The effect of different cutoff schemes in molecular simulations of proteins

Molecular simulations of nanoscale systems invariably involve assumptions and approximations to describe the electrostatic interactions, which are long-ranged in nature. One approach is the use of cutoff schemes with a reaction-field contribution to account for the medium outside the cutoff scheme. Recent reports show that macroscopic properties may depend on the exact choice of cutoff schemes in modern day simulations. In this work, a systematic analysis of the effects of different cutoff schemes was performed using a set of 52 proteins. We find no statistically significant differences between using a twin-range or a single-range cutoff scheme. Applying the cutoff based on charge groups or based on atomic positions, does lead to significant differences, which is traced to the cutoff noise for energies and forces. While group-based cutoff schemes show increased cutoff noise in the potential energy, applying an atomistic cutoff leads to artificial structure in the solvent at the cutoff distance. Carefully setting the temperature control, or using an atomistic cutoff for the solute and a group-based cutoff for the solvent significantly reduces the effects of the cutoff noise, without introducing structure in the solvent. This study aims to deepen the understanding of the implications different cutoffs have on molecular dynamics simulations.     

Multiple time step update schemes for dissipative particle dynamics

In dynamical simulations of coarse-grained models of soft matter in a solvent a considerable amount of the total simulation time is generically spent on updating the solvent particles, although their dynamics is in most cases not of primary interest. In order to speed up such simulations and as a consequence allow to extend them to cover larger system sizes, we propose and examine various multiple, specifically dual, time step update algorithms for dissipative particle dynamics simulations that are based on the velocity-Verlet scheme [Phys. Rev. 159, 98 (1967)]. Common to all update variants is that the solvent beads are updated with a lower frequency than the bonded interactions within the solute. As a test case we consider a coarse-grained model of a lipid bilayer in water. Our results demonstrate that a considerable saving of simulation time can be gained, while the obtained simulation data are within error brackets virtually identical to those obtained for the reference single time step update scheme.     

Effect of long-range interactions on the surface tension

The effect of long-range interactions on the surface tension at a liquid-gas interface was considered. An analytical expression for the correction to the surface tension for the cutoff of the particle interaction potential at the distance r _c was derived based on a step density profile. For the Lennard-Jones fluid, this correction was calculated numerically from the results of computer simulations of the density profiles. It was established that, in the vicinity of the triple point, the correction is as great as ～6% at the potential cutoff radius r _c=6.78 molecular diameters, a quantity insensitive to the form of the density profile in the interfacial layer.     

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.     

An improved method for nonbonded list generation: rapid determination of near-neighbor pairs

Generation of the list of near-neighbor pairs of atoms not bonded to each other is a key feature of many programs for calculating the energy and energy derivatives for large molecules. Because this step can take a significant amount of CPU time, more efficient nonbonded list generation can speed up the energy calculations. In this article, a novel nonbonded list generation algorithm, BYCC, is introduced. It combines certain features of other algorithms and achieves more rapid nonbonded list generation; a factor of approximately 2.5 for a molecule of 5000 atoms with a cutoff in the 10 A range is obtained on Hewlett-Packard (HP) and Alpha processors, without greatly increasing memory requirements.     

Difficulties with multiple time stepping and fast multipole algorithm in molecular dynamics

Numerical experiments are performed on a 36,000-atom protein–DNA–water simulation to ascertain the effectiveness of two devices for reducing the time spent computing long-range electrostatics interactions. It is shown for Verlet-I/r-RESPA multiple time stepping, which is based on approximating long-range forces as widely separated impulses, that a long time step of 5 fs results in a dramatic energy drift and that this is reduced by using an even larger long time step. It is also shown that the use of as many as six terms in a fast multipole algorithm approximation to long-range electrostatics still fails to prevent significant energy drift even though four digits of accuracy is obtained. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1785–1791, 1997     

Inherent speedup limitations in multiple time step/particle mesh Ewald algorithms

