Particle mesh Ewald summation

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... 

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 In the modified dodecamer each phosphodiester has been replaced by a phosphoramidate unit with a N3-P5 intemucleoside linkage. 

A. Toukmaji and D. Paul and J. A. Board, Jr., Distributed Particle-Mesh Ewald A Parallel Ewald Summation Method, Proceedings, International Conference on Parallel and Distributed Processing Techniques and Applications (PDPTA 96), CSREA Press (1996), pp. 33-43. 

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 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... 

Ewald summation presented above calls for the calculation of AP terms for each of the periodic boxes, a computationally demanding requirement for large biomolecular systems. Recently, Darden et al. proposed an N log N method, called particle mesh Ewald (PME), which incorporates a spherical cutoff R. This method uses lookup tables to calculate the direa space sum and its derivatives. The reciprocal sum is implemented by means of multidimensional piecewise interpolation methods, which permit the calculation of this sum and its first derivative at predefined grids with fast Fourier transform methods. The overhead for this calculation in comparison to Coulomb interactions ranges from 16 to 84% of computer time, depending on the reciprocal sum grid size and the order of polynomial used in calculating this sum. 

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... 

For the non-periodic systems it is more common to use a list of non-bonded neighbors and spherical truncation of the non-bonded interactions at some cut off distance in the range 10-15 A, often together with a smoothing of the interaction to zero at the cut off. In one case a fast multipole expansion of the long-range Coulomb interactions was used. For periodic systems there is also the possibility to use Ewald summation, and the simulations of the restriction endonucleases were made with the fast particle mesh Ewald (PME) summation algorithm. 

There is a growing number of approaches to treat the essentially infinite reach of charge-charge interactions. To mention just a few of the more traditional numerical ones which are well adapted to the requirements of MD, we have charge group cut-off [63], Ewald [72] summation, smooth particle Ewald [66] summation and particle-particle-particle-mesh (P M) [73]. There are also several variations of hierarchical methods [74] a few examples are the method of Bames and Hut (BH) [75], the fast multipole method (EMM), with [76] and without [77] multipoles, and the cell multipole method [78]. 

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