Normal mode analysis applications

More traditional applications of internal coordinates, notably normal mode analysis and MC calculations, are considered elsewhere in this book. In the recent literature there are excellent discussions of specific applications of internal coordinates, notably in studies of protein folding [4] and energy minimization of nucleic acids [5]. 

Various techniques exist that make possible a normal mode analysis of all but the largest molecules. These techniques include methods that are based on perturbation methods, reduced basis representations, and the application of group theory for symmetrical oligomeric molecular assemblies. Approximate methods that can reduce the computational load by an order of magnitude also hold the promise of producing reasonable approximations to the methods using conventional force fields. 

Although not discussed in detail here, the normal mode analysis method has been used to calculate the electron transfer reorganization spectrum in / M-modified cytochrome c [65,66]. In this application the normal mode analysis fits comfortably into the theory of electron transfer. 

Despite its obvious limitations, normal mode analysis has found varied and perhaps unexpected applications in the study of the dynamics of biological molecules. In many... 

The result (173) applies to a photon model with the angular momentum h/2n of a boson, whereas the photon radius r would become half as large for the angular momentum h/4 of a fermion. Moreover, the present analysis on superposition of EMS normal modes is applicable not only to narrow linewidth wavepackets but also to a structure of short pulses and soliton-like waves. In these latter cases the radius in Eq. (173) is expected to be replaced by an average value resulting from a spectrum of broader linewidth. 

Another class of techniques monitors surface vibration frequencies. High-resolution electron energy loss spectroscopy (HREELS) measures the inelastic scattering of low energy ( 5eV) electrons from surfaces. It is sensitive to the vibrational excitation of adsorbed atoms and molecules as well as surface phonons. This is particularly useful for chemisorption systems, allowing the identification of surface species. Application of normal mode analysis and selection rules can determine the point symmetry of the adsorption sites./24/ Infrarred reflectance-adsorption spectroscopy (IRRAS) is also used to study surface systems, although it is not intrinsically surface sensitive. IRRAS is less sensitive than HREELS but has much higher resolution. 

The application of normal-mode analysis to the study of vibrational spectra of proteins is in its infancy, and we may expect this area to develop significantly. In this connection, certain general studies will be useful the nature of vibrations in combined a and jS structures the extent of localization of modes in structures which contain hinge regions the correlation of the calculated spectrum of a protein with that obtained from a sum of its constituent secondary structural components. 

The atoms in a molecule undergo vibrations around their equilibrium configuration within the quantum mechanical picture, even at zero temperature. The application of elementary djmamical principles to these small amplitude vibrations leads to normal mode analysis. Crystalline solids can naively be thought of as big molecules but solving the equations becomes impossible imless the periodicity of the unit cell is included whereupon major simplifications of the algebra are introduced. 

G.J. Kearley, J. Totnkinson J. Penfold (1987). Z. Phys. B, 69, 63-65. New constraints for normal-mode analysis of inelastic neutron-scattering spectra application to the HF2 ion. 

Q. Cui and I. Bahar (Eds.), Normal mode analysis Theory and applications to biological and chemical systems, CRC Press, London, 2005. 

The most common method for determining vibrational frequencies is the normal mode analysis, based on the harmonic force constant matrix of energy second derivatives (Hessians). Of course, vibrations are not truly harmonic, and the anharmonicity generally increases as the frequency of the vibration (steepness of the potential) decreases. That is, the more anharmonic a motion is, the less applicable is the traditional approach to... 

Cui, Q., Bahar, I. Normal Mode Analysis Theory and Applications to Biological and Chemical Systems, Boca Raton, FL Chapman HaU/CRC 2006. 

Ma, J., New advances in normal mode analysis of supermolecular complexes and applications to structural refinement. Cum Protein Pept. Sci., 5, 119 (2004). 

Lu, M. and Ma, J., A minimalist network model for coarse-grained normal mode analysis and its application to biomolecular x-ray crystallography, Proc. Natl. Acad. Sci. U.S.A.,105, 15358 (2008). 

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. 

One of the most common applications of protein normal mode analysis is to the study of hinge-bending in certain multi-domain proteins. Because functional sites are often found in the clefts between domains in such proteins, hinge-bending motions have been postulated to be integral to their function. An understanding of the dynamics of such systems is then critical to understanding their mechanism of action. 

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 technique of normal mode analysis has been described as a relatively simple procedure for obtaining an exact solution to the approximate equations of motion for a chemical system. Despite its severe approximation (that the dynamics of a system can be represented by the sum of harmonic terms that are only strictly valid for small displacements), the normal mode technique has proven to perform well at predicting many experimentally observed properties. The preceding applications have illustrated the variety of ways in which normal modes can serve to define the dynamic structure and eneiget-ics of small molecules, proteins, and nucleic acids and to aid in the interpretation and refinement of experimental data. This technique is likely to see increased use in the future. 

There is an extensive literature on the normal mode analysis of polyatomic molecules. The classical treaties by Herzberg (1945) and by Wilson et al. (1955), make use of symmetry properties of polyatomic molecules and the theory of group representations. However, the application of potential functions to normal mode analysis (Lifson and Warshel, 1968) is made much simpler and straightforward, mostly thanks to the availability of fast computers with large memory space. Symmetry properties of the system are neither needed nor assumed. Rather, they are derived for both the equilibrium structure and the normal vibrations around equilibrium. 

Molecular mechanics calculations are an attempt to understand the physical properties of molecular systems based upon an assumed knowledge of the way in which the energy of such systems varies as a function of the coordinates of the component atoms. While this term is most closely associated with the conformational energy analyses of small organic molecules pioneered by Allinger (1), in their more general applications molecular mechanics calculations include energy minimization studies, normal mode calculations, molecular dynamics (MD) and Monte Carlo simulations, reaction path analysis, and a number of related techniques (2). Molecular mechanics... 

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