Molecular dynamics trajectory calculation

A very simple implementation of a molecular dynamics trajectory calculation is achieved by using a velocity Verlet algorithm to calculate the... 

Quantum mechanical calculation of molecular dynamics trajectories can sim ulate bon d breakin g and frtrm ation.. Although you dt) n ot see th e appearance or disappearan ce ofhonds, you can plot the distan ce between two bonded atom s.. A distan ce excccdi n g a theoretical bond length suggests bond breaking. 

Finite difference techniques are used to generate molecular dynamics trajectories with continuous potential models, which we will assume to be pairwise additive. The essential idea is that the integration is broken down into many small stages, each separated in time by a fixed time 6t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interactions with other particles. From the force we can determine the accelerations of the particles, which are then combined with the positions and velocities at a time t to calculate the positions and velocities at a time t + 6t. The force is assumed to be constant during the time step. The forces on the particles in their new positions are then determined, leading to new positions and velocities at time t - - 2St, and so on. 

Most potential energy surfaces are extremely complex. Fiber and Karplus analyzed a 300 psec molecular dynamics trajectory of the protein myoglobin. They estimate that 2000 thermally accessible minima exist near the native protein structure. The total number of conformations is even larger. Dill derived a formula to calculate the upper bound of thermally accessible conformations in a protein. Using this formula, a protein of 150 residues (the approx-... 

Once HyperChem calculates potential energy, it can obtain all of the forces on the nuclei at negligible additional expense. This allows for rapid optimization of equilibrium and transition-state geometries and the possibility of computing force constants, vibrational modes, and molecular dynamics trajectories. 

Figure 3 Calculated X-ray diffuse scattering patterns from (a) a full molecular dynamics trajectory of orthorhombic hen egg white lysozyme and (b) a trajectory obtained by fitting to the full trajectory rigid-body side chains and segments of the backbone. A full description is given in Ref. 13.

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

Figure 2. Temperature dependence of mean square atomic fluctuations of the Z-DNA hexamer. (a) Mean square atomic fluctuations were calculated directly from the molecular dynamics trajectories.

The basic idea underlying AIMD is to compute the forces acting on the nuclei by use of quantum mechanical DFT-based calculations. In the Car-Parrinello method [10], the electronic degrees of freedom (as described by the Kohn-Sham orbitals y/i(r)) are treated as dynamic classical variables. In this way, electronic-structure calculations are performed on-the-fly as the molecular dynamics trajectory is generated. Car and Parrinello specified system dynamics by postulating a classical Lagrangian ... 

Following earlier workby Warshel, Halley and Hautman"" and Curtiss etal presented an approximate numerical scheme to calculate the nonadiabatic electron transfer rate under the above conditions. The method is based on solving Eq. (18) to the lowest order in the coupling F by treating the elements Hj and as known functions of time obtained from the molecular dynamics trajectories. The result for the probability of the system making a transition to the final state at time t, given that it was in the initial state at time fo. is given by... 

Figure 5b. Histoiy of the fluctuations in the hydrogen bond distance H 2g. calculated from a molecular dynamics trajectory.

Figure 7 shows a Dimer calculation for the two-dimensional test problem. The initial configurations for the dimer searches were taken from the extrema of a short high temperature molecular dynamics trajectory (shown as a dashed line). 

Fig. 11. Standard Gibbs energy profile (solid line) for an ion transfer across the sharp interface calculated through a non-Boltzmann sampling using a total of 1.6 ns molecular dynamics trajectories. The crosses denote half of the average ion-solvent electrostatic energy. (After [128]).

In transition state theory, dynamic effects are included approximately by including a transmission coefficient in the rate expression [9]. This lowers the rate from its ideal maximum TS theory value, and should account for barrier recrossing by trajectories that reach the TS (activated complex) region but do not successfully cross to products (as all trajectories reaching this point are assumed to do in TS theory). The transmission coefficient can be calculated by activated molecular dynamics techniques, in which molecular dynamics trajectories are started from close to the TS and their progress monitored to find the velocity at which the barrier is crossed and the proportion that go on to react successfully [9,26,180]. It is not possible to study activated processes by standard molecular dynamics because barrier crossing events occur so rarely. One reason for the... 

We have outlined a new numerical approach to compute approximate long-time molecular dynamics trajectories. We have explained the underlying assumptions and the limitations of the present approach as well as its promise. Numerical examples were shown for relatively small system for which detailed and extensive calculations can be performed. A future direction, the calculation of relative rates, was outlined. The research described in this chapter was supported by grants from the NIH GM59796 and the NSF Grant No. 9982524 to Ron Elber. 

The reader can envision that the accurate calculation of the energy of a many-electron system is a computationally intensive endeavor, particularly when it must be performed thousands or tens of thousands of times, throughout a molecular dynamics trajectory. This section addresses the reasons one would wish to use these methods, in contrast to both experimental methods and molecular dynamics methods based on classical potentials. In its broadest sense, the reason for using Car-Parrincllo methods is to determine properties of systems with as few fitted parameters or a priori assumptions as possible. 

On the other hand, the DFT is a dominant tool to optimize fixed geometrical structures and compute stationary points with their relative energies along a reaction path. It is also a powerful tool through ab initio molecular dynamics (MD) calculations, the methodology being first described in 1985 by Car and Parrinello (CP) [32]. These calculations are of molecular dynamics and involve the movement of nuclei and electrons, whereas standard molecular dynamics illustrates trajectories of... 

C. E. Wozny, B. G. Sumpter, and D. W. Noid, Calculating the vibrational spectra of linear polymers from molecular dynamics trajectories, Trends Polym. Sci. 2 375 (1994) B. G. Sumpter and D. W. Noid, Computational experiments on the migration of internal energy in macromolecular systems, Chem. Phys. 186 323 (1994). 

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