Single-correlation-time model

Deviations from predicted relaxation behavior have been observed for large proteins (3 -7 ), polymers (8y9j and highly associated small molecules (10). Particularly prominent are observations of Ti field dependences and low NOE s within the so-called "extreme spectral narrowing region," where single correlation time models predict field independence of Tp and full NOE s. 

When a log-/2 distribution is invoked, 13C T s and NOE s are affected as in Figures 2 and 3. In those figures, the parameter describes the width of the distribution, with small values corresponding to a broad distribution, and p = 00 corresponding to the single correlation time model. 

C NMR spectroscopy. The T minimum was clearly observed and the 7 value at the minimum increased with increasing isotactic content, whilst the temperature giving the minimum shifted concomitantly to a lower temperature. The data could not be interpreted by a single correlation time model and were analysed by a modified 3r model introducing a three-fold potential minimum around the rotational axis of the methyl group. The activation energy of the a-methyl rotation is 5.9kJ/mol for... 

These data may reflect that PDES retains both ordered and disordered phases in the range of -60 to -10°C. Above 25°C, PDES takes only a disordered phase and the molecular motion is in the fast-motion region for the single correlation-time model based on BPP theory [22], because the Si Ti values increase as the temperature is increased from 25 to 125°C. That is to say, the disordered phase (I) is conformationally disordered but shows rudimentary intermolecular packing and reflect a single motional state. 

As mentioned above, the relaxation phenomena of macromolecules seldom follow the single correlation time theory dictated by eqn (36). In such cases, a wide distribution is usually introduced in the correlation time. However, as discussed elsewhere, the distribution of correlation time not only fails to explain the temperature dependencies of Ti, T2 and the NOE of the non-crystalline components observed by scalar decoupled NMR on linear polyesters and polyethylene, but also overlooks the intrinsic motion of long-chain molecules. On the contrary, the 3r theory dictated by eqn (41) was found to be very effective to describe such temperature dependencies of the relaxation parameters. Irrespectively of whether the motional mode assumed in the 3t model for the C-H vector is really true, the concept that the C-H vector in macromolecules involves plural independent diffusional motions with discretely different correlation times is very useful to explain the magnetic relaxation phenomena of macromolecules, as will be shown later. 

Many different models and correlations have been proposed for the prediction of the heat transfer coefficient at vertical surfaces in FFBs. At time of this writing, no single correlation or model has won general acceptance. The following discussion presents a summary of some potentially useful approaches. It is helpful to consider the total heat transfer coefficient as eomposed of convective contributions from the lean-gas phase and the dense-particle phase plus thermal radiation, as defined by Eqs. (15) and (16). All eorrela-tions based on ambient temperature data, where thermal radiation is negligible, should be considered to represent only the convective heat transfer coefficient hr. 

The temperature dependence of the spin-lattice relaxation rate Ef is usually analyzed using the simple Bloembergen, Purcell and Pound (BPP) model, which assumes a single correlation time describing non-correlated isotropic random motions. The spin-lattice relaxation expressed in terms of the spectral density function J(a>) evaluated at the NMR Larmor frequencies (Do and 2(Oo is... 

Figure 2 shows the effects of this correlation time model on the calculated relaxation parameters. The theoretical curves demonstrate the dependence of the relaxation parameters on the internal motion correlation time for a series ofto values ranging from 10 to 10 s. These curves are calculated for the case of a P nucleus relaxed by three protons, each 2.86 A away, with rotation about an internal rotation axis such that a = 40° with an additional contribution from the CSA mechanism. The shape of the relaxation parameter curves for the single isotropic motion model and the two correlation time models can he compared in Figs. 1 and 2. Although use of only two measured relaxation parameters may or may not permit a distinction between the models, employment of additional parameters will permit a distinction between these two models. An illustration of the influence of the average intemuclear distance on the relaxation when CSA contributions can... 

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. 

In the isotropic model, the overall rotational diffusion is characterized by a single parameter, the overall correlation time zc. The following steps could be used to determine zc. 

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. 

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

Correlation Times for Backbone and Side-Chain Motions in Poly(but-1-ene sulfone) of P = 700 as a 25% w/v Solution in Chloroform-d, Deduced from the Simple Isotropic Single-T Motional Model... 

The simplest motional description is isotropic tumbling characterized by a single exponential correlation time ( ). This model has been successfully employed to interpret carbon-13 relaxation in a few cases, notably the methylene carbons in polyisobutylene among the well studied systems ( ). However, this model is unable to account for relaxation in many macromolecular systems, for instance polystyrene (6) and poly(phenylene oxide)(7,... 

The anisotropy decay of the tryptophan fluorescence of both model peptides and biologically active peptides containing a single tryptophan residue has been determined in various studies. Even in the case of the tripeptide H-Gly-Trp-Gly-OH quenched by acrylamide the anisotropy decay displayed two correlation times with values of 39 and 135 ps. 44 The shorter correlation time was thought to be due to motions of the indole ring relative to the tripeptide. In the case of ACTH(l-24) the fluorescence anisotropy decay of the single tryptophan residue in position 9 of the peptide sequence obtained in phosphate buffer (pH 7, 3.5 °C) was also double-exponential. 29 The shorter rotational correlation time (0 = 92ps)... 

Using single-frequency and noise-modulated resonance and off-resonance proton decoupling, 7] relaxation time measurements, relaxation reagents like Gd (fod)3 and specifically deuterated compounds, all the carbons in retinal isomers, the model compounds a-and /i-ionone, and vitamin A and its isomers [165, 555-557] were assigned. The olefinic ring carbons (C-5 and C-6) could be identified on the assumption that the 13C relaxation times are dominated by intramolecular dipole-dipole interactions with neighboring protons and that the same rotational correlation time characterizes the interactions for both carbons. Consequently the ratio of T/s for C-5 and C-6 can be estimated from eq. (5.1)... 

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