Electrostatic Interactions In Molecular Simulations Research Articles

Molecular simulations of charged complex fluids: A review

Molecular simulation plays an increasingly important role in studying the properties of complex fluid systems containing charges, such as ions, piezoelectric materials, ionic liquids, ionic surfactants, polyelectrolytes, zwitterionic materials, nucleic acids, proteins, biomembranes and etc., where the electrostatic interactions are of special significance. Several methods have been available for treating the electrostatic interactions in explicit and implicit solvent models. Accurate and efficient treatment of such interactions has therefore always been one of the most challenging issues in classical molecular dynamics simulations due to their inhomogeneity and long-range characteristics. Currently, two major challenges remain in the application field of electrostatic interactions in molecular simulations; (i) improving the representation of electrostatic interactions while reducing the computational costs in molecular simulations; (ii) revealing the role of electrostatic interactions in regulating the specific properties of complex fluids. In this review, the calculation methods of electrostatic interactions, including basic principles, applicable conditions, advantages and disadvantages are summarized and compared. Subsequently, the specific role of electrostatic interactions in governing the properties and behaviors of different complex fluids is emphasized and explained. Finally, challenges and perspective on the computational study of charged systems are given.

A Modified Shifted Force Approach to the Wolf Summation.

The Wolf method for calculation of electrostatic interactions in molecular simulations is known to describe the energy well, whereas the forces have discontinuities. For a more reliable description of the forces this method can be extended with a shifted force approach. This leads to a good description of the forces and precise molecular dynamics simulation, but the description of the energy becomes poorer. In this study we propose a modification of a shifted force extension to describe the energy as well as the forces in better agreement to reference data as determined from the Ewald summation. We show that vapor-liquid phase equilibria (VLE) calculated with Monte Carlo simulations in the grand canonical ensemble and dynamic properties calculated with molecular dynamics simulations can be calculated reliably using this modification to describe the electrostatic interactions.

The optimal particle-mesh interpolation basis.

The fast Ewald methods are widely used to compute the point-charge electrostatic interactions in molecular simulations. The key step that introduces errors in the computation is the particle-mesh interpolation. In this work, the optimal interpolation basis is derived by minimizing the estimated error of the fast Ewald method. The basis can be either general or model specific, depending on whether or not the charge correlation has been taken into account. By using the TIP3P water as an example system, we demonstrate that the general optimal basis is always more accurate than the B-spline basis in the investigated parameter range, while the computational cost is at most 5% more expensive. In some cases, the optimal basis is found to be two orders of magnitude more accurate. The model specific optimal basis further improves the accuracy of the general optimal basis, but requires more computational effort in the optimization, and may not be transferable to systems with different charge correlations. Therefore, the choice between the general and model specific optimal bases is a trade-off between the generality and the accuracy.

Drude Polarizable Force Field for Molecular Dynamics Simulations of Saturated and Unsaturated Zwitterionic Lipids.

Additive force fields are designed to account for induced electronic polarization in a mean-field average way, using effective empirical fixed charges. The limitation of this approximation is cause for serious concerns, particularly in the case of lipid membranes, where the molecular environment undergoes dramatic variations over microscopic length scales. A polarizable force field based on the classical Drude oscillator offers a practical and computationally efficient framework for an improved representation of electrostatic interactions in molecular simulations. Building on the first-generation Drude polarizable force field for the dipalmitoylphosphatidylcholine 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC) molecule, the present effort was undertaken to improve this initial model and expand the force field to a wider range of phospholipid molecules. New lipids parametrized include dimyristoylphosphatidylcholine (DMPC), dilauroylphosphatidylcholine (DLPC), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC), dipalmitoylphosphatidylethanolamine (DPPE), 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE), and 1,2-dioleoyl-sn-glycero-3-phosphoethanolamine (DOPE). The iterative optimization protocol employed in this effort led to lipid models that achieve a good balance between reproducing quantum mechanical data on model compound representative of phospholipids and reproducing a range of experimental condensed phase properties of bilayers. A parametrization strategy based on a restrained ensemble-maximum entropy methodology was used to help accurately match the experimental NMR order parameters in the polar headgroup region. All the parameters were developed to be compatible with the remainder of the Drude polarizable force field, which includes water, ions, proteins, DNA, and selected carbohydrates.

