Relaxation in Coupled Spin Systems

3 Relaxation in Coupled Spin Systems. - The ability to transfer coherences between different nuclei in homo- and heteronuclear spin systems is one of the cornerstones of NMR. The transfer of polarization from I = (1/2) to enhance the signal of low-gyromagnetic ratio S = (1/2) nuclei has become routine. In solution state the majority of these methods employ free-precession type techniques such as INEPT where polarization transfer from one spin to another is mediated via the scalar coupling. The majority of these techniques are designed for spin-half systems and have shown particular utility in the study of structure and dynamics of biomolecules labeled with C and N. The use of INEPT for achieving polarization transfer in systems of scalar coupled quadrupolar nuclei has also been investigated.  

The NMR signals of insensitive nuclear spins can be enhanced by transferring polarization from a more sensitive species to which they are coupled. The well-known pulse sequences as the polarization transfer techniques are insensitive nuclei enhanced by polarization transfer (INEPT), distortionless enhancement by polarization transfer (DEPT), and reverse insensitive nuclei enhanced by polarization transfer (RINEPT) The INEPT sequence is an alternative to the nuclear Overhauser effect. The INEPT experiment does not require any particular relaxation mechanism and therefore a better enhancement factor can be obtained. Furthermore it is demonstrated that INEPT sequence can be used to determine the multiplicity of each signal in a NMR spectrum. More recently, the INEPT and DEPT experiments were used for the coherence transfer via heteronuclear J-coupling between spin-1/2 and quadrupolar nuclei in the solids. Fyfe et showed that coherence transfer via the scalar coupling between spin-1/2 and quadrupolar nuclei can be obtained in the solid state by using INEPT experiment. 

Kupriyanov studied relaxation of spin 1/2 in the scalar coupled spin system AMX with quadrupolar nuclei in the presence of cross correlation effects. Khaneja et a/. presented optimal control of spin dynamics in the presence of relaxation. Eykyn et a/. studied selective cross-polarization in solution state NMR of scalar coupled spin 1/2 and quadrupolar nuclei. Tokatli applied the product operator theory to spin 5/2 nuclei. Mahesh et a/. used strongly coupled spins for quantum information processing. Luy and Glaser inves- 

In liquid-state NMR, spin relaxation due to cross-correlation of two anisotropic spin interactions can provide useful information about molecular structure and dynamics. These effects are manifest as differential line widths or line intensities in the NMR spectra. Recently, new experiments were developed for the accurate measurement of numerous cross-correlated relaxation rates in scalar coupled multi-spin systems. The recently introduced concept of transverse relaxation optimized spectroscopy (TROSY) is also based on cross-correlated relaxation. Brutscher outlined the basic concepts and experimental techniques necessary for understanding and exploiting cross-correlated relaxation effects in macromolecules. In addition, he presented some examples showing the potential of cross-correlated relaxation for high-resolution NMR studies of proteins and nucleic acids. 

Prompers and Briischweiler showed by quasiharmonic analysis that the conformational partition function of a globular protein sampled on the ns time scale can be factorized in good approximation into purely reorientational part, which determines heteronuclear NMR spin relaxation, and a remaining part that includes other types of intramolecular motions. Thus a thermodynamic interpretation of NMR relaxation parameters in proteins in the presence of motional correlations can be given. 

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Although the invariant trajectory approach was derived for uncoupled spins i and j, it also reflects qualitatively the cross-relaxation and autorelaxation behavior in coupled spin systems (Griesinger and Ernst, 1988 Bax, 1988a). 

These complications require some carefiil analysis of the spin systems, but fiindamentally the coupled spin systems are treated in the same way as uncoupled ones. Measuring the z magnetizations from the spectra is more complicated, but the analysis of how they relax is essentially the same. 

