Internal motion correlation function

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. 

Up to this point only overall motion of the molecule has been considered, but often there is internal motion, in addition to overall molecular tumbling, which needs to be considered to obtain a correct expression for the spectral density function. Here we apply the model-free approach to treat internal motion where the unique information is specified by a generalized order parameter S, which is a measure of the spatial restriction of internal motion, and the effective correlation time re, which is a measure of the rate of internal motion [7, 8], The model-free approach only holds if internal motion is an order of magnitude (<0.3 ns) faster than overall reorientation and can therefore be separated from overall molecular tumbling. The spectral density has the following simple expression in the model-free formalism ... 

Most FPA studies to date on DNA have lacked sufficient time resolution to observe directly the relaxation of the internal correlation functions. Instead, the initial anisotropy r0 is taken as an adjustable parameter. Equations (4.30) show that such a procedure is completely valid for anisotropic diffusors (i.e., (A ft)2 = (dx(t)2)), provided the rapid internal motion of the transition dipole is isotropic. It has not yet been ascertained whether the internal motion actually is isotropic, so this must be assumed.(83) A recent claim(86) that large amplitudes of polar wobble are required to fit both the small amplitude of initial FPA relaxation 87 and the linear dichroism88 has been refuted. 83j... 

Yamakawa and co-workers have formulated a discrete helical wormlike chain model that is mechanically equivalent to that described above for twisting and bending/79111 117) However, their approach to determining the dynamics is very different. They do not utilize the mean local cylindrical symmetry to factorize the terms in r(t) into products of correlation functions for twisting, bending, and internal motions, as in Eq. (4.24). Instead, they... 

An important difference between the BO and non-BO internal Hamiltonians is that the former describes only the motion of electrons in the stationary field of nuclei positioned in fixed points in space (represented by point charges) while the latter describes the coupled motion of both nuclei and electrons. In the conventional molecular BO calculations, one typically uses atom-centered basis functions (in most calculations one-electron atomic orbitals) to expand the electronic wave function. The fermionic nature of the electrons dictates that such a function has to be antisymmetric with respect to the permutation of the labels of the electrons. In some high-precision BO calculations the wave function is expanded in terms of basis functions that explicitly depend on the interelectronic distances (so-called explicitly correlated functions). Such... 

The value of the jump distance in the )0-relaxation of PIB found from the study of the self-motion of protons (2.7 A) is much larger than that obtained from the NSE study on the pair correlation function (0.5-0.9 A). This apparent paradox can also be reconciled by interpreting the motion in the j8-regime as a combined methyl rotation and some translation. Rotational motions aroimd an axis of internal symmetry, do not contribute to the decay of the pair correlation fimction. Therefore, the interpretation of quasi-elastic coherent scattering appears to lead to shorter length scales than those revealed from a measurement of the self-correlation function [195]. A combined motion as proposed above would be consistent with all the experimental observations so far and also with the MD simulation results [198]. 

If this observation corresponds to the true situation in solution then the internal motions are an order of magnitude faster than the rotational correlation time. Under such circumstances, the spectral density function used in these calculations is incorrect. This aspect requires further investigation, particularly once the data from dynamics calculations specifically including water become available. 

The spectrum of scattered light contains dynamical information related to translational and internal motions of polymer chains. In the self-beating mode, the intensity-intensity time correlation function can be expressed (ID) as... 

The parameter t is given in Eq. A-9 in the Appendix, as a function of the correlation time, t associated with internal motion. One of the input parameters is the angle j3, formed between the relaxation vector (C—H bond) and the internal axis of rotation (or jump axis), namely the C-5—C-6 bond. The others are correlation times t0 and r, of the HWH model, obtained from the fit of the data for the backbone carbons. The fitting parameters for the two-state jump model are lifetimes ta and tb, and for the restricted-diffusion model, the correlation time t- for internal rotation. The allowed range of motion (or the jump range) is defined by 2x for both models (Eqs. A-4 and A-9). 

Polymer Backbone Motion. Alternate descriptions of molecular motion utilize an effectively non-exponential autocorrelation function to describe polymer dynamics. One formalism is the use of a log-/2 distribution of correlation times in place of a single correlation time(14). Such a description may simulate the various time scales for overall and internal motions in polymers. 

From the general considerations presented in the previous section, one can expect that the many-body non-adiabatic wave function should fulfill the following conditions (1) All particles involved in the system should be treated equivalently (2) Correlation of the motions of all the particles in the system resulting from Coulombic interactions, as well as from the required conservation of the total linear and angular momenta, should be explicitly incorporated in the wave function (3) Particles can only be distinguishable via the permutational symmetry (4) The total wave function should possess the internal and translational symmetry properties of the system (5) For fixed positions of nuclei, the wave functions should become equivalent to what one obtains within the Born-Oppenheimer approximation and (6) the wave function should be an eigenfunction of the appropriate total spin and angular momentum operators. 

In the following sections we shall consider the correlation functions which arise in the motion of various model systems. The correlation function of a well-defined model has at least the merit of being physically realizable. However, the simple systems at first discussed serve more or less to represent local conditions in a dielectric. The internal field continues to lurk in the backgroimd, and will not be considered till a later section of this chapter. 

A review of this nature comes to a halt rather than a conclusion. The time-correlation-function technique is proving not merely fashionable but profitable, though there remain problems more easily discussed by analysis in the frequency variable. A fair vocabulary of functions giving consistent description of simple motions is now available, and we can proceed to compound them to describe more realistic situations. The solution of the internal field problem for the Onsager model opens a new field for calculation, and the massive computations of the molecular dynamidsts offer well-defined systems for the testing of more speculative theories. This field of research is in a period of most interesting and fruitful development. 

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