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.

In Eq. (4-62) Wq is the Larmor precessional frequency, and Tc is the correlation time, a measure of the rate of molecular motion. The reciprocal of the correlation time is a frequency, and 1/Tc may receive additive contributions from several sources, in particular I/t, where t, is the rotational correlation time, t, is, approximately, the time taken for the molecule to rotate through one radian. Only a rigid molecule is characterized by a single correlation time, and the value of Tc for different atoms or groups in a complex molecule may provide interesting chemical information. 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

St. Pierre A. G., Steele W. A. Collisional effects upon rotational correlations of symmetrical top molecules. J. Chem. Phys. 57, 4638-48 (1972). 

Dill J. F., Litovitz T. A., Bucaro J. A. Molecular reorientation in liquids by Rayleigh scattering pressure dependence of rotational correlation functions, J. Chem. Phys. 62, 3839-50 (1975). 

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... 

PGSE-NMR provides direct information on the translational mobility of a liquid medium capable of swelling a given CFP. The self-diffusion coefficient of the swelling agent is found to be related to the nanoporosity of the matrix as determined from ISEC and to the rotational correlation time of a suitable paramagnetic probe (ESR) [22]. 

Table 5, Average polymer chain concentration (ape), polymer swellability (S), rotational correlation times of TEMPONE (r) and self-diffusion coefficient of methanol (Zf) in the swollen 2,2% Pd catalysts.

X-ray microprobe Diffusion coefficient Self-diffusion coefficient of a liquid Rotational correlation time... 

The nitroxide moiety of 16-SASL and 16-PC exhibits such a great deal of motion that the rotational correlation time can be calculated (Berliner 1978). The rotational correlation time (assuming isotropic rotational diffusion of the nitroxide fragment) can be calculated from the linear term of the line width parameter ... 

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.)... 

Berliner, L. J. 1978. Spin labeling in enzymology Spin-labeled enzymes and proteins. Rotational correlation times calculation. Methods Enzymol. 49 466 170. 

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. 

Anisotropy describes the rotational dynamics of reporter molecules or of any sensor segments to which the reporter is rigidly fixed. In the simplest case when both the rotation and the fluorescence decay can be represented by single-exponential functions, the range of variation of anisotropy (r) is determined by variation of the ratio of fluorescence lifetime (xF) and rotational correlation time ([Pg.9]

Rotational correlation time is directly coupled to rotational diffusion coefficient ... 

If the g- and hyperfine anisotropies are known from analysis of a solid-state spectrum, the line-width parameters (1, and yt can be used to compute the rotational correlation time, tr, a useful measure of freedom of motion. Line widths in ESR spectra of nitroxide spin labels, for example, have been used to probe the motional freedom of biological macromolecules.14 Since tr is related to the molecular hydrodynamic volume, Ft, and the solution viscosity, r, by a relationship introduced by Debye 15... 

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...

---