Autocorrelation function molecular motion

Fluorescence intensity detected with a confocal microscope for the small area of diluted solution temporally fluctuates in sync with (i) motions of solute molecules going in/out of the confocal volume, (ii) intersystem crossing in the solute, and (hi) quenching by molecular interactions. The degree of fluctuation is also dependent on the number of dye molecules in the confocal area (concentration) with an increase in the concentration of the dye, the degree of fluctuation decreases. The autocorrelation function (ACF) of the time profile of the fluorescence fluctuation provides quantitative information on the dynamics of molecules. This method of measurement is well known as fluorescence correlation spectroscopy (FCS) [8, 9]. 

Formally, S2 represents a decrease in the autocorrelation function caused by the motion S2=0 corresponds to completely unrestricted motion of a bond (N-H in this case), while S2=0 is expected if the bond reorientations are frozen. It was shown recently that the order parameter may be related to the statistical mechanical properties of a protein molecule [33-35] hence, changes in the NMR-derived order parameters can indicate localized contributions to overall molecular entropy. 

For a rod-like probe with its absorption transition moment direction coinciding with the long molecular axis, the rotational motion in this potential well is described by the diffusion coefficient D. The decay of the autocorrelation functions is then shown to be an infinite sum of exponential terms ... 

It shows the clear link between the change of motion of the particle and its diffusion coefficient. In Fig. 50, the velocity autocorrelation function of liquid argon at 90 K (calculated by computer simulation) is shown [451], The velocity becomes effectively randomised within a time less than lps. Further comments on the velocity autocorrelation functions obtained by computer simulation are reserved until the next sub-section. Because the velocity autocorrelation function of molecular liquids is small for times of a picosecond or more, the diffusion coefficient defined in the limit above is effectively established and constant. Consequently, the diffusion equation becomes a reasonable description of molecular motion over times comparable with or greater than the time over which the velocity autocorrelation function had decayed effectively to zero. Under... 

In more complex molecular systems, increased coupling between the translational motion and both rotational and vibrational modes occurs. It is difficult to separate these effects completely. Nevertheless, the velocity autocorrelation functions of the Lennard—Jones spheres [519] (Fig. 52) and the numerical simulation of the carbon tetrachloride (Fig. 39) are quite similar [452a]. 

A much simpler picture emerges in the time domain. The corresponding autocorrelation function, depicted in Figure 8.4, exhibits three well resolved recurrences with very small amplitudes. The recurrence times 2"i, T2, and T3 are incommensurable which indicates that they reflect different types of molecular motion. Since the recurrences are well separated we can write S(t) as a sum S(t) = Si(t), i = 0,..., 3, with So representing the main peak at t = 0. The Fourier transformation is linear so that the absorption cross section also splits into four individual terms,... 

MD simulation is advantageous for obtaining dynamic properties directly, since the MD technique provides not only particle positions but also particle velocities that enable us to utilize the response theory (e.g., the Kubo formula [175,176]) to calculate the transport coefficients from time-dependent correlation functions. For example, we will examine the self-diffusion process of a tagged PFPE molecular center of mass (Fig. 1.49) from the simulation to gain insight into the excitation of translational motion, specifically, spreading and replenishment. The squared displacement of the center mass of a molecule or a bead is used as a measure of translational movement. The self-diffusion coefficient D can be represented as a velocity autocorrelation function... 

T2, and the nuclear Overhauser enhancement, NOE, comprise a set of parameters which characterize molecular motions. In the case of simple isotropic motion, the dependence is in terms of a single correlation time characterizing the exponential decay of the autocorrelation function. However, in many instances, the assumption of isotropic motion is not valid. For rigid systems, the relaxation behavior can then often be predicted by assuming simple anisotropic motion (1 ). Often, superposition of two or more independent motions must be used to satisfactorily interpret observed relaxation behavior. Recently, however, the wide-... 

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. 

In summary, then, autocorrelation functions are useful mathematical devices which, when applied to velocities, tell us to what degree the motion of a particle at a given instant is related to the impelling force of the last collision. Their usefulness is mainly in molecular dynamics, the principal computer-oriented method by which systems are inaeasingly being analyzed (Section 2.17). 

Through time autocorrelation and cross-correlation functions at equilibrium. These can be computed for and among a variety of vectors associated with the molecular motion. They are equiUbrium properties, and the fiuctuation-dissipation theorem relates them to transient properties such as... 

