Quasi-harmonic analysis

Fig. 3. Thioredoxin of E. coli s averaged spectrum calculated using Amber, quasi harmonic analysis. Dashed line- generalized Bom approximation, solid line- MD with explicit water.

The application of normal mode analysis to macromolecules such as proteins and nucleic acids has only recently become more common. Normal modes can be calculated either using harmonic analysis, where the second derivative matrix of the potential energy is calculated for a minimized structure, or using quasi-harmonic analysis, where the matrix of correlations of atomic displacements is calculated from a molecular dynamics (MD) trajectory. At temperatures below about 200 K, protein dynamics are primarily harmonic. Above this temperature there is appreciable non-harmonic motion which can be studied using quasi-elastic scattering techniques. There is evidence that such anharmonic motions are also important for protein function and quasi-harmonic analysis allows them to be incorporated implicitly to some extent within a harmonic model. 

Quasi-harmonic analysis is the computation of the normal modes of a molecule from atomic displacements generated by a molecular dynamics simulation. In this case, the atomic coordinate fluctuations are inversely related to the force constants, which are the second derivatives of the potential function. This formulation allows anharmonic motions, arising either from continuous diffusive motion or from transitions between wells, to be included implicitly within a harmonic representation, Brooks and co-workers " have carried out a comparison of different approaches to calculating the harmonic and quasiharmonic normal modes for the protein bovine pancreatic trypsin inhibitor (BPTI) with different force field and simulation models, Yet another approach, called essential dynamics, differs from quasi-harmonic analysis in that the atomic masses are not considered and motion is not reduced to a harmonic form, ... 

Quasi-harmonic analysis utilizes the atomic fluctuations calculated from a molecular dynamics simulation. Constmc-tion of a fluctuation matrix, which is inversely related to the... 

The idea of the Green s function/principal component analysis is closely related to the essential dynamics approach recently introduced into biomolecular simulations. Other similar works include those by Garcia, Ichiye and Karplus, Go and coworkers,and developers of the quasi-harmonic method. " The basic idea of the essential dynamics approach is to diagonalize a covariance matrix a whose elements are given by the formula... 

This equation can be written as an infinite series as was done previously. It converges for low ac amplitudes. A detailed analytical method of obtaining harmonic elements was described in Ref. [645]. Harmonic analysis was also applied to quasi-reversible reactions on a rotating disk electrode [642]. 

In the case of lanthanide trihalides, we endorse the computational method of heat capacity calculations in a wide interval of temperatures (from about T = 0 to T ), based on the analysis of the experimental values of the low-temperature heat capacities, Cp,exp(71- High-temperature enthalpy increments are used for correcting Cp,cai(T) values in the range of temperature T > 0.57. Such an approach removes the restriction of quasi-harmonic approximation of a heat capacity in this temperature range. 

Expression 11.3 provides the whole simulated spectrum, while a detailed vibrational analysis requires the unambiguous assignment of each mode contribution. Recently, a number of methods appeared in the literature aimed at the extraction of normal-mode-like analysis from ab initio dynamics [58-63]. Some of these [58-60] refer to the quasi-harmonic model introduced by Karplus [64,65] in the framework of classical molecular dynamics and individuate normal-mode directions as main components of the nuclear fluctuations in the NVE or NVT ensemble. The quasinormal model relies on the equipartition of the kinetic energy among normal modes thus problems arise when the simulation time required to obtain such a distribution is computationally too expensive, as is often the case for ab initio dynamics. Other approaches [61-63] carry out the time evolution analysis in the momenta subspace instead of the configurational space. In these approaches the basic consideration is that, at any temperature, generalized normal modes g, correspond to uncorrelated momenta such that [61]... 

Application of the collective relaxation model requires first comparing heteronuclear order parameters for C-H and N-H obtained experimentally (not necessarily a complete set) with those calculated from normal mode or quasi-harmonic normal mode analysis. The agreement between the calculated and experimentally determined order parameters can be improved by (I) a global scaling of the frequencies of the low frequency modes (2) adjustment of individual eigenfrequen-cies and (3) application of an orthogonal rotation matrix to mix the directions of the low frequency modes. Penalty functions are applied to restrict the magnitude of adjustment for the frequencies. 

The calculated harmonic vibrational frequencies in Table 35 show that the NNN antisymmetric stretch at 2257 cm decreases to 2206 cm in H2NNN , while the symmetric stretch shifts from 1131 cm to 1077 cm k It is difficult to interpret these moderate changes upon protonation in terms of bonding and bond length changes, because the normal modes are not pure linear motions of only the heavy atoms. In HNNNH the symmetric stretch increases to 1336 cm relative to HN3 and the antisymmetric stretch decreases to 2173 cm k The intensity of the symmetric stretch mode should decrease substantially for this isomer because of the quasi-inversion symmetry of the cation which is shown, for example, by a zero coefficient for the middle nitrogen atom in the normal mode analysis vector. 

The curve is elevated relative to zero by a constant amount (0.125) and has a contribution of 2o), double the modulation frequency (+, second harmonic). Both these contributions are not included in the experimental reversing heat flow which contains only the contribution to the first harmonic ( , see Sect. 4.4.3). Accepting the present analysis, it is possible to determine y and x from the reversing heat capacity by matching the last term of the equation in Figs. 6.118 and 4.131, and then use the paramters describing the match to compute (A), the actual response of the TMDSC to the quasi-isothermal temperature modulation. 

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