Diffusion anisotropy, rotational

Fluorescence polarization 1) steady state 2) time-resolved emission anisotropy rotational diffusion of the whole probe simple technique but Perrin s Law often not valid sophisticated technique but very powerful also provides order parameters... 

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. 

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.

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

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. 

An estimate of the degree of rotational anisotropy can be obtained from the spread of the experimental values of p, assuming that the minimal and maximal observed values of P (Pmin and pmax) correspond to NH vectors oriented parallel (0 = 0) and perpendicular (8 = 90°) to the rotational diffusion axis. We obtain ... 

In the particular case of prolate and oblate ellipsoids, the number of exponentials is reduced to three because two of the three axes are equivalent. The rotation diffusion coefficients around the axis of symmetry and the equatorial axis are denoted Di and D2, respectively. The emission anisotropy can then be written as... 

For isotropic motions in an isotropic medium, the values of the instantaneous and steady-state emission anisotropies are linked to the rotational diffusion coefficient Dr by the following relations (see Chapter 5) ... 

The major reasons for using intrinsic fluorescence and phosphorescence to study conformation are that these spectroscopies are extremely sensitive, they provide many specific parameters to correlate with physical structure, and they cover a wide time range, from picoseconds to seconds, which allows the study of a variety of different processes. The time scale of tyrosine fluorescence extends from picoseconds to a few nanoseconds, which is a good time window to obtain information about rotational diffusion, intermolecular association reactions, and conformational relaxation in the presence and absence of cofactors and substrates. Moreover, the time dependence of the fluorescence intensity and anisotropy decay can be used to test predictions from molecular dynamics.(167) In using tyrosine to study the dynamics of protein structure, it is particularly important that we begin to understand the basis for the anisotropy decay of tyrosine in terms of the potential motions of the phenol ring.(221) For example, the frequency of flips about the C -C bond of tyrosine appears to cover a time range from milliseconds to nanoseconds.(222)... 

There should exist a correlation between the two time-resolved functions the decay of the fluorescence intensity and the decay of the emission anisotropy. If the fluorophore undergoes intramolecular rotation with some potential energy and the quenching of its emission has an angular dependence, then the intensity decay function is predicted to be strongly dependent on the rotational diffusion coefficient of the fluorophore.(112) It is expected to be single-exponential only in the case when the internal rotation is fast as compared with an averaged decay rate. As the internal rotation becomes slower, the intensity decay function should exhibit nonexponential behavior. 

The long lifetime of phosphorescence allows it to be used for processes which are slow—on the millisecond to microsecond time scale. Among these processes are the turnover time of enzymes and diffusion of large aggregates or smaller proteins in a restricted environment, such as, for example, proteins in membranes. Phosphorescence anisotropy is one method to study these processes, giving information on rotational diffusion. Quenching by external molecules is another potentially powerful method in this case it can lead to information on tryptophan location and the structural dynamics of the protein. 

Fig. 2c. It can be seen that at 530 nm, the fluorescence decays mono-exponentially with the fluorescence lifetime of 3.24 ns. The rise of the emission seen below 50 ps in the corresponding FlUp data is obviously not resolved here. In contrast, the TCSP data at 450 nm is described by a triple-exponential decay whose dominant component has a correlation time well below the time resolution. This component is obviously equivalent to the fluorescence decay observed in the FlUp experiment. A minor contribution has a correlation time of about 3.2 ns and reflects again the fluorescence lifetime that was also detected at 530 nm. The most characteristic component at 450 nm however has a time constant of about 300 ps. It is important to emphasize that this 300 ps decay does not have a rising counterpart when emission near the maximum of the stationary fluorescence spectrum is recorded. In other words, the above mentioned mirror image correspondence of the fluorescence dynamics between 450 nm and 530 nm holds only on time scales shorter than 20 ps. Finally, in contrast to picosecond time scales, the anisotropy deduced from the TCSPC data displays a pronounced decay. This decay is reminiscent of the rotational diffusion of the entire protein indicating that the optical chromophore is rigidly embedded in the core of the 6-barrel protein.

Reticulum ATPase [105,106], Owing to the long-lived nature of the triplet state, Eosin derivatives are suitable to study protein dynamics in the microsecond-millisecond range. Rotational correlation times are obtained by monitoring the time-dependent anisotropy of the probe s phosphorescence [107-112] and/or the recovery of the ground state absorption [113— 118] or fluorescence [119-122], The decay of the anisotropy allows determination of the mobility of the protein chain that cover the binding site and the rotational diffusion of the protein, the latter being a function of the size and shape of the protein, the viscosity of the medium, and the temperature. 

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]. 

From an experimental point of view, if we want to establish whether a loss of anisotropy is due to rotational diffusion or to energy transfer, we must probe very short times when molecular motions are inhibited. This is precisely what we did by observing fluorescence anisotropy on the sub-picosecond time-... 

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. 

In order to illustrate the anisotropy of the rotational diffusion, and hence the influence of the intermolecular interactions, the square roots of the moments of inertia (IJIxxr m = 0.92, (IJIxxr1/2 = 0.95, and (IJI Y1/2 = 1.00 are compared with the quotients of the diffusion constants in Table VI. The values of the... 

First, the diffusional radical reactions in solutions whose rates are proportional to I) sometimes have rate constants that are much smaller than their contact estimate for the isotropic black sphere, ki> = 4naD. It was proved that this is the result of chemical anisotropy of the reactants. Partially averaged by translational and rotational diffusion of reactants, this anisotropy manifests itself via the encounter efficiency w < 1, which enters the rate constant kn = w4nak) [249]. Even the model of white spheres with black spots is more appropriate for such reactions than the conventional Smoluchowski model. 

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