A reoptimization of the five-site water potential (TIP5P) for use with Ewald sums

The five-site transferable interaction potential (TIP5P) for water is most accurate at reproducing experimental data when used with a simple spherical cutoff for the long-ranged electrostatic interactions. When used with other methods for treating long-ranged interactions, the model is considerably less accurate. With small modifications, a new TIP5P-like potential can be made which is very accurate for liquid water when used with Ewald sums, a more physical and increasingly more commonly used method for treating long-ranged electrostatic interactions. The new model demonstrates a density maximum near 4 degrees C, like the TIP5P model, and otherwise is similar to the TIP5P model for thermodynamic, dielectric, and dynamical properties of liquid water over a range of temperatures and densities. An analysis of this and other commonly used water models reveals how the quadrupole moment of a model can influence the dielectric response of liquid water.     

Ewald summation and multiple time step methods for molecular dynamics simulation of biological molecules

Methods by which to determine conditions for a molecular dynamics (MD) simulation of biological molecules were investigated. Derivation of the optimal parameters of the Ewald summation was described so as to give same precision to the real space, the reciprocal space summations and the van der Waals interaction. Later, the procedure by which to determine the condition of the multiple time step method by RESPA (REference System Propagator Algorithm; Tuckerman et al., 1992, J. Chem. Phys., 97, 1990) was described as exemplified by MD simulations of a solvated β-sheet peptide. The conservation of the total energy in a microcanonical ensemble was measured to investigate the stability of the simulation conditions. The most feasible respective combinations of the time steps were: 0.25 fs for bond, angle and torsion interactions; 2 fs for van der Waals interaction and Ewald real-space summation; and 4 fs for Ewald reciprocal-space summation. Though it retained an acceptable accuracy, this condition accelerated the simulation ten-fold compared to that in which a simple velocity-Verlet method with a time step of 0.25 fs was used. The update of the correction term due to excluded neighbors was then investigated. Better results were obtained when the correction was updated with the real-space than when it was updated with the reciprocal-space summation. Finally, an MD simulation as long as 50 ps performed under the optimal Ewald and RESPA parameters was thus determined. The trajectory showed a good stability, indicating the feasibility of the parameters.     

Gaussian split Ewald: A fast Ewald mesh method for molecular simulation

Gaussian split Ewald (GSE) is a versatile Ewald mesh method that is fast and accurate when used with both real-space and k-space Poisson solvers. While real-space methods are known to be asymptotically superior to k-space methods in terms of both computational cost and parallelization efficiency, k-space methods such as smooth particle-mesh Ewald (SPME) have thus far remained dominant because they have been more efficient than existing real-space methods for simulations of typical systems in the size range of current practical interest. Real-space GSE, however, is approximately a factor of 2 faster than previously described real-space Ewald methods for the level of force accuracy typically required in biomolecular simulations, and is competitive with leading k-space methods even for systems of moderate size. Alternatively, GSE may be combined with a k-space Poisson solver, providing a conveniently tunable k-space method that performs comparably to SPME. The GSE method follows naturally from a uniform framework that we introduce to concisely describe the differences between existing Ewald mesh methods.     

A potential model for the study of ices and amorphous water: TIP4P/Ice

The ability of several water models to predict the properties of ices is discussed. The emphasis is put on the results for the densities and the coexistence curves between the different ice forms. It is concluded that none of the most commonly used rigid models is satisfactory. A new model specifically designed to cope with solid-phase properties is proposed. The parameters have been obtained by fitting the equation of state and selected points of the melting lines and of the coexistence lines involving different ice forms. The phase diagram is then calculated for the new potential. The predicted melting temperature of hexagonal ice (Ih) at 1 bar is 272.2 K. This excellent value does not imply a deterioration of the rest of the properties. In fact, the predictions for both the densities and the coexistence curves are better than for TIP4P, which previously yielded the best estimations of the ice properties.     

Fast and spectrally accurate Ewald summation for 2-periodic electrostatic systems

A new method for Ewald summation in planar/slablike geometry, i.e., systems where periodicity applies in two dimensions and the last dimension is "free" (2P), is presented. We employ a spectral representation in terms of both Fourier series and integrals. This allows us to concisely derive both the 2P Ewald sum and a fast particle mesh Ewald (PME)-type method suitable for large-scale computations. The primary results are: (i) close and illuminating connections between the 2P problem and the standard Ewald sum and associated fast methods for full periodicity; (ii) a fast, O(N log N), and spectrally accurate PME-type method for the 2P k-space Ewald sum that uses vastly less memory than traditional PME methods; (iii) errors that decouple, such that parameter selection is simplified. We give analytical and numerical results to support this.     

