation of the two dipolar vectors with respect to the alignment tensor. (Reproduced with permission from N. Tjandra, Structure 1999, 7, R205-R211.)...

A different approach to determining the magnitude and direction of the alignment tensor is reported in the work by Zweckstetter and Bax [65]. This method consists of a systematic search of solute orientations that have the highest tendency to clash with the orienting media, and is based on the molecular shape of the solute. It works well, provided that solute alignment is driven only by steric interactions. The method is being modified to also consider electrostatic interactions. 

Another method to determine the magnitude and rhombicity of the alignment tensor is based on the determination of the Saupe order matrix. The anisotropic parameter of motional averaging is represented by this order matrix, which is diagonalized by a transformation matrix that relates the principal frame, in which the order matrix is diagonal,... 

In a multidomain protein whose domains have fixed orientations relative to each other, a unique alignment tensor will represent the preferred orientation of all the domains in the anisotropic environment. Therefore, structure refinement with dipolar couplings is performed as in one-domain proteins (Sect. 8.4). Several examples are reported in the literature of cases with conformational ambiguity due to the lack of NOE contacts between the domains. One example is the determination of subdomain orientation of the riboso-mal protein S4 z)41 [97]. In this work the lack of NOE contacts between the domains produces an ambiguity in interdomain orientation. The authors use two different anisotropic media to obtain dipolar couplings (DMPC/DHPC bicelles and Pfl filamentous bacteriophages). They conclude that subdomain orientation in solution is similar to the one present in the crystal structure. 

Since the discovery of the nuclear Overhauser effect (NOE, see previous section) [4, 5] and scalar coupling constants [36, 37] decades ago, NMR-derived structure calculations of biomolecules largely depended on the measurement of these two parameters [38]. Recently it became possible to use cross-correlated relaxation (CCR) to directly measure angles between bond vectors [39] (see also Chapt 7). In addition, residual dipolar couplings of weakly aligned molecules were discovered to measure the orientation of bond vectors relative to the alignment tensor (see Sect 16.5). Measurement of cross-correlated relaxation was described experimentally earlier for homonuclear cases [40, 41] and is widely used in solid-state NMR [42 14]. 

In Section 2 it was established theoretically that, relative to some molecule-fixed reference axes, a molecular order (alignment) tensor with five independent parameters was sufficient to describe the molecular orientation upon which the observed dipolar couplings depend. Based on knowledge of the order tensor and the molecular structure, it is possible to predict the corresponding RDCs. From this relationship, one might anticipate that... 

For the reversible parts of the equations some coupling constants have been introduced the flow-alignment tensor... 

In order to derive information on dynamical quantities from such a coincidence experiment, one has to select angles d at which other components of the alignment tensor also contribute, because then the different dependences on the dipole matrix elements, including their relative phases, are involved. As can be seen from... 

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