Particle mesh Ewald electrostatics

Molecular Dynamics Simulations on Nucleic Acid Systems Using the CorneU et al. Force Field and Particle Mesh Ewald Electrostatics... 

Polarizable Atomic Multipole X-Ray Refinement Particle Mesh Ewald Electrostatics for Macromolecular Crystals. 

U. Essmann, L. Perera, M. L. Berkowitz, T. Darden, H. Lee, and L. G. Pedersen. The smooth particle mesh ewald method. J. Chem. Phys., 103 8577, 1995. Brock A. Luty, Ilario G. Tironi, and Wilfried F. van Gunsteren. Lattice-sum methods for calculating electrostatic interactions in molecular simulations. J. Chem. Phys., 103 3014-3021, 1995. 

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]. 

In infinite periodic systems, an attractive alternative to the use of a cut-off distance is the Ewald sum technique, first described for chemical systems by York, Darden and Pedersen (1993). By using a reciprocal-space technique to evaluate long-range contributions, the total electrostatic interaction can be calculated to a pre-selected level of accuracy (i.e., the Ewald sum limit is exact) witli a scaling that, in tlie most favorable case (called Particle-mesh Ewald , or PME), is AlogA. Prior to the introduction of Ewald sums, the modeling of polyelectrolytes (e.g., DNA) was rarely successful because of the instabilities introduced... 

The motions of proteins are usually simulated in aqueous solvent. The water molecules can be represented either explicitly or implicitly. To include water molecules explicitly implies more time-consuming calculations, because the interactions of each protein atom with the water atoms and the water molecules with each other are computed at each integration time step. The most expensive part of the energy and force calculations is the nonbonded interactions because these scale as 77 where N is the number of atoms in the system. Therefore, it is common to neglect nonbonded interactions between atoms separated by more than a defined cut-off ( 10 A). This cut-off is questionable for electrostatic interactions because of their 1/r dependence. Therefore, in molecular dynamics simulations, a Particle Mesh Ewald method is usually used to approximate the long-range electrostatic interactions (71, 72). 

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... 

Within the last decade important progress has been made in the reliability of MD simulations of solvated nucleic acids using improved force fields and, in particular, a better treatment of electrostatics by the particle-mesh Ewald method. For the first time unrestrained simulations have become possible. Starting out firom experimental geometries it is now possible to explore the conformational space in the vicinity of the starting geometry and to study conformational transitions. ... 

From a methodological point of view it has to be noted that the reliability of MD simulations on biopolymers, and in particular on nucleic acids, has been substantially improved since the particle-mesh Ewald summation for an appropriate treatment of longe-range electrostatics has become available in the second half of the nineties. A second point that is worth mentioning refers... 

The description of the mDC method in the present work is supplemented with mathematical details that we Have used to introduce multipolar densities efficiently into the model. In particular, we describe the mathematics needed to construct atomic multipole expansions from atomic orbitals (AOs) and interact the expansions with point-multipole and Gaussian-multipole functions. With that goal, we present the key elements required to use the spherical tensor gradient operator (STGO) and the real-valued solid harmonics perform multipole translations for use in the Fast Multipole Method (FMM) electrostatically interact point-multipole expansions interact Gaussian-multipoles in a manner suitable for real-space Particle Mesh Ewald (PME) corrections and we list the relevant real-valued spherical harmonic Gaunt coefficients for the expansion of AO product densities into atom-centered multipoles. 

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