A flexible model for water based on TIP4P/2005

A new flexible water model, TIP4P/2005f, is developed. The idea was to add intramolecular degrees of freedom to the successful rigid model TIP4P/2005 in order to try to improve the predictions for some properties, and to enable the calculation of new ones. The new model incorporates flexibility by means of a Morse potential for the bond stretching and a harmonic term for the angle bending. The parameters have been fitted to account for the peaks of the infrared spectrum of liquid water and to produce an averaged geometry close to that of TIP4P/2005. As for the intermolecular interactions, only a small change in the σ parameter of the Lennard-Jones potential has been introduced. The overall predictions are very close to those of TIP4P/2005. This ensures that the new model may be used with the same confidence as its predecessor in studies where a flexible model is advisable.     

Ewald summation versus direct summation of shifted-force potentials for the calculation of electrostatic interactions in solids: A quantitative study

The calculated Madelung energies and Madelung forces of the electrostatic interaction for nine crystal structures are reported. The method of direct summation with two different shifted-force potentials is compared to the Ewald summation. There is a considerable difference in the convergence of the energy and the force for the two shifted-force potentials regarding the cutoff radius. The convergence depends not only on the potential itself, but also on the crystal structure. One of the shifted-force potentials used is implemented in the CHARMM force field. The energy calculated with this potential shows a good convergence for small cutoff radii. With the other shifted-force potential, the force shows a better convergence for small cutoff radii. The number of pair interactions for obtaining the Madelung limit using the Ewald summation and the direct summation of a shifted-force potential is also reported. For complex structures like zeolites, the number of relevant pair interactions is smaller using the direct summation of a shifted-force potential. For simple structures such as cesium chloride, the number of significant pair interactions is smaller using the Ewald summation. © 1997 by John Wiley & Sons, Inc.     

Some comments and corrections regarding the calculation of electrostatic potential derivatives using the Ewald summation technique

A review of the literature on the calculation of electrostatic potentials, fields, and field gradients in systems consisting of charges and dipoles using the Ewald summation technique is presented. Discrepancies between the previous formulas are highlighted, and an error in the derivation of the reciprocal contributions to the electrostatic field and field gradient is corrected. The new formulas for the field and field gradient are shown to exhibit a termwise identity with the ones for the electrostatic energy.     

Errors in the Ewald summation method with electrostatic layer correction for Coulomb interactions in slab geometry

New theoretical expressions for cut-off errors in 2D reciprocal-space summation of the electrostatic layer correction (ELC) term in energy and forces are derived, and a procedure to determine optimal parameters of the method is proposed. The procedure is tested in numerical calculations for charges distributed uniformly in a cubic box and charges located in two layers near the box basis. The summation errors for conventional Ewald method can be used to find out optimal values of the convergence parameter, and real- and reciprocal-space cut-off radii, whereas the ELC errors give possibility to choose an optimal value of an empty space gap in the simulation box.     

Structural properties of water: comparison of the SPC, SPCE, TIP4P, and TIP5P models of water

Molecular-dynamics simulations were carried out for the SPC, SPCE, TIP4P, and TIP5P models of water at 298 K. From these results we determine the following quantities: the absolute entropy using the two-particle approximation, the mean lifetime of the hydrogen bond, the mean number of hydrogen bonds per molecule, and the mean energy of the hydrogen bond. From the entropy calculations we find that nearly all contributions to the total entropy originates from the orientation effects. Moreover, we determine the contributions to the total entropy which originate from the first, second, and higher solvation shells. It is interesting that the limits between solvation shells are clearly visible. The first solvation shell (0.22 < r < 0.36 nm) contributes approximately 43 J mol K to the total entropy; the second solvation shell (0.36 < r < 0.60 nm) contributes approximately 12 J mol K, while contributions from the third and other solvation shells are very small, approximately 2 J mol K in summary. This indicates that water molecules are strongly ordered up to 0.55-0.6 nm around the central water molecule, and beyond this limit the ordering diminishes. The results of calculations (entropy and hydrogen bonds) are compared with the experimental data for the choosing of the best water model. We find that the SPC and TIP4P models reproduce the best experimental values, and we recommend these models for computer simulations of the aqueous solution of biomolecules.     

