Relaxation matrix model structure

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. 

Measurement of the accuracy of NMR-derived structures is a much more difficult task than estimating their precision. An absolute measure of the accuracy of an NMR-derived structure is not possible in the absence of any knowledge about the true structure and therefore it has to be measured by some statistic.2 3 One advantage of iterative relaxation matrix analysis (IRMA),204 205 in which the structure is iteratively refined by comparison of the experimental NOESY spectrum with a synthetic spectrum back-calculated from the coordinates of the current structural model, is that it enables an NMR R factor to be calculated,203 205 which is analogous to the R factor (or reliability index) used in crystallography. However, IRMA is not widely used for structure calculations and hence NMR R factors are rarely reported. 

One earlier model was developed by Dannenberg to explain observations of behavior of compounded rubber." Figure 7.33 shows how this model works. Polymer chains are connected with filler particles. Depending on strain, chains remain relaxed, are fully extended, slip, or matrix undergoes structural changes. It is im-... 

Figure 2. The hybrid-hybrid relaxation matrix refinement procedure for 3D NOESY-NOESY data. A starting model is used to simulate NOE data for the first iteration. Experimental 3D data is scaled and merged with simulated 3D data to produce a linear table of 3D experimental volumes and the simulated volumes needed for deconvolution. Deconvoluted 2D data is merged with any available 2D experimental data and then with a simulated, complete 2D volumes matrix. The structures resulting from the standard 2D MORASS refinement are used in subsequent iterations until convergence is reached. (Reproduced with permission from reference 24. Copyright 1996.)...

Program CORMA (Complete Relaxation Matrix Analysis) calculates relaxation rates and NOE intensities for a given structure (28-29) or an ensemble of structures with specified populations (30). Typically for short DNA duplexes, a motional model of overall isotropic tumbling is used for all protons except methyl groups which are assumed to undergo fast internal rotation (31). If experimental NOE intensities are specified, CORMA also calculates a number of indices of agreement between experimental and simulated data (2). 

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]. 

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. 

Let us consider a simple model of a quenched-annealed system which consists of particles belonging to two species species 0 is quenched (matrix) and species 1 is annealed, i.e., the particles are allowed to equlibrate between themselves in the presence of 0 particles. We assume that the subsystem composed of 0 particles has been a usual fluid before quenching. One can characterize it either by the density or by the value of the chemical potential The interparticle interaction Woo(r) does not need to be specified for the moment. It is just assumed that the fluid with interaction woo(r) has reached an equlibrium at certain temperature Tq, and then the fluid has been quenched at this temperature without structural relaxation. Thus, the distribution of species 0 is any one from a set of equihbrium configurations corresponding to canonical or grand canonical ensemble. We denote the interactions between annealed particles by Un r), and the cross fluid-matrix interactions by Wio(r). 

In order to study lattice relaxation effect by ASR we assume a simple model. As a first step we consider the terminal point approximation. Here the distortion of the lattice taken into account is the stretching or the contraction and angular distortion of the bond connecting two sites in a lattice and the effect of neighbouring site is neglected. As a result of such distortion the structure matrix takes the form ... 

A more realistic model for the secondary relaxation needs to consider motions of a molecular group (considered as a rigid object) between two levels. The group may contain N atoms with the scattering length h, at positions r (i=lj ). The associated motion may consist of a rotation aroimd an arbitrary axis, e.g. through the centre of mass depicted by a rotational matrix Q and a displacement by a translational vector . In order to evaluate the coherent dynamic structure factor, scattering amphtudes of the initial (1) and final (2) states have to be calculated ... 

The next two chapters are devoted to ultrafast radiationless transitions. In Chapter 5, the generalized linear response theory is used to treat the non-equilibrium dynamics of molecular systems. This method, based on the density matrix method, can also be used to calculate the transient spectroscopic signals that are often monitored experimentally. As an application of the method, the authors present the study of the interfadal photo-induced electron transfer in dye-sensitized solar cell as observed by transient absorption spectroscopy. Chapter 6 uses the density matrix method to discuss important processes that occur in the bacterial photosynthetic reaction center, which has congested electronic structure within 200-1500cm 1 and weak interactions between these electronic states. Therefore, this biological system is an ideal system to examine theoretical models (memory effect, coherence effect, vibrational relaxation, etc.) and techniques (generalized linear response theory, Forster-Dexter theory, Marcus theory, internal conversion theory, etc.) for treating ultrafast radiationless transition phenomena. 

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