Overall rotation

The total rotations of the molecule (as distinct from internal rotation) are sometimes described as the adiabatic rotations because formation of the transition state occurs without change of angular momentum. 

Formation of the complex is, however, associated with changes of the moments of inertia of the molecule. Consequently, part of the energy of total rotation is exchanged with that of vibration and internal rotation. In the case where the molecule is stretched in the complex, the overall rotational energy is decreased and hence +, IF( +) and the derived k( ) are enhanced. This is known as the centrifugal effect. The equation for the energy balance now has to be modified to 

A full discussion of this effect is beyond the scope of this review. Following Marcus [5], we consider only the mean energy exchange accompanying dissociation of a symmetric top, such as CH3I, along the top axis. 

In such an example, the change of moment of inertia about the symmetry axis is usually small. Let 7 be the moment of inertia of the molecule about the degenerate axis perpendicular to the top axis, and let 7+ be the corresponding moment of inertia of the complex. 

In a state with angular momentum, l, perpendicular to the top axis, the change of rotational energy accompanying formation of the transition state approximates to 

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It must be pointed out that another type of internal motion is the overall rotation of the molecule. The vibration and rotation of the molecule are shown schematically in figure Al.2.2. 

The catalytic subunit then catalyzes the direct transfer of the 7-phosphate of ATP (visible as small beads at the end of ATP) to its peptide substrate. Catalysis takes place in the cleft between the two domains. Mutual orientation and position of these two lobes can be classified as either closed or open, for a review of the structures and function see e.g. [36]. The presented structure shows a closed conformation. Both the apoenzyme and the binary complex of the porcine C-subunit with di-iodinated inhibitor peptide represent the crystal structure in an open conformation [37] resulting from an overall rotation of the small lobe relative to the large lobe. 

Once the electronic Schrodinger equation has been solved for a large number of nuclear geometries (and possibly also for several electronic states), the PES is known. This can then be used for solving the nuclear part of the Schrodinger equation. If there are N nuclei, there are 3N coordinates that define the geometry. Of these coordinates, three describe the overall translation of the molecule, and three describe the overall rotation of the molecule with respect to three axes. Eor a linear molecule, only two coordinates are necessary for describing the rotation. This leaves 3N-6(5) coordinates to describe the internal movement of the nuclei, the vibrations, often chosen to be... 

Another difficulty with the infrared method is that of determining the band center with sufficient accuracy in the presence of the fine structure or band envelopes due to the overall rotation. Even when high resolution equipment is used so that the separate rotation lines are resolved, it is by no means always a simple problem to identify these lines with certainty so that the band center can be unambiguously determined. The final difficulty is one common to almost all methods and that is the effect of the shape of the potential barrier. The infrared method has the advantage that it is applicable to many molecules for which some of the other methods are not suitable. However, in some of these cases especially, barrier shapes are likely to be more complicated than the simple cosine form usually assumed, and, when this complication occurs, there is a corresponding uncertainty in the height of the potential barrier as determined from the infrared torsional frequencies. In especially favorable cases, it may be possible to observe so-called hot bands i.e., v = 1 to v = 2, 2 to 3, etc. This would add information about the shape of the barrier. 

Another early example is nitromethane which is special in two respects. First, the barrier has sixfold symmetry because of the threefold character of the methyl group and the twofold character of the nitro group. Secondly, the barrier turns out to be extremely low, only about 5 small calories. For such a low barrier, it is convenient to treat the coupling between free internal rotation and overall rotation exactly and consider the barrier as a small perturbation. 

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. 

Starburst (TM) dendrimers with DTPA can contain 170 bound Gd(III) ions and have relaxivities (per bound Gd) up to 6 times that of Gd-DTPA (308). Both global and local motion contribute to the overall rotational correlation time. Attempts have been made to increase the re-laxivity of Gd(III) by optimizing the rotational correlation time via binding of Gd(III) to derivatized polysaccharides (309) and by binding lipophilic complexes to albumin in serum (310). The latter approach has achieved relaxivities as high as 44.2 mM l s1 for derivatized 72 (311). 

The overall rotation angle for a particular offset 8 can also be obtained from the rotation matrix defined in Eqs. (38) and (39)... 

In a similar way, the overall rotation axis as a function of the offset can be derived from the rotation matrix... 