Influence of water-protein hydrogen bonding on the stability of Trp-cage miniprotein. A comparison between the TIP3P and TIP4P-Ew water models

We report extensive replica exchange molecular dynamics (REMD) simulations on the folding/unfolding equilibrium of Trp-cage miniprotein using the Amber ff99SB all atom forcefield and TIP3P and TIP4P-Ew explicit water solvent models. REMD simulation-lengths in the 500 ns to the microsecond regime per replica are required to adequately sample the folding/unfolding equilibrium. We observe that this equilibrium is significantly affected by the choice of the water model. Compared with experimental data, simulations using the TIP3P solvent describe the stability of the Trp-cage quite realistically, providing a melting point which is just a few Kelvins above the experimental transition temperature of 317 K. The TIP4P-Ew model shifts the equilibrium towards the unfolded state and lowers the free energy of unfolding by about 3 kJ mol(-1) at 280 K, demonstrating the need to fine-tune the protein-forcefield depending on the chosen water model. We report evidence that the main difference between the two water models is mostly due to the different solvation of polar groups of the peptide. The unfolded state of the Trp-cage is stabilized by an increasing number of hydrogen bonds, destabilizing the α-helical part of the molecule and opening the R-D salt bridge. By reweighting the strength of solvent-peptide hydrogen bonds by adding a hydrogen bond square well potential, we can fully recover the effect of the different water models and estimate the shift in population as due to a difference in hydrogen bond-strength of about 0.4 kJ mol(-1) per hydrogen bond.     

A general purpose model for the condensed phases of water: TIP4P/2005

A potential model intended to be a general purpose model for the condensed phases of water is presented. TIP4P/2005 is a rigid four site model which consists of three fixed point charges and one Lennard-Jones center. The parametrization has been based on a fit of the temperature of maximum density (indirectly estimated from the melting point of hexagonal ice), the stability of several ice polymorphs and other commonly used target quantities. The calculated properties include a variety of thermodynamic properties of the liquid and solid phases, the phase diagram involving condensed phases, properties at melting and vaporization, dielectric constant, pair distribution function, and self-diffusion coefficient. These properties cover a temperature range from 123 to 573 K and pressures up to 40,000 bar. The model gives an impressive performance for this variety of properties and thermodynamic conditions. For example, it gives excellent predictions for the densities at 1 bar with a maximum density at 278 K and an averaged difference with experiment of 7 x 10(-4) g/cm3.     

Fragment-based quantum mechanical methods for periodic systems with Ewald summation and mean image charge convention for long-range electrostatic interactions

We describe an Ewald-summation method to incorporate long-range electrostatic interactions into fragment-based electronic structure methods for periodic systems. The present method is an extension of the particle-mesh Ewald technique for combined quantum mechanical and molecular mechanical (QM/MM) calculations, and it has been implemented into the explicit polarization (X-Pol) potential to illustrate the computational details. As in the QM/MM-Ewald method, the X-Pol-Ewald approach is a linear-scaling electrostatic method, in which the short-range electrostatic interactions are determined explicitly in real space and the long-range Ewald pair potential is incorporated into the Fock matrix as a correction. To avoid the time-consuming Fock matrix update during the self-consistent field procedure, a mean image charge (MIC) approximation is introduced, in which the running average with a user-chosen correlation time is used to represent the long-range electrostatic correction as an average effect. Test simulations on liquid water show that the present X-Pol-Ewald method takes about 25% more CPU time than the usual X-Pol method using spherical cutoff, whereas the use of the MIC approximation reduces the extra costs for long-range electrostatic interactions by 15%. The present X-Pol-Ewald method provides a general procedure for incorporating long-range electrostatic effects into fragment-based electronic structure methods for treating biomolecular and condensed-phase systems under periodic boundary conditions.     

Chemometric study of liquid water simulations. I. The parameters of the TIP4P model potential

The multivariate chemometric techniques two level factorial design (TLFD) and principal component analysis (PCA) were used to investigate the TIP4P model potential behavior with respect to perturbations on all intermolecular interaction parameters. The effects of these perturbations were calculated for the enthalpy of vaporization, the density, the first maximum of the radial distribution functions of the O-H and O-O pairs, and the second maximum of the radial distribution function of the O-H pair obtained from Monte Carlo simulations of liquid water at 25 degrees C. The principal effects were quantified and rationalized in terms of the pair-wise interaction potential of the TIP4P model. They also corroborate previously published sensitivity analysis results using molecular dynamics and other model potentials. In addition, significant interaction effects between some parameters of the TIP4P model potential were observed and quantified, which hardly could be obtained without such a statistic approach. These interaction effects are very regular and systematic, and their behavior has not been encountered in other chemometric studies and cannot be rationalized in terms of the functional form of the pair-wise potential.     

Multiple "time step" Monte Carlo simulations: application to charged systems with Ewald summation

Recently, we have proposed an efficient scheme for Monte Carlo simulations, the multiple "time step" Monte Carlo (MTS-MC) [J. Chem. Phys. 117, 8203 (2002)] based on the separation of the potential interactions into two additive parts. In this paper, the structural and thermodynamic properties of the simple point charge water model combined with the Ewald sum are compared for the MTS-MC real-/reciprocal-space split of the Ewald summation and the common Metropolis Monte Carlo method. We report a number of observables as a function of CPU time calculated using MC and MTS-MC. The correlation functions indicate that speedups on the order of 4.5-7.5 can be obtained for systems of 108-500 waters for n=10 splitting parameter.