Relaxation Rate Dispersion

Chemical exchange rate is also determined by Rip experiments in the rotating frame coordinate in which magnetization is spin-locked by application of a radio-frequency field with near-resonance or off-resonance. CPMG experiments in proteins will be apphcable to chemical exchange processes of the value feex l X 10 s as a result of experimental constraints on 

00 is the spin-lock rf frequency and Wq is population-averaged chemical shift. In the presence of an exchange between the two sites A and B, R2 =R 2 +l ex in which R°2 is the relaxation constant due to relaxation mechanism other than exchange. For two-site exchange that is fast on the chemical shift time scale [20,90,91], 

Slow conformational fluctuation was characterized by the difference decay rates (Rcc) ofDQC and ZQC selected by CPMG experiments [98,99]. The relaxation rates, DQC = ( 2)(N H + N H ) and ZQC =( 2) (N H + N H ), depend on the pulse repetition rate if there are local motions on slow timescale (ps—ms). This dispersion effect occurs when the two nuclei experience slow correlated modulations of the isotropic chemical shifts. The difference of the decay rates, Rcc, is the sum of chemical shift anisotropy cross-correlation (CSA/CSA) and isotropic chemical shift modulation (CSM/CSM) as well as additional dipole/dipole crosscorrelation (DD/DD) contributions due to coupftngs to neighbouring nuclei  

The CSA/CSA rate does not depend on the frequency i cpmg = 1/4t of a CPMG sequence, while the CSM/CSM rate is attenuated with increasing I CPMG. like the Rex contribution in a conventional R2 CPMG dispersion [99]. Therefore, this DQC/ZQC experiment is shown to complement the widely used R2 CPMG experiments. An attenuation of the relaxation rate with increasing pulse repetition rate can be attributed to slow motions. 

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It can be seen that, in all cases, relaxation rates are directly proportional to (Aa). Because Aa reflects the anisotropy of the shielding tensor and because the chemical shift originates from the shielding effect, the terminology Chemical Shift Anisotropy is used for denoting this relaxation mechanism. Dispersion may be disconcerting because of the presence of Bq (proportional to cOq) in the numerator of and R2 (Eq. (49)). Imagine that molecular reorientation is sufficiently slow so that coo 1 for all considered values of coo from (49), it can be seen that R is constant whereas R2 increases when Bq increases, a somewhat unusual behavior. 

At ambient temperature, H, and 0 relaxation is in the extreme narrow range and dispersion curves are perfectly flat (see Fig. 9 bottom) precluding any correlation time determination. Furthermore as inter- and intramolecular contributions to proton relaxation cannot be easily separated and as the deuterium and 0 quadrupole coupling constants are not known with sufficient accuracy, there is a real problem for determining a meaningful correlation time. This problem was solved only in the early 1980s by resorting to the cross-relaxation rate which is purely intramolecular... 

The proton longitudinal relaxation rate of deoxygenated water is 0.3 s at 25°C, with about 25% of this value being attributed to intermolecular dipolar relaxation. In that case, of course, no dispersion occurs. However, for water in equilibrium with air, due to paramagnetic molecular oxygen, the relaxation rate increases by 0.1 at low fields and exhibits a dispersion around 40 MHz (47). 

The MSB-equations were first presented by Connick and Mat (23) and by Reuben et al. (24). A formal derivation of these equations can be found, in a somewhat sketchy form, in the article by Gueron (25) and in a more stringent version in an article by Benetis et al. (26). Since oos 658m/ if / is a proton, and even more if it represents another nuclear spin, the first and third term of the DD part of Eq. (12) can safely be combined into a seven term , dispersing at msXc2 = 1 while the three term disperses at m/Td = 1. Similar equations can also be derived for the nuclear spin-spin relaxation rate in a paramagnetic complex ... 

The Bloembergen-Morgan equations, Eqs. (14) and (15), predict that the electron spin relaxation rates should disperse at around msTy = 1. This will make the correlation times for the dipolar and scalar interaction, %ci and respectively, in Eq. (11) dependent on the magnetic field. A combination of the modified Solomon-Bloembergen equations (12) and (13), for nuclear relaxation rates with the Bloembergen-Morgan equations for the field dependence... 

