Overall tumbling

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. 

Fig. 12.1 Illustration of the temperature sensitivity of 15N relaxation parameters, Rlf R2t and NOE, as indicated. Shown are the relative deviations in these relaxation parameters from their values at 25 °C as a function of temperature in the range of + 3 °C. The expected variations in / ] and R2 due to temperature deviations of as little as +1 °C are already greater than the typical level of experimental precision ( % ) of these measurements (indicated by the dashed horizontal lines). For simplicity, only temperature variation of the overall tumbling time of the molecule (due to temperature dependence of the viscosity of water) is taken into account the effect of temperature variations on local dynamics is not considered here.

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. 

D11/Dj, from 1 to 10. Symbols correspond to synthetic experimental data generated assuming overall tumbling with rc = 5 ns and various degrees of anisotropy as indicated. Model-free parameters typical of restricted local backbone dynamics in protein core, S2=0.87, T oc =20 ps, were used to describe the effect of local motions. The H resonance frequency was set to 600 MHz. The solid lines correspond to the right-hand-side expression in Eq. (10). 

Since local motions of flexible protein domains can randomize the emission dipoles in a manner that does not reflect the overall tumbling time of the entire protein, and since there is an apparent lack of fluorophores with the appropriate properties for larger analytes, most FPIAs are useful only for determination of relatively low-molecular-weight analytes. Nevertheless, a few studies of larger-molecular-weight analytes have been reported. A competitive assay for human... 

The results from fitting the anisotropy decay support the above conclusions. Wells and Lakowicz(200) resolved two exponential components in the anisotropy decay. They obtained ro = 0.11, t r = 0.3 ns, rj = 0.15, and t = 18.5 ns for the sample with no added Mg2+, and ro = 0.05, t R = 0.4 ns, r J = 0.17, and t"R = 17.4 ns for a sample with 10 mM Mg2+. Here r 0 and r o are the amplitudes of the fast and slow components. The longer rotational relaxation time corresponds to overall tumbling of the tRNA, although its amplitude is reduced by much more rapid local motions. The shorter relaxation time corresponds directly to a rapid local motion. Upon addition of Mg2+, the relative amplitude of die rapid local motion decreases, while that of the overall tumbling increases. This implies that the wyebutine base is held in a more rigid or constrained state, such as a 3 stack, in the presence of Mg2+. In that state, the amplitude of local angular motion is substantially diminished in comparison with that in the alternate state that prevails in the absence of Mg2+. As noted before, the exact nature of these conformation(s) is unresolved. 

Clarkson et al. investigated molecular dynamics of vanadyl-EDTA and DTPA complexes in sucrose solution or attached to PAMAM dendrimers by EPR [74,75]. The motion-sensitive EPR data of the dendrimeric system have been fitted to an anisotropic model which is described by an overall spherical rotation combined with a rotation around the axis of the arm branching out of the central core. The motions around the axis of the branch connecting the chelate to the central core were found to be very rapid, whereas the overall tumbling was slow. 

The value of the general order parameter S2 is between 0 and 1. Value 0 corresponds to totally unrestricted motion, while value 1 to fully restricted motion. As these parameters are commonly interpreted within the internal molecular reference frame,c the value of 1 means that the bond is constrained to a fixed orientation and all of its motions correspond to the overall tumbling of the molecule. 

Study of line shapes in solids often provides valuable information on molecular motion—gross phase changes, overall tumbling of molecules, or internal rotations and other motions. For a limited number of spins the dipolar, CSA, or quadrupo-lar interactions may be simulated and compared with experiment, whereas for multispin systems the line shape is often rather featureless and only the overall shape can be characterized. 

The motion of the R1 nitroxide in a protein has contributions from the overall tumbling of the protein, the internal motions of the side chain, and fluctuations in the backbone structure. For membrane proteins such as rhodopsin, the correlation time for molecular tumbling is slow on the EPR time scale defined above and can be ignored. The internal motion of the R1 side chain is due to torsional oscillations about the bonds that connect the nitroxide to the backbone, and the correlation times for these motions lie in the nanosecond regime where the EPR spectra are highly sensitive to changes in rate. 

Heteronuclear NOEs (e.g., H - N or H - C hetNOEs) are obtained by measuring HSQC-type spectra (see the section entitled Two-dimensional heteronuclear correlation experiments ) with and without proton saturation. The hetNOE is extracted from the difference in the signal amplitude of these measurements and reports on the fast dynamics of the heteronuclear bonds (ps to ns timescale). Maximal hetNOE values are observed when the bond vector tumbles at the same frequency as the entire protein, whereas faster motion with respect to overall tumbling leads to smaller hetNOEs. 

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