Local Correlation Times

Instead of relying on chain diffusion from lengthy simulations, it is often more convenient to use shorter local time scales to map between 

In the extreme case where this subchain for high Rouse modes becomes only a single monomer, we end up with the segmental relaxation time, i.e., the reorientation dynamics on the monomer scale. This time scale can almost always be used for mapping, and it can also be used for comparison with and calibration to NMR experiments. If we use the Rouse model for mapping time scales, we should make sure that the Rouse model is a reasonable description for the system under study. 

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Global and local correlation times, generalized order parameter, S... 

The introduction of and/or labels into a protein facilitates the study of dynamic properties and, in particular, localized intramolecular motions. This arises because relaxation of these nuclei is usually dominated by dipole-dipole interactions with the directly bonded proton and this relaxation is dependent upon internuclear distance (which is fixed) and the rotational correlation time, which is only uniform throughout a rigid protein. Proteins, however, usually contain regions that have greater flexibility, such as surface loops, which have different local correlation times that are reflected in heteronuclear relaxation times. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

The most common way of dealing with non-rigid (macro) molecular systems is the model free analysis proposed by Lipari and Szabo and even earlier, in a slightly different formulation, by Wennerstrom et al. The Lipari-Szabo model assumes uncorrelated internal and global motions. The spectral densities are expressed in terms of a global and a local correlation time and a generalized order parameter, measuring the... 

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

Polycarbonate (PC) serves as a convenient example for both, the direct determination of the distribution of correlation times and the close connection of localized motions and mechanical properties. This material shows a pronounced P-relaxation in the glassy state, but the nature of the corresponding motional mechanism was not clear 76 80> before the advent of advanced NMR techniques. Meanwhile it has been shown both from 2H NMR 17) and later from 13C NMRSI) that only the phenyl groups exhibit major mobility, consisting in 180° flips augmented by substantial small angle fluctuations about the same axis, reaching an rms amplitude of 35° at 380 K, for details see Ref. 17). 

Locahzed motion can also lead to local variations in correlation times. Folded peptides with unfolded C- or N-terminal residues, for example, will have varying correlation times for the rigid and flexible parts of the molecule, resulting in different cross-relaxation rates. Such effects can usually be distinguished by the Unewidths and intensities of the corresponding diagonal signals, since the autorelaxation rates also depend on the correlation time. 

Figure 1 Schematic representation of the 13C (or 15N) spin-lattice relaxation times (7"i), spin-spin relaxation (T2), and H spin-lattice relaxation time in the rotating frame (Tlp) for the liquid-like and solid-like domains, as a function of the correlation times of local motions. 13C (or 15N) NMR signals from the solid-like domains undergoing incoherent fluctuation motions with the correlation times of 10 4-10 5 s (indicated by the grey colour) could be lost due to failure of attempted peak-narrowing due to interference of frequency with proton decoupling or magic angle spinning.

Figure 4.9 illustrates time-gated imaging of rotational correlation time. Briefly, excitation by linearly polarized radiation will excite fluorophores with dipole components parallel to the excitation polarization axis and so the fluorescence emission will be anisotropically polarized immediately after excitation, with more emission polarized parallel than perpendicular to the polarization axis (r0). Subsequently, however, collisions with solvent molecules will tend to randomize the fluorophore orientations and the emission anistropy will decrease with time (r(t)). The characteristic timescale over which the fluorescence anisotropy decreases can be described (in the simplest case of a spherical molecule) by an exponential decay with a time constant, 6, which is the rotational correlation time and is approximately proportional to the local solvent viscosity and to the size of the fluorophore. Provided that... 

We should note that the use of the Lipari-Szabo analysis implies that relaxation data are available at multiple magnetic fields. It provides a phenomenological description of the rotational motion that can be very useful for comparing systems with similar structure. Nevertheless, one should be aware of the limits of this approach and avoid direct comparison of local or global rotational correlation times for structurally very different compounds. 

To evaluate veff (f, t) at a particular time z, the adiabatic approximation is introduced. This approximation is local in time, and thus the Coulomb and exchange-correlation potentials are just those of time-independent DFT, evaluated using the density determined at time z. 

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). 

Solid state 2H NMR parameters are almost exclusively governed by the quadrupole interaction with the electric field gradient (EFG) tensor at the deuteron site.1 8 The EFG is entirely intramolecular in nature. Thus molecular order and mobility are monitored through the orientation of individual C-2H bond directions. Therefore, 2H NMR is a powerful technique for studying local molecular motions. It enables us to discriminate different types of motions and their correlation times over a wide frequency range. Dynamics of numerous polymers has been examined by solid state 2H NMR.1 3,7,9 Dynamic information on polypeptides by NMR is however limited,10 26 although the main-chain secondary structures of polypeptides in the solid have been extensively evaluated by 13C and 15N CP/MAS NMR.27,28... 

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