Harmonic dynamics atom motions

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

At normal temperatures the lattice dynamics involves predominantly low amplitude atomic motions that are well described in a harmonic approximation. Therefore, potential models widely used in the theory of molecular vibration, such as a generalized valence force field (GVFF) model, may be of use for such studies. In a GVFF the potential energy of a system is described with a set... 

Periodicity is an important attribute of crystals with significant implications for their properties. Another important property of these systems is the fact that the amplitudes of atomic motions about their equilibrium positions are small enough to allow a harmonic approximation of the interatomic potential. The resulting theory of atomic motion in harmonic crystals constitutes the simplest example for many-body dynamics, which is discussed in this section. 

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. 

Here, we rq>ort related trapped-ion research at NIST on (1) the study of the dynamics of a two-level atomic system coupled to harmonic atomic motion, (2) the creation and characterization of nonclassical states of motion such as Schrodinger-cat superposition states, and (3) quantum logic for the generation of highly entangled states and for the investigation of scaling in a quantum computer. 

Atoms in a crystal are not at rest. They execute small displacements about their equilibrium positions. The theory of crystal dynamics describes the crystal as a set of coupled harmonic oscillators. Atomic motions are considered a superposition of the normal modes of the crystal, each of which has a characteristic frequency a(q) related to the wave vector of the propagating mode, q, through dispersion relationships. Neutron interaction with crystals proceeds via two possible processes phonon creation or phonon annihilation with, respectively, a simultaneous loss or gain of neutron energy. The scattering function S Q,ai) involves the product of two delta functions. The first guarantees the energy conservation of the neutron phonon system and the other that of the wave vector. Because of the translational symmetry, these processes can occur only if the neutron momentum transfer, Q, is such that... 

However, the drawback of ab initio calculations is that they usually refer to the athermal limit (T = 0 K), so that pressure but not temperature effects are included in the simulation. Although in principle the ab initio molecular dynamics approach[13] is able to overcome this limitation, at the present state of the art no temperature-dependent quantum-meehanieal simulations are feasible yet for mineral systems. Thus thermal properties have to be dealt with by methods based on empirical interatomic potential functions, containing parameters to be fitted to experimental quan-tities[14,15, 16]. The computational scheme applied here to carbonates is that based on the quasi-harmonic approximation for representing the atomic motion[17]. 

C. Heavy crystalline moderators. For crystalline materials, the dynamics of the atomic motions is well represented in terms of the quantized, simple-harmonic vibrations of the lattice. These excitations are commonly known as phonons, and are of considerable interest to the solid-state physicist. Since the materials of interest as reactor moderators will occur in polycrystalline form, the use of the incoherent approximation to determine the cross... 

MD simulations have been employed in a wide variety of simulations. In the simplest applications, trajectories are generated for biomolecular systems and analyzed for a variety of structural, dynamic, and thermodynamic properties. The average motions of atoms over time may be observed, as well as time correlations for atomic positions and velocities. This type of information can often provide insight regarding structure-function relations and mechanistic details for biomacromolecules. Root mean square position fluctuations and other rms geometric fluctuations can be computed and compared with experimental observations. For example, in the limit of harmonic and isotropic atomic motion, mean square position fluctuations... 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

To separate the effects of static and dynamic disorder, and to obtain an assessment of the height of the potential barrier that is involved in a particular mean-square displacement (here abbreviated (x )), it is necessary to find a parameter whose variation is sensitive to these quantities. Temperature is the obvious choice. A static disorder will be temperature independent, whereas a dynamic disorder will have a temperature dependence related to the shape of the potential well in which the atom moves, and to the height of any barriers it must cross (Frauenfelder et ai, 1979). Simple harmonic thermal vibration decreases linearly with temperature until the Debye temperature Td below To the mean-square displacement due to vibration is temperature independent and has a value characteristic of the zero-point vibrational (x ). The high-temperature portion of a curve of (x ) vs T will therefore extrapolate smoothly to 0 at T = 0 K if the sole or dominant contribution to the measured (x ) is simple harmonic vibration ((x )y). In such a plot the low-temperature limb is expected to have values of (x ) equal to about 0.01 A (Willis and Pryor, 1975). Departures from this behavior indicate more complex motion or static disorder. 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

Another kind of slowness comes from the approximately 1000-fold disparity between bonded and nonbonded forces among atoms. This means that a typical covalent bond undergoes about 30 smal1-amplitude, nearly-harmon-ic vibrations in the time required for any other significant molecular motion to take place. In doing dynamics calculations, these fast vibrational modes are a nuisance because they force the use of a very short time step, about. 001 psec. or less. Fortunately, they... 

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