Molecular dynamics Ewald summation

Belhadj,M., Alper, H.A., Levy, R.M. Molecular dynamics simulations of wa ter with Ewald summation for the long-range electrostatic interactions. Chem. Phys. Lett. 179 (1991) 13-20. 

In periodic boimdary conditions, one possible way to avoid truncation of electrostatic interaction is to apply the so-called Particle Mesh Ewald (PME) method, which follows the Ewald summation method of calculating the electrostatic energy for a number of charges [27]. It was first devised by Ewald in 1921 to study the energetics of ionic crystals [28]. PME has been widely used for highly polar or charged systems. York and Darden applied the PME method already in 1994 to simulate a crystal of the bovine pancreatic trypsin inhibitor (BPTI) by molecular dynamics [29]. 

For the calculation of the normal mode spectra external and internal coordinates were assumed to be dynamically decoupled. Translational and rotational coordinates were extracted from the trajectories while all vibrational coordinates were set to zero. Dynamical matrices were set up for 50 configurations generated by molecular dynamics simulation. Long-range Coulombic interactions were treated using the Ewald summation technique. In Figure 2 the instantaneous normal mode spectra are depicted while in Table 3 some of their integral properties are compiled. 

In recent years, a number of models have been introduced which permit the inclusion of long-range electrostatic interactions in molecular dynamics simulation. For simulations of proteins and enzymes in a crystalline state, the Ewald summation is considered to be the correct treatment for long range electrostatic interactions (Ewald 1921 Allen and Tildesley 1989). Variations of the Ewald method for periodic systems include the particle-mesh Ewald method (York et al. 1993). To treat non-periodic systems, such as an enzyme in solution other methods are required. Kuwajima et al. (Kuwajima and Warshel 1988) have presented a model which extends the Ewald method to non-periodic systems. Other methods for treating explicitly long-range interactions... 

The reported results for equilibrium properties were obtained by means of the standard Monte Carlo (MC), molecular dynamics (MD), and Gibbs ensemble (GE) simulation methods [23, 24], For the trial systems of a finite range the simple spherical cutoff was used, whereas in simulations of the full systems either the Ewald summation or the reaction field method were used. For further technical details we refer the reader to the original papers. 

In all simulations of clay mineral systems we apply periodic boundary conditions at constant pressure and temperature (constant NPT), This allows the system volume to change freely at 100 kPa (1 bar) external pressure and 298 K. Furthermore we employ Ewald summation to compute both electrostatic potentials and dispersive van der Waals interactions, and the simulations are fully dynamic, using the Discover module and Insight II graphical user interface of the MSI molecular modeling suite (MSI, 1997). The free energy perturbation technique is not implemented in this software per se so that many of the aforementioned calculations have to be performed with spreadsheet software (e.g., Microsoft Excel). 

All of our atomistic simulations were performed using standard Grand Canonical Monte Carlo (GCMC) and Equilibrium Molecular Dynamics (EMD) simulation methods. The RASPA [15] code was employed. Electrostatic energies were calculated using Ewald summation [16, 17] with a relative error of 10 . A 12 A van der Waals cutoff was used for the short-range interactions. Periodic boundary conditions were employed. 

Moving beyond standard DNA and RNA duplexes, simulations of chemically modified DNA provide a further test of the simulation methods. To this end, we performed unrestrained molecular dynamics simulations on a standard d(CGCGAATTCGCG)2 dodecamer duplex in aqueous solution and its phosphoramidate (N3 -P) analog using the particle mesh Ewald summation technique (5) and recent AMBER force field (10). In the modified dodecamer each phosphodiester has been replaced by a phosphoramidate unit with a N3-P5 intemucleoside linkage. 