Multiple time step (MTS) algorithms present an effective integration approach to reduce the computational cost of dynamics simulations. By using force splitting to allow larger time steps for the more slowly varying force components, computational savings can be realized. The Particle-Mesh-Ewald (PME) method has been independently devised to provide an effective and efficient treatment of the long-range electrostatics interactions. Here we examine the performance of a combined MTS/PME algorithm previously developed for AMBER on a large polymerase beta/DNA complex containing 40,673 atoms. Our goal is to carefully combine the robust features of the Langevin/MTS (LN) methodology implemented in CHARMM-which uses position rather than velocity Verlet with stochasticity to make possible outer time steps of 150 fs-with the PME formulation. The developed MTS/PME integrator removes fast terms from the reciprocal-space Ewald component by using switch functions. We analyze the advantages and limitations of the resulting scheme by comparing performance to the single time step leapfrog Verlet integrator currently used in AMBER by evaluating different time-step protocols using three assessors for accuracy, speedup, and stability, all applied to long (i.e., nanosecond) simulations to ensure proper energy conservation. We also examine the performance of the algorithm on a parallel, distributed shared-memory computer (SGI Origin 2000 with 8 300-MHz R12000 processors). Good energy conservation and stability behavior can be demonstrated, for Newtonian protocols with outer time steps of up to 8 fs and Langevin protocols with outer time steps of up to 16 fs. Still, we emphasize the inherent limitations imposed by the incorporation of MTS methods into the PME formulation that may not be widely appreciated. Namely, the limiting factor on the largest outer time-step size, and hence speedup, is an intramolecular cancellation error inherent to PME. This error stems from the excluded-nonbonded correction term contained in the reciprocal-space component. This cancellation error varies in time and introduces artificial frequencies to the governing dynamics motion. Unfortunately, we find that this numerical PME error cannot be easily eliminated by refining the PME parameters (grid resolution and/or order of interpolating polynomial). We suggest that methods other than PME for fast electrostatics may allow users to reap the full advantages from MTS algorithms.     

A molecular-dynamics simulation study on the dependence of Lennard-Jones gas-liquid phase diagram on the long-range part of the interactions

The particle-transfer molecular-dynamics technique is adopted to construct the Lennard-Jones fluid gas-liquid phase diagram. Detailed study of the dependence of the simulation results on the system size and the cutoff distance is performed to test the validity of the simulation technique. Both the traditional cutoff plus long-range correction (CPC) and Ewald summation methods are used in the simulations to calculate the interactions. In the intermediate range of temperatures, the results with the Ewald summation method are almost the same as those with the CPC method. However, in the range close to the critical point, the results with the CPC method deviate from those with the Ewald summation. Compared with the results obtained via the Ewald summation in a smaller system, simply increasing the system size in the CPC scheme may not give better results.     

Application of MDGRAPE-3, a special purpose board for molecular dynamics simulations, to periodic biomolecular systems

We describe the application of a special purpose board for molecular dynamics simulations, named MDGRAPE-3, to the problem of simulating periodic bio-molecular systems. MDGRAPE-3 is the latest board in a series of hardware accelerators designed to calculate the nonbonding long-range interactions much more rapidly than normal processors. So far, MDGRAPEs were mainly applied to isolated systems, where very many nonbonded interactions were calculated without any distance cutoff. However, in order to regulate the density and pressure during simulations of membrane embedded protein systems, one has to evaluate interactions under periodic boundary conditions. For this purpose, we implemented the Particle-Mesh Ewald (PME) method, and its approximation with distance cutoffs and charge neutrality as proposed by Wolf et al., using MDGRAPE-3. When the two methods were applied to simulations of two periodic biomolecular systems, a single MDGRAPE-3 achieved 30-40 times faster computation times than a single conventional processor did in the both cases. Both methods are shown to have the same molecular structures and dynamics of the systems.     

A simple algorithm to accelerate the computation of non-bonded interactions in cell-based molecular dynamics simulations

Cell lists are ubiquitous in molecular dynamics simulations--be it for the direct computation of short-range inter-atomic potentials, the short-range direct part of a long-range interaction or for the periodic construction of Verlet lists. The conventional approach to computing pairwise interactions using cell lists leads to a large number of unnecessary interparticle distance calculations. In this paper, an algorithm is presented which reduces the number of spurious distance calculations by first sorting the particles along the cell pair axis and then only interacting two particles if their distance along the axis is smaller than the cutoff distance of the interaction. This approach is shown to be more efficient than the conventional approach and similar approaches using smaller cells.     

