Correlation functions overall

Since the stochastic Langevin force mimics collisions among solvent molecules and the biomolecule (the solute), the characteristic vibrational frequencies of a molecule in vacuum are dampened. In particular, the low-frequency vibrational modes are overdamped, and various correlation functions are smoothed (see Case [35] for a review and further references). The magnitude of such disturbances with respect to Newtonian behavior depends on 7, as can be seen from Fig. 8 showing computed spectral densities of the protein BPTI for three 7 values. Overall, this effect can certainly alter the dynamics of a system, and it remains to study these consequences in connection with biomolecular dynamics. 

Many researchers have correlated the overall decomposition as an nxh. order reaction, with most paraffins following the first order and most olefins following a higher order. In general, isoparaffin rate constants are lower than normal paraffin rate constants. The rate constants are somewhat dependent on conversion due to inhibition effects that is, the rate constant often decreases with increasing conversion, and the order of conversion is not affected. This has been explained by considering the formation of aHyl radicals (38). To predict the product distribution, yields are often correlated as a function of conversion or other severity parameters (39). 

FIG. 4 Time-resolved fluorescence Stokes shift of coumarin 343 in Aerosol OT reverse micelles, (a) normalized time-correlation functions, C i) = v(t) — v(oo)/v(0) — v(oo), and (b) unnormalized time-correlation functions, S i) = v i) — v(oo), showing the magnitude of the overall Stokes shift in addition to the dynamic response, wq = 1.1 ( ), 5 ( ), 7.5 ( ), 15 ( ), and 40 (O) and for bulk aqueous Na solution (A)- Points are data and lines that are multiexponential fits to the data. (Reprinted from Ref 38 with permission from the American Chemical Society.)... 

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

In the absence of a correlation between the local dynamics and the overall rotational diffusion of the protein, as assumed in the model-free approach, the total correlation function that determines the 15N spin-relaxation properties (Eqs. (1-5)) can be deconvolved (Tfast, Tslow < Tc) ... 

This approach yields spectral densities. Although it does not require assumptions about the correlation function and therefore is not subjected to the limitations intrinsic to the model-free approach, obtaining information about protein dynamics by this method is no more straightforward, because it involves a similar problem of the physical (protein-relevant) interpretation of the information encoded in the form of SD, and is complicated by the lack of separation of overall and local motions. To characterize protein dynamics in terms of more palpable parameters, the spectral densities will then have to be analyzed in terms of model-free parameters or specific motional models derived e.g. from molecular dynamics simulations. The SD method can be extremely helpful in situations when no assumption about correlation function of the overall motion can be made (e.g. protein interaction and association, anisotropic overall motion, etc. see e.g. Ref. [39] or, for the determination of the 15N CSA tensor from relaxation data, Ref. [27]). 

In a very broad overview of the structural categories one can state several statistical correlations with type of function. Hemes are almost always bound by helices, but never in parallel a//3 structures. Relatively complex enzymatic functions, especially those involving allosteric control, are occasionally antiparallel /3 but most often parallel a//3. Binding and receptor proteins are most often antiparallel /3, while the proteins that bind in those receptor sites (i.e., hormones, toxins, and enzyme inhibitors) are most apt to be small disulfide-rich structures. However, there are exceptions to all of the above generalizations (such as cytochrome cs as a nonhelical heme protein or citrate synthase as a helical enzyme), and when one focuses on the really significant level of detail within the active site then the correlation with overall tertiary structure disappears altogether. For almost all of the dozen identifiable groups of functionally similar proteins that are represented by at least two known protein structures, there are at least... 

Overall, the correlation functions discussed in detail in Malek et al. provide valuable structural information at the nanometer scale that allows refining the picture of fhe phase-segregafed cafalysf layer morphology, lonomer... 

The relaxation dynamics (W7 in Fig. 38) is the response of the environment around Trp7 to its sudden shift in charge distribution from the ground state to the excited state. Under this perturbation, the response can result from both the surrounding water molecules and the protein. We separately calculated the linear-response correlation functions of indole-water, indole-protein, and the sum of the two. The results for isomer 1, relative to the time-zero values, are shown in Fig. 42a. The linear response correlation function is accumulated from a 6-ns interval indicated in Fig. 41a during which the protein was clearly in the isomer 1 substate. All three correlation functions show a significant ultrafast component 63% for the total response, 50% for indole-water, and nearly 100% for indole-protein. A fit to the total correlation function beyond the ultrafast inertial decrease requires two exponential decays 1.4 ps (3.6kJ/mol) and 23 ps (2.0kJ/mol). Despite the 6-ns simulation window for isomer 1, the 23-ps long component is not well determined on account of the noise apparent in the linear response correlation function (Fig. 42a) between 30 and 140 ps. The slow dynamics are mainly observed in the indole-water relaxation and the overall indole-protein interactions apparently make nearly no contributions to the slowest relaxation component. 

The functions, and ij/, are called the synchronous and asynchronous 2D intensity correlation functions, respectively. These functions represent the overall similarity and dissimilarity, respectively, between two intensity variations at vi and V2 caused by changing the magnitude of the perturbation. The results are plotted on two orthogonal axes (vi and V2) with the spectral intensity plotted on the third axis normal to the 2D spectral plane. Figures 3-31A and 3-3 IB illustrate schematic contour maps of a synchronous and an asynchronous 2D correlation spectrum, respectively, where + and - signs indicate the directions of the contour peaks relative to the 2D spectral plane. 

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