Automatic Generation of FFT for Translations of Multipole Expansions in Spherical Harmonics

The fast multipole method (FMM) is an efficient algorithm for calculating electrostatic interactions in molecular simulations and a promising alternative to Ewald summation methods. Translation of multipole expansion in spherical harmonics is the most important operation of the fast multipole method and the fast Fourier transform (FFT) acceleration of this operation is among the fastest methods of improving its performance. The technique relies on highly optimized implementation of fast Fourier transform routines for the desired expansion sizes, which need to incorporate the knowledge of symmetries and zero elements in the input arrays. Here a method is presented for automatic generation of such, highly optimized, routines.

Development of a lattice-sum method emulating nonperiodic boundary conditions for the treatment of electrostatic interactions in molecular simulations: A continuum-electrostatics study

Artifacts induced by the application of periodic boundary conditions and lattice-sum methods in explicit-solvent simulations of (bio-)molecular systems are nowadays a major concern in the computer-simulation community. The present article reports a first step toward the design of a modified lattice-sum algorithm emulating nonperiodic boundary conditions, and therefore exempt of such periodicity-induced artifacts. This result is achieved here in the (more simple) context of continuum electrostatics. It is shown that an appropriate modification of the periodic Poisson equation and of its boundary conditions leads to a continuum-electrostatics scheme, which, although applied under periodic boundary conditions, exactly mimics the nonperiodic situation. The possible extension of this scheme to explicit-solvent simulations is outlined and its practical implementation will be described in more details in a forthcoming article.

On the Ewald Artifacts in Computer Simulations. The Test-Case of the Octaalanine Peptide With Charged Termini

The treatment of electrostatic interactions in molecular simulations is of fundamental importance. Ewald and related methods are being increasingly used to the detriment of cutoff schemes, which are known to produce several artifacts. A potential drawback of the Ewald method is the spatial periodicity that is imposed to the system, which could produce artifacts when applied in the simulation of liquids. In this work we analyze the octaalanine peptide with charged termini in explicit solvent, for which severe effects due to the use of Ewald sums were predicted using continuum electrostatics. Molecular Dynamics simulations for a total of 158 nanoseconds were performed in cells of different sizes. From the comparison of the results of different system sizes, no significant periodicity-induced artifacts were observed. It is argued that in current biomolecular simulations, the incomplete sampling is likely to affect the results to a larger extent than the artifacts induced by the use of Ewald sums.

Combining the lattice-sum and reaction-field approaches for evaluating long-range electrostatic interactions in molecular simulations

A new scheme, the lattice-sum-emulated reaction-field (LSERF) method, is presented that combines the lattice-sum (LS) and reaction-field (RF) approaches for evaluating electrostatic interactions in molecular simulations. More precisely, the LSERF scheme emulates a RF calculation (based on an atomic cutoff) via the LS machinery. This is achieved by changing the form of the electrostatic interactions in a standard LS calculation (Coulombic) to the form corresponding to RF electrostatics (Coulombic plus quadratic reaction-field correction term, truncated at the cutoff distance). It is shown (both analytically and numerically) that in the limit of infinite reciprocal-space accuracy, (i) the LSERF scheme with a finite reaction-field cutoff and a given reaction-field permittivity is identical to the RF scheme with the same parameters (and an atomic cutoff), and (ii) the LSERF scheme is identical to the LS scheme in the limit of an infinite reaction-field cutoff, irrespective of the reaction-field permittivity. This new scheme offers two key advantages: (i) from a conceptual point of view, it shows that there is a continuity between the RF and LS schemes and unifies them into a common framework; (ii) from a practical point of view, it allows us to perform RF calculations with arbitrarily large reaction-field cutoff distances for the same computational costs as a corresponding LS calculation. The optimal choice for the cutoff will be the one that achieves the best compromise between artifacts arising from the dielectric heterogeneity of the system (short cutoff) and its artificial periodicity (long cutoff). The implementation of the LSERF method is extremely easy, requiring only very limited modifications of any standard LS code. For practical applications to biomolecular systems, the use of the LSERF scheme with large reaction-field cutoff distances is expected to represent a significant improvement over the current RF simulations involving comparatively much shorter cutoffs.