Dynamically raised processes in the dispersion, such as Brownian molecular motion, cause variations in the intensities of the scattered light with time, which is measured by PCS. Smaller the particle, higher the fluctuations by Brownian motion. Thus, a correlation between the different intensities measured is only possible for short time intervals. In a monodisperse system following first-order kinetics, the autocorrelation function decreases rather fast. In a half logarithmic plot of the auto correlation function, the slope of the graph enables the calculation of the hydrodynamic radius by the Stokes-Einstein equation. With the commercial PCS devices the z-average is determined, which corresponds to the hydrodynamic radius. 

Successful first-principles molecular dynamics simulations in the Car-Paxrinello framework requires low temperature for the annealed electronic parameters while maintaining approximate energy conservation of the nuclear motion, all without resorting to unduly small time steps. The most desirable situation is a finite gap between the frequency spectrum of the nuclear coordinates, as measured, say, by the velocity-velocity autocorrelation function. 

Time-resolved optical experiments rely on a short pulse of polarized light from a laser, synchrotron, or flash lamp to photoselect chromophores which have their transition dipoles oriented in the same direction as the polarization of the exciting light. This non-random orientational distribution of excited state transition dipoles will randomize in time due to motions of the polymer chains to which the chromophores are attached. The precise manner in which the oriented distribution randomizes depends upon the detailed character of the molecular motions taking place and is described by the orientation autocorrelation function. This randomization of the orientational distribution can be observed either through time-resolved polarized fluorescence (as in fluorescence anisotropy decay experiments) or through time-resolved polarized absorption. 

The experimental anisotropy contains information about molecular motion, but is independent of the excited state lifetime. Equation 4 indicates that the orientation autocorrelation function can be obtained directly and unambiguously (within the multiplicative constant r(0)) from the results of a transient grating experiment. 

Molecular motions in solids are incoherent processes, and as such are best described by autocorrelation functions. If fit) describes the time-dependent position (orientation) of a molecule in a sample of many such molecules, the autocorrelation function, G(t) is defined by... 

The correlation function, <-P2[am(0) ( )]>. provides a measure of the internal motions of particular residues in the protein.324 333 Figure 46 shows the results obtained for Trp-62 and Trp-63 from the stochastic boundary molecular dynamics simulations of lysozyme used to analyze the displacement and velocity autocorrelation functions. The net influence of the solvent for both Trp-62 and Trp-63 is to cause a slower decay in the anisotropy than occurs in vacuum. In vacuum, the anisotropy decays to a plateau value of 0.36 to 0.37 (relative to the initial value of 0.4) for both residues within a picosecond. In solution there is an initial rapid decay, corresponding to that found in vacuum, followed by a slower decay (without reaching a plateau value) that continues beyond the period (10 ps) over which the correlation function is ex-... 

The intensity I of the light scattered from a dilute macromolecular or supra-molecular solution is a fluctuating quantity due to the Brownian motion of the scattering particles. These fluctuations can be analysed in terms of the normalised autocorrelation function y1 ( t ) of the scattered electrical field Es, which contains information about the structure and the dynamics of the scattering particles [80]. 

There is a large class of relaxation mechanisms which operate on molecules in motion in non-metailic samples. All but one, the spin-rotation interaction, depend on the fact that the change in the molecular orientation or translation modulates the field due to that particular interaction and creates a randomly varying field at the site of the nucleus in question. Any such random motion can have associated with it a special form of an autocorrelation function G(t), expressed in terms of a scalar product of the local field h(t) and the same local field at an earlier time h(0), which is a measure of... 

Figure 3. Dipolar J, for a CH carbon as a function of an effective correlation time for molecular tumbling, t. Key A, isotropic or pseudoisotropic tumbling characterized by an exponential autocorrelation function G(t) and B, complex behavior resulting from a nonexponential decay of G(t). This requires that motion be represented by a set of t s, or a distribution around some mean. (Reproduced, with permission, from Ref. 20. Copyright 1981, Adenine Press.)...

Rigorous statistical mechanical analysis indicates that the dielectric relaxation function 0(f) of an isotropic system in the linear response regime is equivalent to an autocorrelation function of a microscopic polarization p(f) fluctuating through the molecular motion at equilibrium (Cole, 1967 Kubo, 1957) ... 

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