Figure 3. Librational OH modes in hydrogen bonded alcohol clusters may be correlated with overall rotation (bottom left) and torsion (top left) of the monomer (illustrated for methanol), but methyl rotation is actually decoupled from OH torsion by hydrogen bonding. Note that the wavenumbers of monomer rotation (fa 4 cm-1) and torsion (fa 280 cm-1) are much lower than that of the cluster libration (fa 600cm ) [93].

In the absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

The overall tumbling of a protein molecule in solution is the dominant source of NH-bond reorientations with respect to the laboratory frame, and hence is the major contribution to 15N relaxation. Adequate treatment of this motion and its separation from the local motion is therefore critical for accurate analysis of protein dynamics in solution [46]. This task is not trivial because (i) the overall and internal dynamics could be coupled (e. g. in the presence of significant segmental motion), and (ii) the anisotropy of the overall rotational diffusion, reflecting the shape of the molecule, which in general case deviates from a perfect sphere, significantly complicates the analysis. Here we assume that the overall and local motions are independent of each other, and thus we will focus on the effect of the rotational overall anisotropy. 

The anisotropy of the overall tumbling will result in the dependence of spin-relaxation properties of a given 15N nucleus on the orientation of the NH-bond in the molecule. This orientational dependence is caused by differences in the apparent tumbling rates sensed by various internuclear vectors in an anisotropically tumbling molecule. Assume we have a molecule with the principal components of the overall rotational diffusion tensor Dx, Dy, and l)z (x, y, and z denote the principal axes of the diffusion tensor), and let Dx< Dy< Dz. 

Since the characterization of the overall rotational diffusion is a prerequisite for a proper analysis of protein dynamics from spin-relaxation data, we first focus on the theoretical basis of the method being used. 

Several approaches to determination of the overall rotational diffusion tensor from 15N relaxation data were suggested in the literature [15, 47, 49, 51-53]. The approach described here uses the orientational dependence of the ratio of spin-relaxation rates [49]... 

In the absence of accurate structural information, the analysis based on anisotropic diffusion as discussed above cannot be applied. The use of the isotropic overall model is still possible (see below) because it does not require any structural knowledge. However, the isotropic model has to be validated, i.e. the degree of the overall rotational anisotropy has to be determined prior to such an analysis. 

The isotropic model is justified when the estimated degree of the overall rotational anisotropy is small. A D /D l ratio of less than 1.1-1.2 could probably be considered as a reasonable value for the isotropic model, although an anisotropy as small as 1.17 can be reliably determined from 15N relaxation measurements, as demonstrated in Ref. [15]. 

In the isotropic model, the overall rotational diffusion is characterized by a single parameter, the overall correlation time zc. The following steps could be used to determine zc. 

Here we describe the model selection algorithm that is used to derive microdynamic (model-free) parameters for each NH group from 15N relaxation data. It is implemented in our program DYNAMICS [9]. Given the overall rotational diffusion tensor parameters (isotropic or anisotropic) derived as described above, this analysis is performed independently for each NH-group in order to characterize its local mobility. 

Additional limitations in the accuracy of the derived dynamic parameters could be related to the limitations in the analytical approaches. For example, neglect of the overall rotational anisotropy could lead to considerable errors in the model-free parameters, as illustrated earlier [46]. As also shown in Ref. [6], the model-free parameters could be in error if the site-specific variations in 15N CSA are not properly taken into account, particularly at higher fields (>600 MHz 111 frequency). 

Chemists pay much less attention to the NMR relaxation rates than to the coupling constants and chemical shifts. From the point of view of the NMR spectroscopist, however, the relaxation characteristics are far more basic, and may mean the difference between the observation or not of a signal. For the quadrupolar nucleides such as 14N the relaxation characteristics are dominated by the quadrupole relaxation. This is shown by the absence of any nuclear Overhauser effect for the 14N ammonium ion despite its high symmetry, which ensures that the quadrupole relaxation is minimized. Relaxation properties are governed by motional characteristics normally represented by a correlation time, or several translational, overall rotational and internal rotational, and thus are very different for solids, liquids and solutions. 

When it comes to polyatomic molecules, there are two problems that complicate the issue, as already discussed in Note 1 of Chapter 3. One is the separation of the overall rotation of the molecule (Jellinek and Li, 1989). The other is that, depending on the choice of internal coordinates, certain coupling terms can be assigned to be kinetic or potential terms. A simple and familiar case is a linear triatomic, when one uses bond coordinates versus Jacobi coordinates. The case for Fermi coupling for a bending motion is discussed in Sibert, Hynes, and Reinhardt (1983). 

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