Effective correlation time

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

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.

A related experiment TOCSY (Total Correlation Spectroscopy) gives similar information and is relatively more sensitive than the REIAY. On the other hand, intensity of cross peak in a NOESY spectrum with a short mixing time is a measure of internuclear distance (less than 4A). It depends on the correlation time and varies as . It is positive for small molecules with short correlation time (o r <<1) and is negative for macromolecules with long correlation time (wr >>l) and goes through zero for molecules with 1 Relaxation effects should be taken into consideration for quantitative interpretation of NOE intensities, however. 

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. 

The relaxation data for the anomeric protons of the polysaccharides (see Table II) lack utility, inasmuch as the / ,(ns) values are identical within experimental error. Obviously, the distribution of correlation times associated with backbone and side-chain motions, complex patterns of intramolecular interaction, and significant cross-relaxation and cross-correlation effects dramatically lessen the diagnostic potential of these relaxation rates. 

The rate at which dipole-dipole relaxation occurs depends on several factors (a) the nature of the nucleus, (b) the internuclear distance, r, and (c) the effective correlation time, t of the vector joining the nuclei (which is inversely proportional to the rate at which the relevant segment of the... 

In addition to the dipole-dipole relaxation processes, which depend on the strength and frequency of the fluctuating magnetic fields around the nuclei, there are other factors that affect nOe (a) the intrinsic nature of the nuclei I and S, (b) the internuclear distance (r,s) between them, and (c) the rate of tumbling of the relevant segment of the molecule in which the nuclei 1 and S are present (i.e., the effective molecular correlation time, Tf). 

Lowering the temperature has a similar effect on the deuterium spectra as does increased loadings. In Figure 3, spectra for benzene-d6/(Na)X at 0.7 molecules/supercage over the temperature range 298 to 133 K are shown. It is observed that both benzene species are detected simultaneously between 228 and 188 K. Below this temperature the oriented benzene species becomes the predominant form. A similar situation occurs for polycrystalline benzene-dg in which two quadrupole patterns, one static and the other motionally narrowed due to C rotation, are observed to coexist at temperatures between 110 and 130 K (7). This behavior has been attributed to sample imperfections (8) which give rise to a narrow distribution in correlation times for reorientation about the hexad axis. For benzene in (Na)X and (Cs,Na)X such imperfections may result from the ion/benzene interaction, and a nonuniform distribution of benzene molecules and ions within the zeolite. These factors may also be responsible for producing the individual species. However, from the NMR spectra it is not possible to... 

Locahzed motion can also lead to local variations in correlation times. Folded peptides with unfolded C- or N-terminal residues, for example, will have varying correlation times for the rigid and flexible parts of the molecule, resulting in different cross-relaxation rates. Such effects can usually be distinguished by the Unewidths and intensities of the corresponding diagonal signals, since the autorelaxation rates also depend on the correlation time. 

A parameterized model for gas relaxation as described can be easily modified to include the effect of wall collisions by utilizing the known additivity of correlation times... 

Figure 16 Correlation of effective pore radius of the paracellular route of rat alveolar epithelial cell monolayers with transepithelial electrical resistance and time in culture. Pore radii were calculated from the data shown in Figure 14.

FIGURE 10.4 Effective rotational correlation time of 16-SASL in DMPC membranes plotted as a function of mole fraction of zeaxanthin at different temperatures (x2B (O) and x2C ( )). (From Subczynski, W.K. et al., Biochim. Biophys. Acta, 1105, 97, 1992. With permission.)... 

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

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...

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... 

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