Longitudinal relaxation rates

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

Thus, the relevant spectral density, here equal to the longitudinal relaxation rate, is given by... 

The crucial point arises from X+ which may become very small, thus leading to the little known property of long lived states. For this purpose let us assume that R , in our case the proton longitudinal relaxation rate, is much greater than Ri and oAb, a common situation which, for X+, leads to a quantity which can be very small... 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

Here R 2 and R i are the transverse and longitudinal relaxation rates modified to subtract the contributions from high-frequency motions (see e.g. [20, 49]) ... 

In Fig. 3b, L2 and L4 have substantially different STD values, with L4 showing significantly smaller effect, even though these two protons are equidistant from the P3 proton. This is a simple consequence of the differences in the relaxation rates for these L2 and L4 protons due to differences in their local environments (e.g., in the asymmetric model, the L4-L5 distance is shorter than the L3-L2 distance, and thus L4-L5 protons experience a faster longitudinal relaxation rate than the L4-L5 protons). These observations suggest that caution is needed in quahtative attempts to relate the magnitudes of steady state STDs to spatial proximity of ligand protons to the protein protons. 

The last point is about equivalent spins (or like spins as the two protons of the water molecule). Referring to Solomon equations (see (13)), we can notice that, because of this equivalence, the effective longitudinal relaxation rate is obtained by adding the cross-relaxation rate to the specific longitudinal relaxation rate ... 

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). 

Fig. 14. Circles typical NMRD data (water longitudinal relaxation rates in a protein aqueous solution adapted from ref. (55)).The curve corresponds to a lorentzian function with tq deduced from the half-height of the experimental data. Note...

The longitudinal relaxation rate inversely decreases with the residence time of water molecules inside the agglomerate. This effect was demonstrated thanks to a controlled and chemically induced process of agglomeration amongst ferrite nanomagnets coated by polyelectrolyte polymers. The NMRD profile becomes flatter on increasing agglomeration (Pig. 10). 

Ferritin The longitudinal relaxation rate of ferritin is not significantly influenced by the pH of the aqueous solution (47,48). However, after subtraction of the apoferritin contribution, the effect is more significant (Fig. 18). The transverse relaxation of ferritin solutions is almost pH independent (Pig. 16). 

Frye, J. S., Comparison of Inversion-Recovery Methods for Measuring Longitudinal Relaxation Rates, Concepts in Magnetic Resonance An Educational Journal, 1989, ... 

While keeping in mind the general picture of nuclear relaxation in paramagnetic systems as described in Section 3.1, it is appropriate to consider first the simple case of dipolar coupling between two point-dipoles as if the unpaired electrons were localized on the metal ion. The enhancement of the nuclear longitudinal relaxation rate Rim due to dipolar coupling with unpaired electrons can be calculated starting from the Hamiltonian for the system ... 

Fig. 3.16. (A) Plot of the nuclear longitudinal dipolar relaxation rate in the presence of a field dependent electron relaxation time (5 = sh) for rl0 = 10-9 s and r = 1,5, 10,50 x 10 12 s. The amplitude of the peak that appears in the high field part of the profile increases with increasing r . (B) Plot of the longitudinal dipolar relaxation rate in the presence of a field dependent electron relaxation time (S = V2) for A, = 0.047 cm-1 and r = 2,5, 10, 20 x 10-12 s. (C) Plot of the longitudinal relaxation rate in the presence of a field dependent electron relaxation time with A/h = 0 and 1 MHz (dotted lines) and of the transverse relaxation rate with A/h = 0 (dashed line) and 1 MHz (solid line). Conditions 5 = %, A, = 0.047 cm-1, r = 2 x 10-12 s (r,o = 10-9 s) and tr = 10-9 s.

As anticipated in Section 3.2, a nuclear longitudinal relaxation rate R can be defined only when relaxation is an exponential process. This is at variance with nuclear transverse relaxation, which is always exponential and always defined by the transverse relaxation rate / 2- As far as longitudinal relaxation is concerned, when the return to the equilibrium value Mz (oo) of longitudinal nuclear magnetization after a 180° pulse is exponential, we can write (see Section 1.7.4)... 

The longitudinal relaxation rate enhancement R p goes asymptotically from zero to fMRiM with increasing r 1 (Fig. 4.3A), according to the following equation ... 

Therefore, by knowing the longitudinal relaxation rate constant of the nucleus in the B site Rf in the absence of exchange and by measuring the fractional change in intensity of signal B (called saturation transfer), the rate constants can be obtained. We recall that the two rate constants are related by the fractional populations of the two sites (Section 4.2), in turn proportional to the equilibrium intensities of the two signals... 

From the functional form of Eq. (4.30) it is easy to predict the behavior of saturation transfer as a function of the exchange rate. When the rate constant for the B - A transformation (k i) is much smaller than the longitudinal relaxation rate of the nucleus in the B site (Rf), the saturation transfer tends to zero. When the rate constant is much higher, the saturation transfer tends to —1, i.e. there is a total transfer of magnetization to the B site when the A site is kept saturated. Note that these conditions are referred to as fast exchange, even if the exchange is still slow with respect to the chemical shift separation. Fast exchange conditions on the... 

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