Harmonic dynamics

Early evidence for motion in the interior of proteins or their fragments came from infrared vibrational spectroscopy.36 It is usually assumed in interpreting such data that a harmonic potential and the resulting normal-mode description of the motions is adequate (see Chapt. IV.F).200-201 Although it is now known that this approximation is not generally applicable to the atomic motions in proteins (see above), the normal mode description is nevertheless useful for understanding certain aspects of the dynamics. It is most likely to be correct for the mainchain atoms of tightly bonded secondary structural elements, like a-helices and /3-sheets. 

The root-mean-square (rms) atom fluctuations for BPTI were calculated from the normal modes by evaluation of the classical expression given in Eq. 

The results for the mainchain and sidechain averages for each residue as a function of residue number are given. For the mainchain fluctuations, the molecular dynamics and normal-mode values are very similar for the side-chains, there is some correspondence, although the differences are considerably more pronounced. This is in accord with the results on anharmonicity found in the molecular dynamic simulations (see above). 

From the results of the normal-mode dynamics it is evident that different residues contribute in varying degrees to the different modes of BPTI. This suggests that mutations can affect the internal motions of proteins in specific ways. Thus site-directed mutagenesis may alter not only the structure but also the dynamics of a protein molecule. 

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Brooks B and M Karplus 1983. Harmonic Dynamics of Proteins Normal Modes and Fluctuations in Bovine Pancreatic Trypsin Inhibitor. Proceedings of the National Academy of Sciences USA 80 6571-6575. 

Inelastic Incoherent Scattering Intensity. For a system executing harmonic dynamics, the transform in Eq. (4) can be performed analytically and the result expanded in a power series over the nonnal modes in the sample. The following expression is obtained [26] ... 

The reason we work with the derivatives is related to the fact that the mode displacements are measured from their instantaneous, t = 0, position rather than some hypothetical harmonic minimum (48). We know that the global potential energy surface of a liquid is far from harmonic, so such a minimum would be a rather unphysical construct. What happens instead is that our modes do obey simple harmonic dynamics, but subject to somewhat unusual initial conditions (49) ... 

Independent of whether or not a well-defined crossover temperature can be observed in NS data above Tg, it has been well known for a considerable time that on heating a glass from low temperatures a strong decrease of the Debye-Waller factor, respectively Mossbauer-Lamb factor, is observed close to Tg [360,361], and more recent studies have confirmed this observation [147,148,233]. Thus, in addition to contributions from harmonic dynamics, an anomalously strong delocalization of the molecules sets in around Tg due to some very fast precursor of the a-process and increases the mean square displacement. Regarding the free volume as probed by positron annihilation lifetime spectroscopy (PALS), for example, qualitatively similar results were reported [362-364]. 

The harmonic dynamical treatment of a model of the -alkanes as a zigzag chain of point mass beads with unconstrained ends and an infinitely strong stretching force constant (so that only CCC bending occurs) has been presented [13]. The analytic result for the infinite chain for which the frequencies of the LAMs are ... 

Parenthetically, we would like to use this opportunity to correct a misunderstanding that is common in enzymatic literature. Highly anharmonic potentials do not necessarily exclude harmonic dynamics An example is water the interatomic potential is extremely anharmonic (hard spheres), but water supports harmonic waves (sound). The resolution of the paradox is that the variable that describes sound waves (density) is not the variable that enters the anharmonic interatomic potentials, so it is possible for equilibrium fluctuations, like sound, to have harmonic dynamics. 

All of the simulation approaches, other than harmonic dynamics, include the basic elements that we have outlined. They differ in the equations of motion that are solved (Newton s equations, Langevin equations, etc.), the specific treatment of the solvent, and/or the procedures used to take account of the time scale associated with a particular process of interest (molecular dynamics, activated dynamics, etc.). For example, the first application of molecular dynamics to proteins considered the molecule in vacuum.15 These calculations, while ignoring solvent effects, provided key insights into the important role of flexibility in biological function. Many of the results described in Chapts. VI-VIII were obtained from such vacuum simulations. Because of the importance of the solvent to the structure and other properties of biomolecules, much effort is now concentrated on systems in which the macromolecule is surrounded by solvent or other many-body environments, such as a crystal. 

An alternative to harmonic dynamics which incorporates some effects due to the anharmonic nature of the forces is called quasi-harmonic dynamics and... 

Modes of protein motion determined by harmonic dynamics or principal component analysis (PCA) serve to describe essential collective motions and they can be introduced directly into the docking procedure. Most conformational variations are captured approximately by a few essential modes. Included as additional variables in protein-ligand docking, they can be optimized and thus the collective protein motions and ligand conformations are simultaneously explored [91-93]. 

Brooks, B. and Karplus, M. (1983) Harmonic dynamics of proteins normal modes and fluctuations in bovine pancreatic trypsin inhibitor. Proceedings of the National Academy of Sciences of the United States of America, 80 (21), 6571-6575. 

Tatsumi, R., Fukunishi, Y., and Nakamura, H. (2004) A hybrid method of molecular dynamics and harmonic dynamics for docking of flexible ligand to flexible receptor. Journal of Computational Chemistry, 25 (16), 1995-2005. 

From the examples that I have discussed, it is clear that the internal motions of macromolecules are intimately involved in ligand binding reactions. It is of interest, therefore, to examine the general characteristics of these internal motions. This can be done theoretically by means of molecular and harmonic dynamic techniques. 

A harmonic dynamic analysis of the BPTI using the CHARMM program potential function has recently been performed in the full conformational space of the molecule [28] that is, all bond lengths and angles, as well as dihedral angles, are included for the 580 atom system consisting of all heavy atoms and polar hydrogens. 

B. Rudolph and D. Case, Biopolymers, 28, 851 (1989). Harmonic Dynamics of a DNA Hexamer in the Absence and Presence of the Intercalator Ethidium. 

For the lattice dynamical evaluation of external contributions to crystal heat capacities, see Filippini, G. Gramaccioli, C. M. Simonetta, M. Suffritti, G. B. Thermodynamic functions for crystals of rigid hydrocarbon molecules a derivation via the Born-von Karman procedure, Chem. Phys. 1975, 8, 136-146. Harmonic dynamics works for crystals thanks to reduced molecular mobility. By contrast, liquids exhibit so-called instantaneous modes (Stratt, R. M. The instantaneous normal modes of liquids, Acc. Chem. Res. 1995, 28, 201-207) the eigenvalues of an instantaneous hessian for a liquid has a spectrum of imaginary frequencies, since any instantaneous frame of liquid stmcture is far from mechanical equilibrium because of collisions. Therefore, it is impossible to estimate heat capacities of liquids in this way, and dynamic simulation is necessary. 

Normal mode calculations have been routinely applied to nucleic acids in order to interpret spectroscopic and neutron diffraction data and to understand the impact of allomorphic form, base sequence and environment on harmonic dynamics. The lowest frequency modes of duplex DNA (which extend down to roughly 20 cm , corresponding to characteristic times of 20 ps) are responsible for main component... 

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