Rotational correlation states

While the comparison of the OMTS and the (CH2)12 spectra helped to learn something about the kind of information solid state chemical shifts can provide, we can obtain much more detailed data about the correlation of chemical shifts and the rotational isomeric states from the spectra of larger cycloalkanes. Usually conformational shift variations are discussed by (i) the so called y-gauche effect and (ii) the vicinal gauche effect, Vg 15) ... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

We can calculate the thermal rate constants at low temperatures with the cross-sections for the HD and OH rotationally excited states, using Eqs. (34) and (35), and with the assumption that simultaneous OH and HD rotational excitation does not have a strong correlated effect on the dynamics as found in the previous quantum and classical trajectory calculations for the OH + H2 reaction on the WDSE PES.69,78 In Fig. 13, we compare the theoretical thermal rate coefficient with the experimental values from 248 to 418 K of Ravishankara et al.7A On average, the theoretical result... 

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. 

If the g- and hyperfine anisotropies are known from analysis of a solid-state spectrum, the line-width parameters (1, and yt can be used to compute the rotational correlation time, tr, a useful measure of freedom of motion. Line widths in ESR spectra of nitroxide spin labels, for example, have been used to probe the motional freedom of biological macromolecules.14 Since tr is related to the molecular hydrodynamic volume, Ft, and the solution viscosity, r, by a relationship introduced by Debye 15... 

Figure 6 Effect of the increased rotational correlation time on the proton relaxivity of MP2269, a Gd111 chelate capable of noncovalent protein binding (Scheme 2). The lower NMRD curve was measured in water, whereas the upper curve was obtained in a 10%w/v bovine serum albumin solution in which the chelate is completely bound to the protein. The rotational correlation times calculated are rR=105ps in the nonbound state, and rR= 1,000 ps in the protein-bound state (t=35°C). For this chelate, the water exchange...

Beside changes in the rotational correlation time, modifications in the hydration state of the chelate were also used to follow enzymatic activity. Gd(HP-D03A)-derivative complexes bearing a galactopyranose moiety were studied as reporters for... 

Fig. 11. HN(CO)CANH-TROSY experiment for establishing sequential 13Ca(t—1), 15N(/), Hn(0 connectivity in 13C/15N/2H labelled proteins. Delay durations A = l/ (4JHn) 2Tn = 23 33 ms, 2Ta = 22-28 ms, depending on rotational correlation time of the protein Tc= 1/(47C/C ) S = gradient + field recovery delay 0 < k < Ta/t2jmax-Phase cycling i = x = x, — x + States-TPPI 03 = x 4>rec. = x, — x.

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

This relation shows that the rotational correlation time is uncoupled from the excited-state lifetime, in contrast to classical steady-state or time-resolved fluorescence polarization measurements (see Chapter 5). The important consequence is the possibility of observing slow rotations with fluorophores of short lifetime. This is the case for biological macromolecules labeled with fluorophores (e.g. rhodamine) whose lifetime is of a few nanoseconds. 

Steady-state and time-resolved emission anisotropy measurements also allows distinction of single molecules on the basis of their rotational correlation time. 

The Perrin equation 48 describes the relationship between the rotational correlational time, 0, and the fluorescence lifetime, r, for steady-state measurements ... 

The mobility of tyrosine in Leu3 enkephalin was examined by Lakowicz and Maliwal/17 ) who used oxygen quenching to measure lifetime-resolved steady-state anisotropies of a series of tyrosine-containing peptides. They measured a phase lifetime of 1.4 ns (30-MHz modulation frequency) without quenching, and they obtained apparent rotational correlation times of 0.18 ns and 0.33 ns, for Tyr1 and the peptide. Their data analysis assumed a simple model in which the decays of the anisotropy due to the overall motion of the peptide and the independent motion of the aromatic residue are single exponentials and these motions are independent of each other. 

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. 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

Cotton-Mouton effect), NMR chemical shift and coupling constants, the optical rotation of polarized light and correlation coefficients between different properties. Extensions to incorporate long-range interactions have also been elaborated11 and it has even been possible to adapt RIS theory for the description of the dynamics of transitions between rotational isomeric states.12,13... 

---