Wobbling-in-a-cone

In the first model,55 depicted schematically in Fig. 14, internal motion is described as wobbling in a cone, so that the C—H vector moves freely at a given rate, tw, inside the conical boundary defined by an angle 6 but has zero probability of being found outside the boundary. The director of the cone makes the angle /3 with the z-axis, while r, and rx — Ty are the correlation times for rotation about the z- and... 

The experimental multifield relaxation data (T T2, n.O.e.) for amylose were nicely reproduced by using these two models.11 However, the wobbling-in-a-cone model is favored for two reasons first, it requires fewer adjustable parameters than the bistable jump model to fit the data second, the jump model requires /32 = 48.9°... 

Molecules in general are not rigid bodies and NMR relaxation is an important source of information on internal motions. To obtain this information, one has to assume a motional model of some kind. One such model, proposed early by Wang and Pecora" is called the wobbling-in-a-cone or diffusion-in-a-cone model. Recently, Sitnitsky" discussed the analytic treatment of this model. 

Now consider rotational diffusion of a rigid symmetric top in an oriented environment such as a lipid bilayer or other liquid crystal. The top (e.g., cylinder) could represent one of the lipids, a probe, or a protein. It may be useful to think of the vector wobbling in a cone , as sketched in Figure 5. The correlation function clearly will decay to a nonzero plateau value, because angular averaging is incomplete. One may also anticipate that the relaxation will be more rapid than free diffusion due to effects of the ordering potential. 

Fig. 4. Restricted diffusion models for internal motions, (a) Bond rotation limited in angular amplitude to 7o. i is the angle between the rotational axis and the intemuclear vector, just as in the free internal diffusion model, (b) With the wobble-in-a-cone model, the diffiision is limited to the conic angle and the cone is tilted at angle from some reference axis.

The second type of restricted difiiision is illustrated in Fig. 4b and has sometimes been dubbed the wobble-in-a-cone model because the orientation of the interaction vector (either DD vector or CSA tensor principal axis) is permitted free diffusion within the cone of half angle yo fhe cone is disposed toward a reference axis at a fixed angle fi. This model can be adapted for internal motions but is especially useful for characterizing segmental motions, in which case the reference axis can be the helix axis of the nucleic acid. Librational motions, such as predicted by molecular dynamics calculations on proteins (McCammon et al., 1977), are readily accommodated by the wobble-in-a-cone model (Howarth, 1979 Richarz et al, 1980), but the model has also accommodated other studies of dynamics, for example, the situation of halide ions bound to proteins (Bull etai, 1978). 

Lipari and Szabo (1981b) compared the 6t of experimental relaxation data from the literature with various models having no internal motion, twisting motion, wobbling-in-a-cone, and a two-state jump. It was assumed that only dipolar contributions from the two H-S and the H-3 protons contribute to the P relaxation. It was concluded that none of the models was substantially superior to the others. For each model, however, the experimental data could be ht only if a large amplitude motion in the nanosecond time range existed. 

The observation that r(t) decays to the finite value rro soon led to the recognition that the fluorophore DPH has an orientational motion which is restricted due to the surrounding lipid chains/35-37 From this the wobbling-in-cone model(37 43 44) was developed. In this model, DPH was assumed to wobble within a cone of half angle 6C (which relates to the degree of orientational constraint, order, or rm) with a wobbling diffusion constant D (which relates to the rate of motion) with... 

The advances in time resolved techniques have fostered a reexamination of theories of the rotational motions of molecules in liquids. Models considered include the anisotropic motion of unsymmetrical fluorophores the internal motions of probes relative to the overall movement with respect to their surroundings, the restricted motion of molecules within membranes (e.g., wobbling within a cone), and the segmental motion of synthetic macromolecules [8]. Analyses of these models point to experimental situations in which the anisotropy can show both multi-exponential and none-exponential decay. Current experimental techniques are capable in principle of distinguishing between these different models. It should be emphasized, however, that to extract a single average rotational correlation time demands the same precision of data and analysis as fluorescence decay experiments which exhibit dual exponential decays. Multiple or non-exponential anisotropy experiments are thus near the limits of present capabilities, and generally demand favourable combinations of fluorescence and rotational diffusion times [48]. 

In the membrane lipid alkyl chains of n-SASL and n-PC spin labels undergo rapid rotational motion about the long axis of the spin label and wobble within the confines of a cone imposed by the... 

It is interesting to note that rc(l — r.y,/r0) is exactly the area A under [r(t) — r, /ro. Therefore, even if the anisotropy decay is not a single exponential, Dw can be determined by means of Eq. (5.50) in which tc(1 — roo/ro) is replaced by the measured area A. An example of application of the wobble-in-cone model to the study of vesicles and membranes is given in Chapter 8 (Box 8.3). More general theories have also been developed (see Box 5.4). 

Fig. B8.3.1. Fluorescence anisotropy decays at 4 °C of PPL A in phospholipid vesicles (PC PI, 95 5 mol %). B in Torpedo membranes. From the best fit of the /(t) and l (t) components, and by using the wobble-in-cone model, the...

Fig. 5.132 Anisotropy decay of a Dil molecule in a PMMA matrix. The molecule has a limited rotational degree of freedom. A fit delivers a final anisotropy of 0.683+0.003, a cone of wobbling of 12.4 0.3°, and a rotational correlation time of 2.7T0.4 ns [419]...

A fluorophore that absorbs incoming electromagnetic radiation will undergo rotational diffusion. This rotational diffusion is best described by a wobbling rotation model (Fig. 3) [6]. The fluorophore will rotate and trace a cone that can be described by the angle between the absorption dipole and the axis of rotation (9 ) and the angle between the emission dipole and the axis of rotation (9e) [3]. If the rate of rotational motion of a fluorophore in a matrix or fluid is... 

The vortex of a cyclone will precess (or wobble) about the center axis of the cyclone. This motion can bring the vortex into close proximity to the wall of the cone of the cyclone and pluck off and reentrain the collected solids flowing down along the wall of the cone. The vortex may also cause erosion of the cone if it touches the cone wall. Sometimes an inverted cone or a similar device is added to the bottom of the cyclone in the vicinity of the cone and dipleg to stabilize and fix the vortex. If it is placed correctly, the vortex will attach to the cone and the vortex movement will be stabilized, thus minimizing the efficiency loss due to plucking the solids off the wall and erosion of the cyclone cone. 

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