To lessen experimental time, the null-point method may be employed by locating the pulse spacing, tnun, for which no magnetization is observed after the 180°-1-90° pulse-sequence. The relaxation rate is then obtained directly by using the relationship / , = 0.69/t n. In this way, a considerable diminution of measuring time is achieved, which is especially desirable in measurements of very low relaxation-rates, or for samples that are not very stable. In addition, estimates of relaxation rates for overlapping resonances can often be achieved. However, as the recovery curves for coupled spin-systems are, more often than not, nonexponential, observation of the null point may violate the initial-slope approximation. Hence, this method is best reserved for preliminary experiments that serve to establish the time scale for spin-lattice relaxation, and for qualitative conclusions. 

The relaxation theory used in the Appendix to describe the principle of TROSY clearly tells us what to expect, but it is always a little more satisfying if one can obtain a simple physical picture of what is happening. We consider a system of two isolated scalar coupled spins of magnitude %, 1H (I) and 15N (S), with a scalar coupling constant JHN. Transverse relaxation of this spin system is dominated by the DD coupling between spins XH and 15N and by the CSA of each individual spin. The relaxation rates of the individual multiplet components of spin 15N are now discussed assuming an axially symmetric 15N CSA tensor with the axial principal component parallel to the 15N-XH vector as shown in Fig. 10.2. 

On the basic of relaxation theory the concept of TROSY is described. We consider a system of two scalar coupled spins A, I and S, with a scalar coupling constant JIS, which is located in a protein molecule. Usually, I represents H and S represents 15N in a 15N-1H moiety. Transverse relaxation of this spin system is dominated by the DD coupling between I and S and by CSA of each individual spin. An additional relaxation mechanism is the DD coupling with a small number of remote protons, / <. The relaxation rates of the individual multiplet components in a single quantum spectrum may then be widely different (Fig. 10.3) [2, 9]. They can be described using the single-transition basis opera-... 

For a spin-1/2 nucleus, such as carbon-13, the relaxation is often dominated by the dipole-dipole interaction with directly bonded proton(s). As mentioned in the theory section, the longitudinal relaxation in such a system deviates in general from the simple description based on Bloch equations. The complication - the transfer of magnetization from one spin to another - is usually referred to as cross-relaxation. The cross-relaxation process is conveniently described within the framework of the extended Solomon equations. If cross-correlation effects can be neglected or suitably eliminated, the longitudinal dipole-dipole relaxation of two coupled spins, such... 

Their theory, based on the classical Bloch equations, (31) describes the exchange of non-coupled spin systems in terms of their magnetizations. An equivalent description of the phenomena of dynamic NMR has been given by Anderson and by Kubo in terms of a stochastic model of exchange. (32, 33) In the latter approach, the spectrum of a spin system is identified with the Fourier transform of the so-called relaxation function. 

Meakin and Jesson (48) used the Bloch equations in part of their work on the computer simulation of multiple-pulse experiments. They find that this approach is efficient for the effect upon the magnetization vector of any sequence of pulses and delays in weakly coupled spin systems. However, relaxation processes and tightly coupled spin systems cannot be dealt with satisfactorily in this way and require the use of the density matrix. 

Even in the absence of relaxation, Hartmann-Hahn transfer depends on a large number of parameters pulse sequence parameters (multiple-pulse sequence, irradiation frequency, average rf power, etc.) and spin system parameters (size of the spin system, chemical shifts, /-coupling constants). For most multiple-pulse sequences, these parameters may be destilled into effective coupling tensors, which completely determine the transfer of polarization and coherence in the spin system. This provides a general classification scheme for homo- and heteronuclear Hartmann-Hahn experiments and allows one to characterize the transfer properties of related... 

Very large micelles may also form in binary surfactant systems. These are long wormlike micelles that become entangled at higher concentrations, giving rise to rheological properties similar to those in polymer solutions. Such systems have been examined by H band shape analysis [52,53]. The protons of the surfactant hydrocarbon chain form a very large dipolar coupled spin system with an essentially continuous distribution of transverse relaxation rates. The distribution of relaxation rates is related to the distribution of order... 

W.P. RothweU, J.S. Waugh, Transverse relaxation of dipolar coupled spin systems under rf irradiation detecting motions in solids, J. Chem. Phys. 74 (1981) 2721-2732. 

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