NOESY dynamic matrix

Fig. 7. Buildup rate analysis of NOESY spectra at T = 298 K (Xn, = 0.3 s, 0.6 s, 1 s) and T = 233 K (Tm = 10 ms, 20 ms, 30 ms, 40 ms, 80 ms, 160 ms, 320 ms, 640 ms, 1.2 s). Lines are drawn according to eq. (8) with a dynamic matrix from the energy minimized model (table 1). High-temperature data (negative slopes) are linearly fitted according to eq. (30). Low-temperature data (positive slopes) are fitted by eq. (30) only points with aij /ajj < 0.2 were used (inset).

If we have tiie relaxation matrix and an approximate structure, we can back calculate the NOESY spectra. The problem with the relaxation matrix method is that some of the cross relaxation rates are not observed due to spectral overlap, dynamic averaging and exchange. Boelens et al. (1988 1989) attempted to solve the problem by supplementing the imobserved NOEs with those calculated from a model structure. From a starting structure, the authors use NOE build-ups, stereospedfic assignments and model-calculated order parameters to construct the relaxation matrix. An NOE matrix is then calculated. This NOE matrix is used to calculate the relaxation matrix and it is in turn used to calculate the new distances. The new distances are then used to calculate a new model structure. The new structure can be used again to construct a new NOE matrix and the process can be iterated to improve the structures. The procedure is called IRMA or iterated relaxation matrix analysis. 

With an algorithm for calculating the gradient of NOE intensities, one can fit the structure to a NOESY spectrum, minimizing the difference between the observed NOE intensities and those calculated based on a full relaxation matrix. This takes into account spin difiusion in the calculation of structures but it reqmres at least some tmderstanding of the dynamics of protein. 

As mentioned above, the cross peak intensities from NOESY spectra taken at long mixing times caimot be related in a simple and direct way to distances between two protons due to spin diffusion effects that mask the actual proton distances. A possibiUty to extract such information is provided by relaxation matrix analysis that accounts for all dipolar interactions of a given proton and hence takes spin diffusion effects explicitly into consideration. Several computational procedures have been developed which iteratively back-calculate an experimental NOESY spectrum, starting from a certain molecular model that is altered in many cycles of the iteration process to fit best the experimental NOESY data. In each cycle, the calculated structures are refined by restrained molecular dynamics and free energy minimization [42,43]. 

Figure 19.6 Arrangement of the three consecutive base pairs G2-C71, G3-U70, and G4-C69 a) in the starting structure (as created with the BIOPOLYMER module of INSIGHT II) obtained from a regular Watson-Crick A-RNA duplex with a G3-C70 base pair after subsequent replacement of the C70 residue by a uridine. No further adjustment of the U70 base position has been made and hence no correct hydrogen bonding pattern is found b) in the final (average) structure derived from relaxation matrix analysis of the 300 ms NOESY spectrum using the IRMA procedure [45] and restrained molecular dynamics calculation. Note that G3 and U70 now are forming a regular base pair. The bases G3 and U70 are indicated by thick lines.

Greater accuracy can be achieved by methods that involve calculation of a full relaxation matrix from the NOESY data to generate interproton distances. A model protein structure can then be iteratively refined by back calculation until differences in the empirical and calculated data are minimized. The resulting distances can be used as restraints for further refining the protein structure by distance geometry or molecular dynamics methods. 

In this context, it is also worth noting that alternative methods exist for quantitative comparisons to NOESY intensities, based mainly on iterative refinements of the rate matrix itself. These models provide a translation from observed intensities to cross-relaxation rates, providing a useful intermediate step in the generation of structural and dynamic models that fit observed data. Much work remains to be done to determine the best approach to refinement and the benefits of going beyond the more common conventional level of structure determinations that uses distance bounds derived by empirical calibration from short-mixing time experiments. 

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