Tensor rotational diffusion

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. 

Key Words Carbon-13 spin relaxation, T, Measurements, Nuclear Overhauser effect, Rotation-diffusion tensor, HOESY experiments... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

It can be noticed that at least two independent relaxation parameters in the symmetric top case, and three in the case of fully anisotropic diffusion rotation are necessary for deriving the rotation-diffusion coefficients, provided that the relevant structural parameters are known and that the orientation of the rotational diffusion tensor has been deduced from symmetry considerations or from the inertial tensor. 

Figure 15 The model molecule used to demonstrate the possibilities of HOESY experiments in terms of carbon-proton distances and reorientational anisotropy. To a first approximation, the molecule is devoid of internal motions and its symmetry determines the principal axis of the rotation-diffusion tensor. Note that H, H,., H,-, H,/ are non-equivalent. The arrows indicate remote correlations.

Another rotational diffusion model known as the anisotropic viscosity model156,157 is very similar to the above model, and its main feature is to diagonalize the rotational diffusion tensor in the L frame defined by the director. A similar (but not the same) expression as Eq. (71) is J R(r)co)... 

How Can We Derive the Rotational Diffusion Tensor of a Molecule from Spin-Relaxation Data 293 ... 

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. 

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 most general case of a completely anisotropic diffusion tensor, six parameters have to be determined for the rotational diffusion tensor three principal values and three Euler angles. This determination requires an optimization search in a six-dimensional space, which could be a significantly more CPU-demanding procedure than that for an axially symmetric tensor. Possible efficient approaches to this problem suggested recently include a simulated annealing procedure [54] and a two-step procedure [55]. 

Fig. 12.4 The results of the determination of the rotational diffusion tensor of the /7ARK PH domain A fit of the orientational dependence of the experimental values ofp for the /iARK PH domain and B a ribbon representation of the 3D structure of the protein, with the orientation of the diffusion axis indicated by a rod. Shown in A are the experimental (symbols) and the best fit (line) values ofp (Eq.

In addition, it is worth mentioning that, even in the case of a uniform distribution, the number of vectors oriented parallel to any given direction is small, so that pmax could be underestimated. Here we assumed that the rotational diffusion tensor is axially symmetric. The presence of a rhombic component could be identified by the shape of the distribution of the values of p (see e.g. Refs. [55, 57]). 

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. 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

The connection between anisotropic molecular motion and nuclear relaxation was derived by Woessner as early as 1962 [161]. Accordingly, the dipole-dipole relaxation time of a carbon nucleus is a function of the diagonal components R, R2, and R3 of the rotational diffusion tensor and the cosines X, p, and v of the angles assumed by the C —H bonds relative to the principal axes of this tensor ... 

If the position of the principal axes of the rotational diffusion tensor were known with respect to the molecular coordinates, then the motion of the molecule could be calculated from the measured relaxation times. With simple molecules, however, it is possible to interpret the Tt values qualitatively in terms of an anisotropic motion. 

The first quantitative estimate of the rotational diffusion tensor for simple molecules was accomplished by Grant et al. [163], By solving the Woessner equations, they were able to show e.g. for trans-decalin that the molecule rotates preferentially like a propeller, i.e. about the axis perpendicular to the plane of the molecule. The values given as a measure of the rotational frequencies do not correlate with the moments of inertia, but instead with the ellipticities of the molecule as defined [163]. They are accessible from the ratios of the interatomic distances perpendicular to the axes of rotation, and can be adopted as a measure of the number of solvent molecules that have to be displaced on rotation about each of the three axes. 

For less symmetric molecules one has to resort to computer programs [164] to solve the Woessner equations. The orientation of the rotational diffusion tensor is usually defined by assuming that its principal axes coincide with those of the moment of inertia tensor. This assumption is probably a good approximation for molecules of low polarity containing no heavy atoms, since under these conditions the moment of inertia tensor roughly represents the shape of the molecule. 

A rather sophisticated application of Woessner s theory has been accomplished for all-frtws-retinal and its isomers [165], After determination of the components of the rotational diffusion tensor in retinal for various dihedral angles between the olefinic chain... 

A more quantitative interpretation of methyl relaxation requires a knowledge of the motional anisotropy of the entire molecule. Thus the activation energy of methyl rotation can be estimated from 7] data if the rotational diffusion tensor of the molecule, mentioned in Section 3.3.3.3, is known [164]. 

