Isotropic rotation

Isotropic rotational diffusion with one internal degree of freedom... 

Time Resolved Fluorescence Depolarization. In Equation 3, it is assumed that the polarization decays to zero as a single exponential function, which is equivalent to assuming that the molecular shape is spherical with isotropic rotational motion. Multiexponential decays arise from anisotropic rotational motion, which might indicate a nonspherical molecule, a molecule rotating in a nonuniform environment, a fluorophore bound to tbe molecule in a manner that binders its motion, or a mixture of fluorophores with different rotational rates. 

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 ... 

Note that in the entire discussion on spin labels, above, we have assumed isotropic rotational diffusion. In several real situations of biochemical relevance this is not a tenable assumption. Perhaps the most significant one occurs when the nitroxide... 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

The same result has been obtained by Lipari and Szabo in their model-free approach.158 The first term in the above equation accounts for the effect of local order on the isotropic rotation through the factor 1 — S2, and the second term is due to the slower fluctuations in the local ordered clusters. This model-free approach has become popular among workers in the area of lyotropics and biomembranes. 

We have considered spherical molecules so far, but it should be noted that isotropic rotations can also be observed in the case of molecules with cylindrical symmetry and whose absorption and emission transition moments are parallel and oriented along the symmetry axis. In fact, any rotation around this axis has no effect on the fluorescence polarization. Only rotations perpendicular to this axis have an effect. A typical example is diphenylhexatriene whose transition moment is very close to the molecular axis (see Chapter 8). 

Equations (8.25) to (8.28) are no longer valid in the case of hindered rotations occurring in anisotropic media such as lipid bilayers and liquid crystals. In these media, the rotational motions of the probe are hindered and the emission anisotropy does not decay to zero but to a steady value rc0 (see Chapter 5). For isotropic rotations (rod-like probe), assuming a single correlation time, the emission anisotropy can be written in the following form ... 

From a practical point of view, the steady-state technique (continuous illumination) is far simpler than the time-resolved technique, but it can only be used in the case of isotropic rotations in isotropic media (Eqs 8.26 and 8.28) provided that the probe lifetime is known. Attention should be paid to the fact that the variations in steady-state anisotropy resulting from an external perturbation (e.g. temperature) may not be due only to changes in rotational rate, because this perturbation may also affect the lifetime. 

The rx term is the anisotropy at times long compared to the fluorescence lifetime, whereas in Eq. (5.9) 2 will be long. If there is no rM, then Eq. (5.8) reduces to the familiar Perrin equation for an isotropic rotator. Earlier, some confusion existed in this field since it was not recognized that an rro term was required for the case of membrane lipid bilayers. For the most part, time-resolved anisotropy measurements have a short rotational correlation time and an term. However, it has been recognized that a more adequate description may be to use two rotational correlation times, where the second may be quite long but not infinite as the rm implies/35 36 ... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

In the previous discussion, the electron-nucleus spin system was assumed to be rigidly held within a molecule isotropically rotating in solution. If the molecule cannot be treated as a rigid sphere, its motion is in general anisotropic, and three or five different reorientational correlation times have to be considered 79). Furthermore, it was calculated that free rotation of water protons about the metal ion-oxygen bond decreases the proton relaxation time in aqua ions of about 20% 79). A general treatment for considering the presence of internal motions faster than the reorientational correlation time of the whole molecule is the Lipari Szabo model free treatment 80). Relaxation is calculated as the sum of two terms 8J), of the type... 

The phase transformations of Py-D3C were also monitored by NMR spectroscopy. Distinct 29Si NMR spectra were observed for all three phases as reported in references 10 and 20. 2H NMR spectra of deuterated Py-D3C samples showed rapid isotropic rotation of the guest molecules in the ambient and high temperature phases but restricted rotation in the low temperature phase. 

Isotropic rotation correlation time measured for the V02+-substituted DTPA complex [4]. 

The main chain exhibits, at room temperature, an isotropic rotational motion with 10° rms at a frequency around 105 Hz, also showing a wide distribution of frequencies. 

Figure 15.9 Nitroxide dynamics and cw-EPR spectral line-shape. (A) Simulated X-band EPR spectra of nitroxides undergoing isotropic rotation at different rotational correlation time x. (B) The three modes of motion that contribute to nitroxide dynamics. Adopted from Sowa and Qin (2008) with permission.

It can be shown4 for the simple case of isotropic rotational diffusion that the TCF is a single exponential function of time, decaying with a time constant, rc, the molecular correlation time ... 

Fig. 28 Typical 13C NMR spectra measured in TDAE-C60 (upper row). Simulations of the 13C NMR spectra assuming fast isotropic rotations of Cg0 ion at 250 K static Cg0 ions but fast pseudorotations of the Jahn-Teller distortion axes at 50 K and the completely static case at 5 K (bottom row)...

Typical 13C NMR spectra measured in TDAE-C60 are shown on the first row of Fig. 28. At high temperatures the 13C NMR line is very narrow and has an approximately Lorentzian lineshape. This indicates fast and nearly isotropic rotations of the Qo ions effectively averaging out all anisotropic interactions. Isotropic contact interaction is of course still present and is about 40 ppm (the room-temperature shift of the 13C NMR line with respect to TMS is 188 ppm and the chemical shift is about 145 ppm), which is comparable with the shifts mea-... 

Unlike the situation with the dipolar interaction, which averages to zero in liquids due to the rapid molecular tumbling motion, the isotropic rotation leaves a residual chemical-shift Hamiltonian ... 

Figure 24.5c Temperature dependence of peak width for the two highest modes of cation rotation within tetramethylammonium dicyanamide. The data show that the onset of rotations is not sudden but occurs over a broad temperature range. As the temperature increases the number of species undergoing isotropic rotations increases until a phase treinsition occurs.

It is instinctive to first consider isotropic rotational diffusion of an atom, which is a simple but somewhat accurate description of a molecular hquid like liquid methane just below rmeiting, a Situation wherein the centers of mass of the molecules remain stationary, but the molecules rotate freely about their centers of mass. If we use the polar angle — 9, (p) to define the orientation of the vector d, the probability distribution function, G (see equation 12), which... 

It is seen that NMR properties, specific to polymeric systems, are not due to the absence of motions but to the non-isotropic character of these motions. It is worth emphasizing that the search for a characterisation of dynamical properties leads actually to the observation of non-isotropic rotations of monomeric units which result from the presence of entanglements. As3Tnmetry properties are observed instead of dynamic ones. Equation (17) is general and applies also to polyisobutylene solutions or to polyisoprene by using different values of the parameters. 

A sharp analysis of relaxation functions is obtained by using equations (11) and (12). They yield an exponential time function for < >r(t), associated with fast and non-isotropic motions whereas Mx (t) is found to depend on deviations from isotropic rotations, only. This is an illustration of the dual behaviour of the spin-spin response. 

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