Molecular dynamics simulations simulated time trajectory

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

Molecular dynamics simulations can produce trajectories (a time series of structural snapshots) which correspond to different statistical ensembles. In the simplest case, when the number of particles N (atoms in the system), the volume V,... 

One drawback to a molecular dynamics simulation is that the trajectory length calculated in a reasonable time is several orders of magnitude shorter than any chemical process and most physical processes, which occur in nanoseconds or longer. This allows yon to study properties that change w ithin shorter time periods (such as energy finctnations and atomic positions), but not long-term processes like protein folding. 

Successful molecular dynamics simulations should have a fairly stable trajectory. Instability and lack of ec uilibratioii can result from a large time step, treatment of long-range cutoffs, or unrealistic coiiplin g to a temperature bath. ... 

Once a PES has been computed, it is often fitted to an analytic function. This is done because there are many ways to analyze analytic functions that require much less computation time than working directly with ah initio calculations. For example, the reaction can be modeled as a molecular dynamics simulation showing the vibrational motion and reaction trajectories as described in Chapter 19. Another technique is to fit ah initio results to a semiempirical model designed for the purpose of describing PES s. 

In many molecular dynamics simulations, equilibration is a separate step that precedes data collection. Equilibration is generally necessary to avoid introducing artifacts during the heating step and to ensure that the trajectory is actually simulating equilibrium properties. The period required for equilibration depends on the property of interest and the molecular system. It may take about 100 ps for the system to approach equilibrium, but some properties are fairly stable after 10-20 ps. Suggested times range from 5 ps to nearly 100 ps for medium-sized proteins. 

Figure 5. Molecular dynamics simulation of the decay forward and backward in time of the fluctuation of the first energy moment of a Lennard-Jones fluid (the central curve is the average moment, the enveloping curves are estimated standard error, and the lines are best fits). The starting positions of the adiabatic trajectories are obtained from Monte Carlo sampling of the static probability distribution, Eq. (246). The density is 0.80, the temperature is Tq — 2, and the initial imposed thermal gradient is pj — 0.02. (From Ref. 2.)...

This sequence of states is a discrete representation of the continuous dynamical trajectory starting from zo at time t = 0 and ending at z at time t = . Such a discrete trajectory may, for instance, result from a molecular dynamics simulation, in which the equations of motion of the system are integrated in small time steps. A trajectory can also be viewed as a high-dimensional object whose description includes time as an additional variable. Accordingly, the discrete states on a trajectory are also called time slices. 

The strategy in a molecular dynamics simulation is conceptually fairly simple. The first step is to consider a set of molecules. Then it is necessary to choose initial positions of all atoms, such that they do not physically overlap, and that all bonds between the atoms have a reasonable length. Subsequently, it is necessary to specify the initial velocities of all the atoms. The velocities must preferably be consistent with the temperature in the system. Finally, and most importantly, it is necessary to define the force-field parameters. In effect the force field defines the potential energy of each atom. This value is a complicated sum of many contributions that can be computed when the distances of a given atom to all other atoms in the system are known. In the simulation, the spatial evolution as well as the velocity evolution of all molecules is found by solving the classical Newton equations of mechanics. The basic outcome of the simulation comprises the coordinates and velocities of all atoms as a function of the time. Thus, structural information, such as lipid conformations or membrane thickness, is readily available. Thermodynamic information is more expensive to obtain, but in principle this can be extracted from a long simulation trajectory. 

Which model provides the best representation for local mobility in a particular group remains unclear, as a detailed picture of protein dynamics is yet to be painted. This information is not directly available from NMR measurements that are necessarily limited by the number of experimentally available parameters. Additional knowledge is required in order to translate these experimental data into a reliable motional picture of a protein. At this stage, molecular dynamic simulations could prove extremely valuable, because they can provide complete characterization of atomic motions for all atoms in a molecule and at all instants of the simulated trajectory. This direction becomes particularly promising with the current progress in computational resources, when the length of a simulated trajectory approaches the NMR-relevant time scales [23, 63, 64]. 