This study is the first where semiquantitative use of relaxation data was made for conformational questions. A similar computer program was written and applied to the Tl data of several small peptides and cyclic amino acids (Somorjai and Deslauriers, 1976). The results, however, are questionable since in all these calculations it is generally assumed that the principal axis of the rotation diffusion tensor coincides with the principal axis of the moment of inertia tensor. Only very restricted types of molecules can be expected to obey this assumption. There should be no large dipole moments nor large or polar substituents present. Furthermore, the molecule should have a rather rigid backbone, and only relaxation times of backbone carbon atoms can be used in this type of calculation. 

In Figure 2.10 we show a selection of results, in which experimental and calculated spectra are compared at 292 and 155K. The results are quite satisfactory, especially when considering that no fitted parameters, but only calculated quantities (via QM and hydrodynamic models) have been employed. The overall satisfactory agreement of the spectral line shapes, particularly at low temperatures, is a convincing proof that the simplified dynamic modelling implemented in the SLE through the purely rotational stochastic diffusive operator f, and the hydrodynamic calculation of the rotational diffusion tensor, is sufficient to describe the main slow relaxation processes. 

When the overall motion is not isotropic, the diagonal elements of the rotational diffusion tensor are no longer equivalent and rotation about the three principal axes of the diffusion tensor may be described by different diffusion coefficients or correlation times. For anisotropic motion, the correlation time in Eqs. 16 and 25 is an effective correlation time, r ff, containing contributions from the various modes of reorientation. Partitioning of the various components of rff can be achieved through appropriate dynamic models. The simplest case of anisotropic motion is that for a symmetric-top molecule. The r ff of a rigid ellipsoid is expressed in terms of two parameters, Dn and DL these two parameters respectively describe the rotational diffusion about the C3 symmetry axis (major axis) and the two perpendicular axes (minor axes), which are assumed to be equivalent25-44 (Fig. 4) ... 

For asymmetric-top molecules, all three principal values of the rotational diffusion tensor are required to describe the molecular dynamics hence at least three different T, values of geometrically nonequivalent carbons are required to solve the three independent simultaneous equations derived by Woessner45 ... 

A more rigorous treatment of the variable-temperature, 3C Tl data has been used to describe the overall molecular motion of the glucose derivative, namely 1,6-an-hydro-/3-D-glucopyranose (31) in solution.147 This rigid molecule contains a number of nonequivalent 13C- H vectors and is amenable to a rigorous quantitative treatment by means of Eq. 29. Table VI summarizes the diagonal elements of the rotational diffusion tensor, that is, the rotational diffusion constants Dx,Dy, and Dz obtained upon diagonalization of the rotational diffusion tensor with respect to... 

The calculated Euler angles (a = 50°, /3 = 60°, and y = 40°), which determine the relative orientation between the principal-axis system of the rotational diffusion tensor and that of the moment of inertia tensor, indicate a significant shift between the two tensors. This result is expected because of the fact that molecule 31 contains a number of polar groups and hydrogen-bonding centers, leading to strong intermolecular interactions. 

For the set of the Euler angles a = 50°, /3 = 60°, y = 40°, the order of the rotational diffusion constants is Dy> Dz> Dx. This trend is also reflected in the quotients of the rotational diffusion constants, DJDX and DJDX (Table VI), which describe the anisotropy of the rotational diffusion in solution. The principal axes of the rotational diffusion tensor corresponding to the aforementioned set of the Euler angles is shown graphically in Fig. 12. 

FIG. 12.—Orientation of the principal-axis system of inertia tensor (x. y. z1) and that of the rotational diffusion tensor (x,y,z) for compound 31. The principal diffusion axis x is perpendicular to the plane of the drawing. [Reproduced with permission from P. Dais, Carbohydrate Res., 263 (1994) 13-24, and Elsevier Science B.V.]... 

In all of the above descriptions and equations, it was assumed that molecular tumbling is isotropic in the solution. This is an idealized case that can be approximated by real proteins only in fortunate instances. In general, proteins exhibit totally anisotropic tumbling7 but in special cases, the simplification of only an axially symmetric motion is used, for which two of the molecular axes are assumed to be identical and thus the corresponding rotational diffusion tensor can be simplified. 

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