Isotropic correlation time

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. 

Figure 1. Schematic representation of dependence of the T, and Tt relaxation times on isotropic correlation time (tc) of motion for a C-H fragment assuming dipole-dipole relaxation and 7 T magnetic field.

The dynamic RIS model developed for investigating local chain dynamics is further improved and applied to POE. A set of eigenvalues characterizes the dynamic behaviour of a given segment of N motional bonds, with v isomeric states available to each bond. The rates of transitions between isomeric states are assumed to be inversely proportional to solvent viscosity. Predictions are in satisfactory agreement with the isotropic correlation times and spin-lattice relaxation times from 13C and 1H NMR experiments for POE. 

In polymers, due to the constraint resulting from the connectivity of the chain, the local motions are usually too complicated to be described by a single isotropic correlation time x, as discussed in chapter 4. Indeed, fluorescence anisotropy decay experiments, which directly yield the orientation autocorrelation function, have shown that the experimental data obtained on anthracene-labelled polybutadiene and polyisoprene in solution or in the melt cannot be represented by simple motional models. To account for the connectivity of the polymer backbone, specific autocorrelation functions, based on models in which conformational changes propagate along the chain according to a damped diffusional process, have been derived for local chain... 

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... 

Figure 10 displays the isotropic correlation times computed as a function of the reciprocal of solvent viscosity, using... 

Fig. 10. Isotropic correlation times Xcon as a function of reciprocal solvent viscosity for polyoxyethylene in dilute solution. Circles represent expoimental data, the line is n-edicted by the DRIS model...

The mobility of spin labels grafted on neurotoxin II and the degree of their exposure towards solvent environment containing nonspecific paramagnetic probes K3Fe(CN) (42) or Ni(Ac)2,were examined both for AChR bonded AchR and non-bonded states (Table IV). Isotropic correlation times for spin labeled compounds (x j g) within the 50 ps - 1 ns interval were determined from the EPR spectra as described in(54). For slower rates the x g values were evaluated by comparison with calculated EPR spectra(55), assuming isotropic rotation. 

Section IV,B,2 introduces the idea that chemical-shift anisotropy could become an important relaxation mechanism for phosphorus at hi field. Vogel et al. (1982) have published the field dependence of the linewidth of the histidine 3-iV-phosphate of succinyl-CoA synthetase and the serine phosphate of glycogen phosphorylase a. They found that the CSA mechanism did, indeed, dominate at 6.3- and 9.3-T fields, contributed substantially at 4.7 T, and contributed about 25% of the total relaxation at 2.1 T (36.4 MHz). Vogel etal. 1982) estimated for Escherichia coli sucdnyl-CoA synthetase (MW 140,(XX)) a hydrated radius of 37 A and an isotropic correlation time of 44 ns. These numbers are referenced to the phosphorus linewidth (sensitive only to overall motion) using an analysis very similar to that mentioned in Section IV,B on DNA phosphorus relaxation. In contrast, the E. coli phosphoryl carrier protein HPr (MW 9(KX)) has a calculated hydrated radius of 17 A and a correlation time of 4.2 ns, according to Vogel et al. (1982). 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Here te, tc are the correlation times of rotational and vibrational frequency shifts. The isotropic scattering spectrum corresponding to Eq. (3.15) is the Lorentzian line of width Acoi/2 = w0 + ydp- Its maximum is shifted from the vibrational transition frequency by the quantity coq due to the collapse of rotational structure and by the quantity A due to the displacement of the vibrational levels in a medium. 

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

The nitroxide moiety of 16-SASL and 16-PC exhibits such a great deal of motion that the rotational correlation time can be calculated (Berliner 1978). The rotational correlation time (assuming isotropic rotational diffusion of the nitroxide fragment) can be calculated from the linear term of the line width parameter ... 

In the case of extreme narrowing in which fast isotropic molecular motions dominate as in the solution state, the spectral density is written by a single correlation time,... 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

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. 

Equations (8.25) to (8.28) are no longer valid in the case of hindered rotations occurring in anisotropic media such as lipid bilayers and liquid crystals. In these media, the rotational motions of the probe are hindered and the emission anisotropy does not decay to zero but to a steady value rc0 (see Chapter 5). For isotropic rotations (rod-like probe), assuming a single correlation time, the emission anisotropy can be written in the following form ... 

Figure 5.2. Simulated fluorescence anisotropy decays for (a) an isotropic system, (b) a system such as a lipid bilayer with short and long rotational correlation times, and (c) a system in which one of the rotational correlation times is infinite, and there is therefore a residual anisotropy or r .

The rx term is the anisotropy at times long compared to the fluorescence lifetime, whereas in Eq. (5.9) 2 will be long. If there is no rM, then Eq. (5.8) reduces to the familiar Perrin equation for an isotropic rotator. Earlier, some confusion existed in this field since it was not recognized that an rro term was required for the case of membrane lipid bilayers. For the most part, time-resolved anisotropy measurements have a short rotational correlation time and an term. However, it has been recognized that a more adequate description may be to use two rotational correlation times, where the second may be quite long but not infinite as the rm implies/35 36 ... 

P(Qol, t) is the conditional probability of the orientation being at time t, provided it was Qq a t time zero. The symbol — F is the rotational diffusion operator. In the simplest possible case, F then takes the form of the Laplace operator, acting on the Euler angles ( ml) specifying the orientation of the molecule-fixed frame with respect to the laboratory frame, multiplied with a rotational diffusion coefficient. Dr. Equation (44) then becomes identical to the isotropic rotational diffusion equation. The rotational diffusion coefficient is simply related to the rotational correlation time introduced earlier, by tr = 1I6Dr. 

---