Influence of cut-off truncation and artificial periodicity of electrostatic interactions in molecular simulations of solvated ions: A continuum electrostatics study

A new algorithm relying on finite integration is presented that solves the equations of continuum electrostatics for truncated (and possibly reaction-field corrected) solute–solvent and solvent–solvent interactions under either nonperiodic or periodic boundary conditions. After testing and validation by comparison with existing methods, the algorithm is applied to investigate the effect of cut-off truncation and artificial periodicity in explicit-solvent simulations of ionic solvation and ion–ion interactions. Both cut-off truncation and artificial periodicity significantly alter the polarization around a spherical ion and thus, its solvation free energy. The nature and magnitude of the two perturbations are analyzed in details, and correction terms are proposed for both effects. Cut-off truncation is also shown to induce strong alterations in the potential of mean force for ion–ion interaction. These observations help to rationalize artifacts previously observed in explicit–solvent simulations, namely spurious features in the radial distribution functions close to the cut-off distance and alterations in the relative stabilities of contact, solvent-separated and free ion pairs.

Optimal charge-shaping functions for the particle–particle—particle–mesh (P3M) method for computing electrostatic interactions in molecular simulations

The application of the particle–particle—particle–mesh (P3M) method for computing electrostatic interactions in molecular simulations relies on the use of a charge-shaping function to split the potential into two contributions, evaluated in real and reciprocal space, respectively. Although the charge-shaping function is traditionally taken to be a Gaussian, many other choices are possible. In the present study, we investigate the accuracy of the P3M method employing, as charge-shaping functions, polynomials truncated to a finite spacial range (TP functions). We first discuss and test analytical estimates of the P3M root-mean-square force error for both types of shaping functions. These estimates are then used to find the optimal values of the free parameters defining the two types of charge-shaping function (width of the Gaussian or coefficients of the TP function). Finally, we compare the accuracy properties of these optimized functions, using both analytical estimates and numerical results for a model ionic system. It is concluded that the use of specific TP functions instead of the traditional Gaussian function leads to improvements in terms of computational speed, simplicity of use, and accuracy of results.

Lattice-sum methods for calculating electrostatic interactions in molecular simulations

When simulating a periodic molecular system, lattice sum methods may be used to evaluate the electrostatic interactions without resorting to an unphysical truncation of the potential. In a previous publication, we compared the computational efficiency of the Ewald and Particle-Particle Particle-Mesh (PPPM) lattice-sum methods. Because the PPPM method discretizes the field equations and utilizes the highly efficient fast Fourier transform algorithm, it requires significantly less computational effort than the Ewald method and scales almost linearly with system size. In this paper, we take a more detailed look into the theory behind the lattice-sum methods to clarify the underlying similarities between the Ewald method and the PPPM method. We also investigate the errors introduced by different approximation used in the discretization of the field equations in the PPPM method.

A local reaction field method for fast evaluation of long-range electrostatic interactions in molecular simulations

One of the major problems in molecular-dynamics simulations of polar fluids or macromolecular systems is the evaluation of electrostatic interactions. A system of N atoms demands an amount of work proportional to N2 for such calculations. Truncation procedures that neglect a significant part of the long-range effects are often necessary for computational feasibility, though such procedures may introduce serious errors in the simulations. This work introduces a simple and very effective approach for treating the long-range electrostatic forces. The present method, which is referred to as the local reaction field method, follows some of the ideas of the previously developed generalized Ewald method but then develops into a much simpler method. This is done by dividing the system into M groups of atoms and evaluating separately the short- and long-range contributions to the potential of each group. The short-range potential is evaluated explicitly as in any standard truncation method, while the long-range potential is approximated by the first four terms in a multipole expansion. In addition, the long-range potential is updated only once in every L time steps leading to a method of the order of N×M×q/L+N×P where q is related to the number of expansion terms and P is the average number of atoms inside the cutoff range. The speed, accuracy, and precision of the present method is assessed by evaluating the self-energy of a sodium ion in water and the self-energies of the acidic residues of bovine pancreatic trypsin inhibitor using an adiabatic charging free-energy perturbation approach. It is found that the present method reproduces accurately the corresponding results obtained without any cutoff but an order of magnitude faster. Furthermore, at the limit of very large systems, the speed of this method can be 2 orders of magnitude faster than that of the no-cutoff method even when each group contains only a small number of atoms. It is also found that the method gives much better results for electrostatic energies in proteins than those obtained by truncation methods. The stability and speed of the local reaction field method provides a powerful tool for the microscopic evaluation of electrostatic energies in macromolecules.