Relaxivity rotational correlation

Small molecules in low viscosity solutions have, typically, rotational correlation times of a few tens of picoseconds, which means that the extreme narrowing conditions usually prevail. As a consequence, the interpretation of certain relaxation parameters, such as carbon-13 and NOE for proton-bearing carbons, is very simple. Basically, tlie DCC for a directly bonded CH pair can be assumed to be known and the experiments yield a value of the correlation time, t. One interesting application of the measurement of is to follow its variation with the site in the molecule (motional anisotropy), with temperature (the correlation... 

N-protonation the absolute magnitude of the Ad values is larger than for Af-methylation <770MR(9)53>. Nuclear relaxation rates of and have been measured as a function of temperature for neat liquid pyridazine, and nuclear Overhauser enhancement has been used to separate the dipolar and spin rotational contributions to relaxation. Dipolar relaxation rates have been combined with quadrupole relaxation rates to determine rotational correlation times for motion about each principal molecular axis (78MI21200). NMR analysis has been used to determine the structure of phenyllithium-pyridazine adducts and of the corresponding dihydropyridazines obtained by hydrolysis of the adducts <78RTC116>. 

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.

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

For folded proteins, relaxation data are commonly interpreted within the framework of the model-free formalism, in which the dynamics are described by an overall rotational correlation time rm, an internal correlation time xe, and an order parameter. S 2 describing the amplitude of the internal motions (Lipari and Szabo, 1982a,b). Model-free analysis is popular because it describes molecular motions in terms of a set of intuitive physical parameters. However, the underlying assumptions of model-free analysis—that the molecule tumbles with a single isotropic correlation time and that internal motions are very much faster than overall tumbling—are of questionable validity for unfolded or partly folded proteins. Nevertheless, qualitative insights into the dynamics of unfolded states can be obtained by model-free analysis (Alexandrescu and Shortle, 1994 Buck etal., 1996 Farrow etal., 1995a). An extension of the model-free analysis to incorporate a spectral density function that assumes a distribution of correlation times on the nanosecond time scale has recently been reported (Buevich et al., 2001 Buevich and Baum, 1999) and better fits the experimental 15N relaxation data for an unfolded protein than does the conventional model-free approach. 

It is clear from the above equations that numerous parameters (proton exchange rate, kcx = l/rm rotational correlation time, tr electronic relaxation times, 1 /rlj2e Gd proton distance, rGdH hydration number, q) all influence the inner-sphere proton relaxivity. Simulated proton relaxivity curves, like that in Figure 3, are often used to visualize better the effect of the... 

Figure 3 Effect of the water exchange rate, kex, and the rotational correlation time, rR, on inner-sphere proton relaxivity. The plot was simulated for a particular value of the longitudinal electron spin relaxation rate, 1/Tie — 5.28xlOss 1. The marketed contrast agents all have relaxivities around 4—5mM 1s 1 in contrast to the theoretically attainable values over lOOrnM-1 s 1, and this is mainly due to their fast rotation...

Table 3 Comparison of rotational correlation times and proton relaxivities for low molecular weight and...

Figure 6 Effect of the increased rotational correlation time on the proton relaxivity of MP2269, a Gd111 chelate capable of noncovalent protein binding (Scheme 2). The lower NMRD curve was measured in water, whereas the upper curve was obtained in a 10%w/v bovine serum albumin solution in which the chelate is completely bound to the protein. The rotational correlation times calculated are rR=105ps in the nonbound state, and rR= 1,000 ps in the protein-bound state (t=35°C). For this chelate, the water exchange...

Gd(DTPA-bisamide)-PEG copolymers52 are also flexible, which explains their low relaxivity. A combined nO NMR, EPR, and NMRD study performed on [Gd(DTPA-BA)-PEG]x concluded that the effective rotational correlation time in the long polymer chain is not higher than that of the same Gd111 monomer unit, Gd(DTPA BMA) restricted to rotate around a single axis (tr for the polymer equals three times the rR for the monomer).52 An interesting feature of this... 

Fig. 2. Calculated relaxivities as a function of the water exchange rate for various proton Larmor frequencies and rotational correlation times, tr. The simulations have been performed by using the common Solomon-Bloembergen-Morgan theory of paramagnetic relaxation.

We should note that the use of the Lipari-Szabo analysis implies that relaxation data are available at multiple magnetic fields. It provides a phenomenological description of the rotational motion that can be very useful for comparing systems with similar structure. Nevertheless, one should be aware of the limits of this approach and avoid direct comparison of local or global rotational correlation times for structurally very different compounds. 

Fig. 4. Inner sphere contribution to the proton relaxivity as a function of the proton Larmor frequency. The curves were calculated on the basis of the Solomon-Bloembergen-Morgan theory for different values of the rotational correlation time, tr, and q — 1, kex — 10 x 106 s-1, tv = 20 ps, A2 = 0.1 x 102Os-2.

For Gdin-based agents, the relaxivity response is most often related to a change in water accessibility or to the variation of the size and consequently of the rotational correlation time of the complex. In addition to Gdm complexes, PARACEST agents are... 

Their Gdm complexes were found to show significantly increased relaxivity upon interaction with Zn11, Ca11, and Mg11 ions. The relaxivity increase was related to an increase of the rotational correlation time, and the mechanism was ascribed to the formation of coordination oligomers or polymers that are typical for bisphosphonate complexes. 

Starburst (TM) dendrimers with DTPA can contain 170 bound Gd(III) ions and have relaxivities (per bound Gd) up to 6 times that of Gd-DTPA (308). Both global and local motion contribute to the overall rotational correlation time. Attempts have been made to increase the re-laxivity of Gd(III) by optimizing the rotational correlation time via binding of Gd(III) to derivatized polysaccharides (309) and by binding lipophilic complexes to albumin in serum (310). The latter approach has achieved relaxivities as high as 44.2 mM l s1 for derivatized 72 (311). 

Fig. 5. Transverse relaxation times of carbonyl carbon as a function of rotational correlation time, rc (a), at four polarizing magnetic field strengths, and as a function of field strength, B0, against four different correlation times (b), according to Eq. (1).

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