Torsional correlation times

Figure 18 Temperature dependence of the C-H vector (selected, filled symbols) and torsional correlation (open symbols) times for PB from simulation. Also shown is the mean waiting time between transitions for the cis-allyl, trans-allyl, and (3 torsions in PB. The solid lines are VF fits, whereas the dashed lines assume an Arrhenius temperature dependence.

Lakowicz et al.(]7] VB) examined the intensity and anisotropy decays of the tyrosine fluorescence of oxytocin at pH 7 and 25 °C. They found that the fluorescence decay was best fit by a triple exponential having time constants of 80, 359, and 927 ps with respective amplitudes of 0.29, 0.27, and 0.43. It is difficult to compare these results with those of Ross et al,(68) because of the differences in pH (3 vs. 7) and temperature (5° vs. 25 °C). For example, whereas at pH 3 the amino terminus of oxytocin is fully protonated, at pH 7 it is partially ionized, and since the tyrosine is adjacent to the amino terminal residue, the state of ionization could affect the tyrosine emission. The anisotropy decay at 25 °C was well fit by a double exponential with rotational correlation times of 454 and 29 ps. Following the assumptions described previously for the anisotropy decay of enkephalin, the longer correlation time was ascribed to the overall rotational motion of oxytocin, and the shorter correlation time was ascribed to torsional motion of the tyrosine side chain. 

There remains an interpretation of ta to be found, ta exhibits an activation energy of about 0.43 0.1 eV, about three times as high as the C-C torsional barrier of 0.13 eV. The discrepancy must reflect the influence of the interactions with the environment and therefore ta appears to correspond to relaxation times most likely involving several correlated jumps. The experimental activation energy is in the range of that for the NMR correlation time associated with correlated conformational jumps in bulk PIB [136] (0.46 eV) and one could tentatively relate ta to the mechanism underlying this process (see later). 

At present the body of data on reactions in clusters is insufficient to test the above two microcanonical approaches. For electron transfers in solution it seems clear that the vibrational assistance approach, stemming from Eq. (1.2), with its extensions mentioned earlier, is the one that has been the most successful [27-30]. For slow isomerizations Sumi and Asano have pointed out that an analysis based on Eq. (1.2) was again needed [40]. An approach based on Eq. (1.1) or on its extension to include a frequency-dependent friction, they noted, led to unphysical correlation times [40]. In investigations of fast isomerizations the most commonly studied system has been the photoex-cited trans-stilbene [5, 41-43,46]. Difficulties encountered by a one-coordinate treatment for that system have been reported [4, 8]. Indeed, coherence results for photoexcited cw-stilbene have shown a coupling of a phenyl torsional mode to the torsional mode about the C=C bond [42, 47]. 

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. 

In this expression r(/) is the time-dependent anisotropy, 0 the correlation times and g, the fraction of the total anisotropy (r ) which decays with this correlation time. In general we expect one component (tf,) due to rotational diffusion of the protein, and one due to torsional motions of the tryptophan residue, if such motions are significant. In proteins which contain more than a single fluorescent residue there can be energy transfer among the residues, which can appear as a component in the anisotropy decay. The timescale of energy transfer depends upon the distance and orientation between the residues, but there is little information on the timescale of energy transfer between intrinsic fluorophores in proteins. 

For both proteins a multi-exponential anisotropy decay was necessary to explain the data, and in both cases a short correlation time (< 1 nsec) was indicated by the data. In the case of nuclease only 12% of the anisotropy decays by this rapid process, indicating that the torsional motions have a limited ampUtude. In contrast, 75% of the melittin anisotropy decays by the rapid process, which indicates considerable free motion of the tryptophan residue. 

Isotropic correlation times and spin lattice relaxation times measured by and H-NMR for polyoxyethylene (POE) solutions in a variety of solvents have been computed using the DRIS formalism for isolated polymer chains [8]. For this purpose, the conformational kinetics of POE has been analyzed and kinetic schemes of rotameric transitions have been estimated for the three distinct types of bond pairs (CO, OC), (OC, CC) and (CC, CO) on the backbone. The effective friction coefficient is deduced from the viscosity of the solvent, irrespective of the size of the kinetic unit, assuming environmental effects and chain connectivity constraints to be of secondary importance compared to torsional energy barriers. The reader is referred to [8] for explicit expressions of... 

Fig. 9 Left panel-, time autoCF of the central torsion of T6 in the smectic (590 K), nematic (650 K) and isotropic (685 K) phase inset shows the Arrhenius plot of the trasional correlation time, with different regimes observed in the smectic (circles) and nematic-isotropic (squares) phases [2]. Right panel time autoCF of molecular axes for alkoxy-substilnted phthalocyanines in the rectangular (300 K) and hexagonal (425 K) columnar phase inset shows an Arrhtmius plot of the correlation times for the in-plane rotation, revealing the phase transition at about 330 K. Reprinted with permission from [34]. Copyright 2009 American Chemical Society...

To optimize force fields for long time scale motions Aliev et al. propose a new robust approach to use NMR spin-lattice relaxation times Ti of both backbone and sidechain carbons. This allows a selective determination of both overall molecular and intramolecular motional time scales. In addition they use motionally averaged experimental/ coupling constants for torsional FF parameters. The force constants in the FFs and the correlation times are fitted in an Arrhenius-type of equation. 

The previous sections indicate the ability of chemically detailed simulations to explain experimental data and the potential for predictions in unknown systems. All-atomic simulation often bridges the gap between indirect structural information obtained from spectroscopic, thermal, and mechanical measurements and molecular level details. The local dynamics of polymer chains [193] agrees with C-H NMR relaxation data on a relative scale [2] however, absolute correlation times in the simulation appear to be 2.5 times the real value. This mismatch could be associated with overestimated torsion barriers in atomistic models, such as an eclipsed barrier of 5-6kcalmol for n-butane (from ab initio calculations) in comparison to the experimental value of 4.0 kcal mol [228] such differences can... 

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