Relaxation matrix

Borgias B A, Gochin M, KenA/ood D J and James T L 1990 Relaxation matrix analysis of 2D NMR data Prog. NMR Spectrosc. 22 83-100... 

Erom a given structure, the NOE effect can be calculated more realistically by complete relaxation matrix analysis. Instead of considering only the distance between two protons, the complete network of interactions is considered (Eig. 8). Approximately, the... 

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

AMU Bonvm, R Boelens, R Kaptem. Determination of biomolecular structures by NMR Use of relaxation matrix calculations. In WF van Gunsteren, PK Weiner, AI Wilkinson, eds. Computer Simulation of Biomolecular Systems Theoretical and Experimental Applications, Vol 2. Leiden ESCOM, 1993, pp 407-440. 

The relaxation matrix of Eq. 5 is symmetric and contains two classes of elements the diagonal elements, p, values, which represent the self-relaxa-tion terms, and the off-diagonal elements, ay values, which are the cross-... 

Here, the W j represent the probability per unit time that the system performs a transition from state i to state j. In the present case of relaxation between the two states denoted as LS and HS, we may associate the matrix elements W 2 and 1 21 with the rate constants /clh and Ichl fc)r the conversion process LS HS and HS LS, respectively. The diagonal elements of the relaxation matrix W... 

Then, the first order relaxation matrix results in an uncoupled system of differential equations with the diagonal form ... 

The symbols Rauto and Rcross within the relaxation matrix are the auto- and cross-relaxation rates, respectively. and (l2Z) are the longitudinal magnetizations of spin 1 and 2, respectively, and the brackets indicate averaging over the whole ensemble of spins. Rcross in terms of the spectral densities is given by... 

In general, a relatively direct and straightforward means of analysis may be performed in the case of slow exchange on the chemical shift time-scale by combining the relaxation matrices of the free and bound state with the kinetic matrix to describe the effect of exchange [12]. For the two spin systems described above the expanded relaxation matrix R can be written as ... 

Alternatively, the much more common situation in trNOE studies involves fast exchange on the chemical shift time scale where the observed resonance shifts are weighted averages of the corresponding shift in the free and bound state [13]. A full account of the complete relaxation matrix and conformational exchange effects for n spins has been performed by London et al. [13], and a similar treatment was later incorporated into the programme CORCEMA [14]. 

The symbol (Oaa denotes the energy difference between the two eigenstates, converted into angular frequency. The first term on the right-hand side (rhs) of Eq. (18) vanishes for the populations (oo a = 0) and describes the preces-sional motion for coherences. Rota pp is an element of the relaxation matrix (also called relaxation supermatrix) describing various decay and transfer processes in the spin system. Under certain conditions (secular approximation), one neglects the relaxation matrix elements unless the condition = pp is fulfilled. 

Selected entries from Methods in Enzymology [vol, page(s)] Anisotropy effects, 261, 427-430 determination by dynamic laser light scattering (quasi-elastic light scattering), 261, 432-433 determination for nucleic acids by NMR [accuracy, 261, 432-433 algorithms, 261, 11-13, 425, 430 carbon-13 relaxation, 261, 11-12, 422-426, 431, 434-435 cross-relaxation rates, 261,419-422, 435 error sources, 261, 430-432 phosphorus-31 relaxation, 261, 426-427, 431 proton relaxation, 261,51,418-422 relaxation matrix calculations, 261,12] deuterium solvent viscosity effects, 261,433 effect... 

In this equation, A is the relaxation matrix in eq. (14), A is the diagonal matrix of the eigenvalues, and U is the matrix with the eigenvectors of A as its columns. Note that this is identical to eq. (8), except that now all the frequencies are zero. 

The AECS scaling model is based on the results of the ECS approximation. The elements of the relaxation matrix / for an isotropic Q branch CARS experiment on N2 are given as ... 

Here p is the density matrix for all molecular states in the three-level system depicted in Fig. 4, and all incoherent relaxation terms caused, for example, by collisions, spontaneous emission, or decay in a (quasi)continuum are incorporated in the relaxation matrix rreiax. 

Powell et al. give an excellent review of several approaches to interpret the frequency dependence of Tle and T2e in these systems [71]. One convenient approach is that developed by Hudson and Lewis [72], who showed that the eigenvalues of the relaxation matrix R as defined in Bloch-Wangsness-Red-field (BWR) theory [73] are functions of rv and the experimental frequency co, and are related to the relaxation time T2ei of the i-th allowed electron spin transition by the expression ... 

The components of the relaxation matrix in the basis set adopted are real numbers. They may be computed from the correlation functions of the molecular rotations or, more precisely, from their Fourier transforms. 

In our further considerations we assume that the components of the relaxation matrix are known in advance. This is a realistic assumption provided that unsaturated NMR spectra are being dealt with. One should note that equation (35) can be used for the description of motion of the system during time spans which are much longer than the typical correlation times for molecular rotation in liquids (about 10-11—10—12 sec). The processes of intra- and inter-molecular exchange which are considered here are characterized by half-life times longer than 10 5 sec. It seems justified to consider these processes independently of molecular rotations, in spite of the fact that they all participate in the relaxation. 

Needless to say, the so-called combination transitions are also considered in this subspace. Lineshape equations for special forms of the relaxation matrix can also be written in terms of the Hilbert space. However, the notation becomes quite involved. This is probably the source of some erroneous simplifications which consist of neglecting combination transitions in the equations of lineshape. (50)... 

In a typical situation we are interested in the absorption mode of a dynamic spectrum,/abs (co), which equals the real part of the complex function f(co) given by equation (145). In most cases of unsaturated spectra the relaxation matrix which describes single-quantum transitions can be replaced by a constant — E/T2(effective) which is characteristic of the experimental conditions involved and reflects the inhomogeneity of the external magnetic field B0. The absorption mode spectrum is given by ... 

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