Exponential correlation time

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,... 

It was soon realized that a distribution of exponential correlation times is required to characterize backbone motion for a successful Interpretation of both carbon-13 Ti and NOE values in many polymers (, lO). A correlation function corresponding to a distribution of exponential correlation times can be generated in two ways. First, a convenient mathematical form can serve as the basis for generating and adjusting a distribution of correlation times. Functions used earlier for the analysis of dielectric relaxation such as the Cole-Cole (U.) and Fuoss-Kirkwood (l2) descriptions can be applied to the interpretation of carbon-13 relaxation. Probably the most proficient of the mathematical form models is the log-X distribution introduced by Schaefer (lO). These models are able to account for carbon-13 Ti and NOE data although some authors have questioned the physical insight provided by the fitting parameters (], 13) ... 

To compare the time scales of the dynamics characterization produced by each model, the spectral density or correlation function can be written as a distribution of exponential correlation times. For a correlation function, (t), the general expression is CO... 

Up to now it has been tacitly assumed that each molecular motion can be described by a single correlation time. On the other hand, it is well-known, e.g., from dielectric and mechanical relaxation studies as well as from photon correlation spectroscopy and NMR relaxation times that in polymers one often deals with a distribution of correlation times60 65), in particular in glassy systems. Although the phenomenon as such is well established, little is known about the nature of this distribution. In particular, most techniques employed in this area do not allow a distinction of a heterogeneous distribution, where spatially separed groups move with different time constants and a homogeneous distribution, where each monomer unit shows essentially the same non-exponential relaxation. Even worse, relaxation... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

If Kj(t) vanishes exponentially at times tj (Fig. 1.3), then the correlation function Km behaves in a similar fashion, though opposite in sign (Fig. 1.4). 

In the impact approximation (tc = 0) this equation is identical to Eq. (1.21), angular momentum relaxation is exponential at any times and t = tj. In the non-Markovian approach there is always a difference between asymptotic decay time t and angular momentum correlation time tj defined in Eq. (1.74). In integral (memory function) theory Rotc is equal to 1/t j whereas in differential theory it is 1/t. We shall see that the difference between non-Markovian theories is not only in times but also in long-time relaxation kinetics, especially in dense media. 

Since relaxation is initially non-exponential, the true correlation time (1.69) does not coincide with r = l/ny, but is equal to... 

Though Kj(t) decays from 1 to 0 it is in general non-exponential relaxation. Its conventually defined correlation time... 

Anisotropy describes the rotational dynamics of reporter molecules or of any sensor segments to which the reporter is rigidly fixed. In the simplest case when both the rotation and the fluorescence decay can be represented by single-exponential functions, the range of variation of anisotropy (r) is determined by variation of the ratio of fluorescence lifetime (xF) and rotational correlation time ([Pg.9]

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... 

Fig. 4.9. Schematic of time-resolved fluorescence anisotropy sample is excited with linearly polarized light and time-resolved fluorescence images are acquired with polarization analyzed parallel and perpendicular to excitation polarization. Assuming a spherical fluorophore, the temporal decay of the fluorescence anisotropy, r(t), can be fitted to an exponential decay model from which the rotational correlation time, 6, can be calculated.

For an exponential dependence we then have T(f) = exp(—f/xc). For the considered one-dimensional Markov process the correlation time may be found in the following way. Alternatively to (5.15) the correlation function may be written in the form... 

In order to obtain kinetic parameters for the electron transfer of [11] /K +, the dephasing time tm of the electron-spin echo near the phase-transition temperature Tt was measured. These experiments gave a correlation time tc of 100 ns for the electron transfer at Tg = 170 K. From the assumption of an exponential decrease of c in solution, a value of 100 ps was estimated for tc at room temperature (Rautter, 1989 Rautter et al., 1992). 

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. 

The limiting value of A0 is never achieved in practice, and partial depolarization can result from molecular motion. For a chromophore which moves with the motion of a rigid spherical macromolecule to which it is attached, the observed anisotropy will decay exponentially as a function of , the rotational correlation time, according to... 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes ... 

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