Long-range Coulomb Forces. - Coulombic interactions are present in many molecular liquids and these play a key role in determining the molecular structure and the physical properties of these systems. It is therefore important to represent them as accurately as possible. The problem is that for charge-charge coulomb interactions, which decay as r simple truncation is not possible and if carried out leads to unrealistic distortions in the structure and the dynamics. Traditionally this has been most often avoided by implementation of the Ewald summation method. The original summation is... 

The Ewald summation method was originally developed as an efficient way to calculate the long-range interactions in ionic crystals, and it has become one of the most common methods for modeling electrostatic properties in periodic structures,particularly for molecular dynamics simulations. ... 

The traditional Ewald summation approach is generally presented in terms of the potential energy of the system. However, the force acting on a given particle is the quantity used by computational approaches, such as molecular dynamics and Brownian dynamics. Therefore, the derivation of the forces (instead of the potential energies) is required, and how these forces are determined is described below. 

Extensive molecular dynamics simulations of dilute and semidilute polyelectrolyte solutions of chains with degree of polymerization N ranging from 16 up to 300 were recently performed by Stevens and Kremer [146-148] and by Liao et al, [149], In these simulations the long-range electrostatic interactions were taken into account by the Ewald summation method, including interactions with all periodic images of the system, Stevens and Kremer [146-148] have used a spherical approximation of Adams and Dubey [150] for the Ewald sum while Liao et al, [149] have applied the PME method [110], In addition to Coulombic interactions, all particles, including monomers and counterions, interacted via a shifted Lennard-Jones potential with cutoff rcui = 2 a... 

In the case of polar liquids computer experiments whether by molecular dynamics or Monte Carlo methods present difficulties arising from the long range character of the dipole interaction energy and resolution of the problems for the modest sizes of systems in simulations is a task for statistical mechanics rather than ingenious computation methods. Vertheim (35) has summarized the results of the two main kinds of effort to improve on truncation, whether by cutoff of the potential or by periodic boundary conditions namely use of mean field approximations for longer distances and use of Ewald or other summation methods. There is much of interest and instruction value in comparison of results so far... 

Molecular dynamics was performed at constant temperature with AMBER 4.1 all-atom force field [121] and Particle Mesh Ewald method (PME) was used for the calculation of electrostatic interactions [122]. This is a fast implementation of the Ewald summation method for calculating the full electrostatic energy of a unit cell in a macroscopic lattice of repeating images. The PME grid spacing was 1.0A. It was interpolated on a cubic B-spline, with the direct set tolerance set to 0.000001. Periodic boundary conditions were imposed in all directions. All solute-solute non-bonded interactions were calculated without jmy cut-off distance, while a non-bonded residue based cutoff distance of 9A was used for the solvent-solvent and for the solute-solvent interactions. The non-bonded pair list was updated every 20 steps and the... 

Computer simulations have been applied to studies of the structure of molten salts along two lines one is the fi ee standing application of the computer simulation to obtain the partial pair correlation functions, the other is the refining of x-ray and neutron diffraction and EXAFS measurements by means of a suitable model. In both cases a suitable potential function for the interactions of the ions must be employed, as discussed in Sect. 3.2.4. Such potential functirms are employed in both the Monte Carlo (MC) and the molecular dynamics (MD) simulation methods. A further aspect that has been considered in the case of molten salts is the long range coulombic interaction that exceeds the limits of the periodic simulation boxes usually involved (for 1000 ions altogether), requiring the Ewald summation that is expensive in computation time and is prone to truncation errors if not applied carefully. 

The past 20 years have seen a renewal of interest in lattice summation methods, catalyzed by the advances in high-performance computing and the ability thereby provided to approach molecular dynamics and condensed-phase structural problems that had previously seemed inaccessible. In this respect, an important development was the so-called fast multipole method (FMM) [5]. With its help, the electrostatic energies of arrays of charged particles can be evaluated in computing times that are nearly linear in the number of particles. One of the strengths of FMM is that the charge distribution need not be periodic, and methods of Ewald character can be combined with FMM concepts for studies of periodic systems [6]. 

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