The most striking feature of AIMD and the novelty over traditional molecular dynamics simulations is the fact that the electronic structure is available all the time. Therefore, the analysis of the wavefunction in the course of the simulation or along the trajectory is one of the major subjects that can contribute to the understanding on the molecular and on the electronic level. The importance of analyzing the wavefunction in bioinorganic systems has been shown in many applications, see also Section... 

Figure 2.4 Simulated trajectories of helium, oxygen and methane molecules during a 200-ps time period in a poly(dimethylsiloxane) matrix [9]. Reprinted with permission from S.G. Charati and S.A. Stern, Diffusion of Gases in Silicone Polymers Molecular Dynamic Simulations, Macromolecules 31, 5529. Copyright 1998, American Chemical Society...

In conclusion, these gas-phase measurements provide new elues to the role of solvation in ion-moleeule reaetions. For the first time, it is possible to study intrinsie reactivities and the extent to which the properties of gas-phase ion-moleeule reaetions relate to those of the eorresponding reactions in solution. It is clear, however, that gas-phase solvated-ion/moleeule reaetions in which solvent moleeules are transferred into the intermediate elusters by the nucleophile cannot be exaet duplieates of solvated-ion/ molecule reactions in solution in which solvated reactants exchange solvent molecules with the surrounding bulk solvent [743]. For a selection of more recent experimental [772] and theoretical studies of Sn2 reactions in gas phase and solution by classical trajectory simulations [773], molecular dynamics simulations [774, 775], ab initio molecular orbital calculations [776, 777], and density functional theory calculations [778, 779], see the references given. For studies of reactions other than Sn2 ion-molecule processes in the gas phase and in solution, see reviews [780, 781]. 

The general principle of BD is based on Brownian motion, which is the random movement of solute molecules in dilute solution that result from repeated collisions of the solute with solvent molecules. In BD, solute molecules diffuse under the influence of systematic intermolecular and intramolecular forces, which are subject to frictional damping by the solvent, and the stochastic effects of the solvent, which is modeled as a continuum. The BD technique allows the generation of trajectories on much longer temporal and spatial scales than is feasible with molecular dynamics simulations, which are currently limited to a time of about 10 ns for medium-sized proteins. 

Processes at detonation fronts occur on such short time scales (subpicosecond) and length scales (subnanometer) that they are ideal for molecular dynamics simulations. Such simulations, which follow individual atomic trajectories, not only have the potential to probe how detonations are initiated but are also capable of studying the subsequent chemistry at the shockfront. The approach might also clarify how the discrete shock-induced chemistry relates to the continuum theory of detonations. 

A comprehensive test of computational protocols applied for the short time dynamics of the photolysed Mb-CO complex is presented by Meller and Elber [110]. 270 different 10 ps molecular dynamics simulations were carried out using two different solvation boxes, two differenc types of electrostatic cutoffs and two different treatments of the photodissociated ligand. In addition, both the wild-type and the Leu29Phe mutant were treated. 9 different setups were combined from the variables described and 30 trajectories were generated for all. Results presented are averages over these 30 trajectories. Calculations were performed using a combination of the AMBER [111] and OPES [112] force fields, the heme model of Kuczera et al. [16] and approximately 2700 TIP3P... 

As has been mentioned above, a new method for the treatment of the dynamics of mixed classical quantum system has been recently suggested by Jung-wirth and Gerber [50,51]. The method uses the classically based separable potential (CSP) approximation, in which classically molecular dynamics simulations are used to determine an effective time-dependent separable potential for each mode, then followed by quantum wave packet calculations using these potentials. The CSP scheme starts with "sampling" the initial quantum state of the system by a set of classical coordinates and momenta which serve as initial values for MD simulations. For each set j (j=l,2,...,n) of initial conditions a classical trajectory [q (t), q 2(t),..., q N(t)] is generated, and a separable time-dependent effective potential V (qj, t) is then constructed for each mode i (i=l,2,...,N) in the following way ... 

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