Tumbling rates correlation time

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. 

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). 

In order for relaxation to occur through Wj, the magnetic field fluctuations need to correspond to the Larmor precession frequency of the nuclei, while relaxation via requires field fluctuations at double the Larmor frequency. To produce such field fluctuations, the tumbling rate should be the reciprocal of the molecular correlation time, i.e., f), so most efficient relaxation occurs only when voT, approaches 1. In very small, rapidly tumbling molecules, such as methanol, the concentration of the fluctuating magnetic fields spectral density) at the Larmor frequency is very low, so the relaxation processes Wj and do not occur efficiently and the nuclei of such molecules can accordingly relax very slowly. Such molecules have... 

Up to this point only overall motion of the molecule has been considered, but often there is internal motion, in addition to overall molecular tumbling, which needs to be considered to obtain a correct expression for the spectral density function. Here we apply the model-free approach to treat internal motion where the unique information is specified by a generalized order parameter S, which is a measure of the spatial restriction of internal motion, and the effective correlation time re, which is a measure of the rate of internal motion [7, 8], The model-free approach only holds if internal motion is an order of magnitude (<0.3 ns) faster than overall reorientation and can therefore be separated from overall molecular tumbling. The spectral density has the following simple expression in the model-free formalism ... 

Lee et al. evolved a comprehensive analytical-theoretical treatment, based on the solution of the reorientational isotropic diffusion equation, for an ensemble of high-spin systems under motion. These authors developed an analytical expression for the slow-tumbling motional region that relates the orientational-motion correlation time t (in s), or the corresponding tumbling rate t, with the step separation bB, of the ESR fine structure of a quartet by Eq. 8,... 

An attempt is often made to relate T] and T2 to the molecular dynamics of a system. For this purpose a relationship is sought between T1 or T2 and the correlation time tc of the nuclei under investigation. The correlation time is the time constant for exponential decay of the fluctuations in the medium that are responsible for relaxation of the magnetism of the nuclei. In general, l/xc can be thought of as a rate constant made up of the sum of all the rate constants for various independent processes that lead to relaxation. One of the most important of these (1 /t2) is for molecular tumbling. 

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... 

For a globular protein of approximately spherical shape, the isotropic tumbling rate can be characterized by the rotational diffusional correlation time, tc, as described above. Assuming that the protein fits in a sphere of radius r, then the viscosity (rf) and temperature (T) of the sample determines rc. 

The motion of the R1 nitroxide in a protein has contributions from the overall tumbling of the protein, the internal motions of the side chain, and fluctuations in the backbone structure. For membrane proteins such as rhodopsin, the correlation time for molecular tumbling is slow on the EPR time scale defined above and can be ignored. The internal motion of the R1 side chain is due to torsional oscillations about the bonds that connect the nitroxide to the backbone, and the correlation times for these motions lie in the nanosecond regime where the EPR spectra are highly sensitive to changes in rate. 

Spin-lattice relaxation is a process by which the excited spins give up energy to the surroundings (the lattice). This type of relaxation is most efficient when the molecule tumbles at a rate that is very close to the resonance frequency of the nucleus being studied. The rate of tumbling of a molecule is described by the correlation time The correlation time can be approximated by... 

The size of the dipolar interaction depends primarily on the distance and orientation between the two dipoles, not on the correlation time. By contrast, the rate of change of the dipolar interactions depends on and hence is relevant to the efficiency of relaxation. The total amount of fluctuating fields is independent of Tc, although Tc determines the upper limit of the frequencies of the fields. The three curves in Figure A5-1 must enclose the same area, but their upper limits vary. In curve (a), molecular tumbling is very rapid and the spectral density is low. In curves (b) and (c), the upper limit of frequencies decreases with the lengthening of Tc, so the spectral density increases proportionately to maintain a constant area. 

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