Longitudinal and transverse nuclear relaxation profiles differ in the high field part. In fact, the equation for the transverse nuclear relaxation rates contains a non-dispersive term, depending only on Xd. Therefore the transverse relaxation does not go to zero at high fields, as longitudinal relaxation does, but increases because Tie increases (until it increases to the point where it becomes longer than x or Xm)-... 

In the presence of contact contribution to nuclear relaxation, the NMRD profile results as a sum of the dipolar and contact relaxation rates. The profile of contact relaxation as a function of field is characterized by the presence of only one dispersion (Fig. 3), corresponding to the (OsT e dispersion (Eqs.(5) and (6)), in the hypothesis that Xg, = T,e (see Section II.B of... 

The water proton NMRD of the pseudooctahedral Co(H20)g (reported in Fig. 13) shows almost field-independent water proton relaxation rate values in the 0.01-60 MHz region (47). Therefore, the (Os c = 1 a nd of course the co/Cc = l dispersions must occur at fields higher than 60 MHz. This provides an upper limit value for Tig equal to 4 x 10 s. Such a low Tig value is consistent with the low water proton relaxation rate values. By using the SBM theory, Tie at 298 K can be estimated to be about 10 s. It can be larger, if the presence of a probable static ZFS is taken into account (47). When measurements are performed in highly viscous ethyleneglycol the observed rates are similar to those obtained in water. This suggests that Tig is also similar and, therefore, it is rotation-independent (47). 

Table II reports the contact coupling constant for different aqua ion systems at room temperature. The contact coupling constant is a measure of the unpaired spin density delocalized at the coordinated protons. The values were calculated from the analysis of the contact contribution to the paramagnetic enhancements of relaxation rates in all cases where the correlation time for dipolar relaxation is dominated by x and Tig > x. In fact, in such cases the dispersion due to contact relaxation occurs earlier in frequency than the dispersion due to dipolar relaxation. In metalloproteins the contact contribution is usually negligible, even for metal ions characterized by a large contact contribution in aqua ion systems. This is due to the fact that the dipolar contribution is much larger because the correlation time increases by orders of magnitude, and x becomes longer than Tig. Under...

The presence of ZFS may also cause the occurrence of a further dispersion in the plot of relaxation rate as a function of the magnetic field, corresponding to the transition from the dominant ZFS limit to the dominant Zeeman limit (see Section I.C.2.a). 

Fig. 19. The magnetic relaxation dispersion for water proton in a Sephadex G-25 sample swollen to equilibrium at different values of pH at 298 K. The open circles are the relaxation rates for the methyl protons of dimethyl sulfoxide. The solid lines were computed from a two-stage exchange model 100).

For a fixed value of Tc, the frequency dependence of either term is a Lorentzian centred at zero frequency. In the tc dependence two regimes are distinguished In the fast motion regime (coiTc spectral density is proportional to tc and does not depend on the measuring frequency a>i, whereas in the slow motion regime (a>iTc > l) it is proportional to ( Tc) i.e. the relaxation rate exhibits dispersion. 

The piezoelectric constant of polymer films is usually a function of the frequency of the applied strain, and the constant is expressed by a complex quantity. In other words, the open-circuit voltage across the film surfaces is not in phase with the applied strain and the short-circuit current is not in phase with the strain rate. This effect, first pointed out by Fukada, Date and Emura (1968) and designated piezoelectric relaxation or dispersion, will be discussed in this review in terms of irreversible thermodynamics and composite-system theory. 

In commercial zeolites the proton relaxation of adsorbed molecules is controlled by magnetic interaction with paramagnetic impurities (Fe8+) of the zeolite lattice (9). The relaxation rate 1/Ti is proportional to the number of these paramagnetic centers. If the lattice is covered with diamagnetic metal atoms, this interaction should be reduced according to the amount and dispersity of the metal. 

The structural information derived from relaxation enhancement studies depends somewhat on the model chosen to describe the interaction of solvent protons with the protein molecules. For example even if the experiments indicated that the dispersion of Tfpr were essentially determined by the correlation time for rotational tumbling of the protein the tumbling of the hydration waters would not necessarily have to be restricted to that of the entire hydrated protein. Evidence was found that fast intramolecular tumbling about an axis fixed with respect to the surface of the hydrated species reduced the proton and O17 nuclear relaxation rates even in extremely stable aquocomplexes of Al3+ and other metal ions (Connick and Wiithrich (21)). The occurrence of similar... 

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