Essential dynamics,

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

G., Berendsen, H.J.C. The essential dynamics of thermolysin Confirmation of the hinge-bending motion and comparison of simulations in vacuum and water. Proteins 22 (1995) 45-54. 

Van Aalten, D.M.F., Findlay, J.B.C., Amadei, A., Berendsen,H.J.C. Essential dynamics of the cellular retinol-binding protein. Evidence for ligand-induced conformational changes. Protein Engin. 8 (1995) 1129-1136. 

The essential slow modes of a protein during a simulation accounting for most of its conformational variability can often be described by only a few principal components. Comparison of PGA with NMA for a 200 ps simulation of bovine pancreatic trypsic inhibitor showed that the variation in the first principal components was twice as high as expected from normal mode analy-si.s ([Hayward et al. 1994]). The so-called essential dynamics analysis method ([Amadei et al. 1993]) is a related method and will not be discussed here. 

Amadei et al. 1993] Amadei, A., Linssen, A.B.M., Berendsen, H.J.C. Essential Dynamics of Proteins. Proteins 17 (1993) 412-425 [Balsera et al. 1997] Balsera, M., Stepaniants, S., Izrailev, S., Oono, Y., Schiilten, K. Reconstructing Potential Energy Functions from Simulated Force-Induced Unbinding Processes. Biophys. J. 73 (1997) 1281-1287 [Case 1996] Case, D.A. Normal mode analysis of protein dynamics. Curr. Op. Struct. Biol. 4 (1994) 285-290... 

Essential Dynamics In most applications details of individual MD trajectories are of only minor interest. An illustrative example due to Grubmuller [10] is documented in Figure 3. It describes the dynamics of a polymer chain of 100 CH2 groups. Possible stepsizes for numerical integration are confined... 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

A Amadei, ABM Lmssen, FIJC Berendsen. Essential dynamics of proteins. Proteins 17 412-425, 1993. 

One of the main attractions of normal mode analysis is that the results are easily visualized. One can sort the modes in tenns of their contributions to the total MSF and concentrate on only those with the largest contributions. Each individual mode can be visualized as a collective motion that is certainly easier to interpret than the welter of information generated by a molecular dynamics trajectory. Figure 4 shows the first two normal modes of human lysozyme analyzed for their dynamic domains and hinge axes, showing how clean the results can sometimes be. However, recent analytical tools for molecular dynamics trajectories, such as the principal component analysis or essential dynamics method [25,62-64], promise also to provide equally clean, and perhaps more realistic, visualizations. That said, molecular dynamics is also limited in that many of the functional motions in biological molecules occur in time scales well beyond what is currently possible to simulate. 

In this way, B preserves the essential dynamical properties of / by preserving orbits i.e., y oB x) = Bof [x). In particular, there is a one-to-one correspondence between repeating symbol sequences in F and periodic orbits of... 

We now describe a relatively simple MD model of a low-index crystal surface, which was conceived for the purpose of studying the rate of mass transport (8). The effect of temperature on surface transport involves several competing processes. A rough surface structure complicates the trajectories somewhat, and the diffusion of clusters of atoms must be considered. In order to simplify the model as much as possible, but retain the essential dynamics of the mobile atoms, we will consider a model in which the atoms move on a "substrate" represented by an analytic potential energy function that is adjusted to match that of a surface of a (100) face-centered cubic crystal composed of atoms interacting with a Lennard-Jones... 

Quite often we are face with the task of reducing the order of a transfer function without losing essential dynamic behavior of the system. Many methods have been proposed for model reduction, however quite often with unsatisfactory results. A reliable method has been suggested by Luus (1980) where the deviations between the reduced model and the original one in the Nyquist plot are minimized. 

Daidone, I., Amadei, A., Roccatano, D., Nola, A. D., Molecular dynamics simulation of protein folding by essential dynamics sampling folding landscape of horse heart cytochrome c, Biophys. J. 2003, 85, 2865-2871... 

Chen, C., Xiao, Y., Zhang, L., A directed essential dynamics simulation of peptide folding, Biophys. J. 2005, 88, 3276-3285... 

Unlike the case of enhancement of yield of product in a chemical reaction, control of qubit state transfers in a quantum computer is useful only if the control does generate sensibly perfect fidelity of population transfer. Fortunately, a typical qubit has a spectrum of states that is much simpler than that of a polyatomic molecule, so that control protocols that focus attention on the dynamics of population transfer in two- and three-level systems are likely to capture the essential dynamics of population transfer in a real qubit system. A large fraction of the theoretical effort devoted to describing such transfers has been confined to those simple cases. To a certain extent, many of these studies are analogous to... 

Information is also lost if we replace the ((1),(2),(3)) basis by the (X, Y, Z) basis for the internal gauge space. The reason is that the former basis is essentially dynamical and the latter is essentially static. This is again a self-consistent result, because electrodynamics, by definition, requires the movement of charge. The misnamed subject of magnetostatics also requires the movement of charge, and so is not static. 

A more detailed decomposition of macromolecular dynamics that can be used not only for assessing convergence but also for other purposes is principal components analysis (PCA), sometimes also called essential dynamics (Wlodek et al. 1997). In PCA the positional covariance matrix C is calculated for a given trajectory after removal of rotational and translational motion, i.e., after best overlaying all structures. Given M snapshots of an N atom macromolecule, C is a 3N X 3A matrix with elements... 

This type of energy exchange in an autoionization process may correspond with the behavior of a kicked rotator in classical mechanics, which is known to exhibit chaos. It would be worthwhile to consider an autoionization process of a simple diatomic molecule in its Rydberg states to understand experimentally the essential dynamics of a quantum system, whose classical counterpart exhibits chaos. 

Obviously, much information on the resonance dynamics should normally be gained by close examination of the structure of the wavefunction in the internal region of the configuration space. However, it is often possible also to infer either the dominant roles or the secondary roles of the interaction potentials in resonance processes without going into details of the wave-function. The inspection of the potentials can often lead to transparent visual understanding of the essential dynamics. 

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