Using one‐step perturbation to predict the effect of changing force‐field parameters on the simulated folding equilibrium of a β‐peptide in solution

Computer simulation using molecular dynamics is increasingly used to simulate the folding equilibria of peptides and small proteins. Yet, the quality of the obtained results depends largely on the quality of the force field used. This comprises the solute as well as the solvent model and their energetic and entropic compatibility. It is, however, computational very expensive to perform test simulations for each combination of force‐field parameters. Here, we use the one‐step perturbation technique to predict the change of the free enthalpy of folding of a β‐peptide in methanol solution due to changing a variety of force‐field parameters. The results show that changing the solute backbone partial charges affects the folding equilibrium, whereas this is relatively insensitive to changes in the force constants of the torsional energy terms of the force field. Extending the cut‐off distance for nonbonded interactions beyond 1.4 nm does not affect the folding equilibrium. The same result is found for a change of the reaction‐field permittivity for methanol from 17.7 to 30. The results are not sensitive to the criterion, e.g., atom‐positional RMSD or number of hydrogen bonds, that is used to distinguish folded and unfolded conformations. Control simulations with perturbed Hamiltonians followed by backward one‐step perturbation indicated that quite large perturbations still yield reliable results. Yet, perturbing all solvent molecules showed where the limitations of the one‐step perturbation technique are met. The evaluated methodology constitutes an efficient tool in force‐field development for molecular simulation by reducing the number of required separate simulations by orders of magnitude. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Cutoff radius effect of isotropic periodic sum method for transport coefficients of Lennard-Jones liquid

Molecular dynamics simulations of a Lennard-Jones (LJ) liquid were applied to compare the isotropic periodic sum (IPS) method [X. Wu and B. R. Brooks, J. Chem. Phys. 122, 044107 (2005)], which can reduce the calculation cost of long-range interactions, such as the Lennard-Jones and Coulombic ones, with the cutoff method for the transport coefficients which includes the self-diffusion coefficient, bulk viscosity, and thermal conductivity. The self-diffusion coefficient, bulk viscosity, and thermal conductivity were estimated with reasonable accuracy if the cutoff distance of the LJ potential for the IPS method was greater than 3sigma. The IPS method is an effective technique for estimating the transport coefficients of the Lennard-Jones liquid in a homogeneous system.     

A comparison of neighbor search algorithms for large rigid molecules

Fast determination of neighboring atoms is an essential step in molecular dynamics simulations or Monte Carlo computations, and there exists a variety of algorithms to efficiently compute neighbor lists. However, most of these algorithms are general, and not specifically designed for a given type of application. As a result, although their average performance is satisfactory, they might be inappropriate in some specific application domains. In this article, we study the case of detecting neighbors between large rigid molecules, which has applications in, e.g., rigid body molecular docking, Monte Carlo simulations of molecular self-assembly or diffusion, and rigid body molecular dynamics simulations. More precisely, we compare the traditional grid-based algorithm to a series of hierarchy-based algorithms that use bounding volumes to rapidly eliminate large groups of irrelevant pairs of atoms during the neighbor search. We compare the performance of these algorithms based on several parameters: the size of the molecules, the average distance between them, the cutoff distance, as well as the type of bounding volume used in the culling hierarchy (AABB, OBB, wrapped, or layered spheres). We demonstrate that for relatively large systems (> 100,000 atoms) the algorithm based on the hierarchy of wrapped spheres shows the best results and the traditional grid-based algorithm gives the worst timings. For small systems, however, the grid-based algorithm and the one based on the wrapped sphere hierarchy are beneficial.     

Long-range electrostatic interactions in hybrid quantum and molecular mechanical dynamics using a lattice summation approach

A method based on a lattice summation technique for treating long-range electrostatic interactions in hybrid quantum mechanics/molecular mechanics simulations is presented in this article. The quantum subsystem is studied at the semiempirical level, whereas the solvent is described by a two-body potential of molecular mechanics. Molecular dynamics simulations of a (quantum) chloride ion in (classical) water have been performed to test this technique. It is observed that the application of the lattice summations to solvent-solvent interactions as well as on solute-solvent ones has a significant effect on solvation energy and diffusion coefficient. Moreover, two schemes for the computation of the long-range contribution to the electrostatic interaction energy are investigated. The first one replaces the exact charge distribution of the quantum solute by a Mulliken charge distribution. The long-range electrostatic interactions are then calculated for this charge distribution that interacts with the solvent molecule charges. The second one is more accurate and involves a modified Fock operator containing long-range electron-charge interactions. It is shown here that both schemes lead to similar results, the method using Mulliken charges for the evaluation of long-range interactions being, however, much more computationally efficient.     

A long-range electrostatic potential based on the Wolf method charge-neutral condition

Molecular simulations rely heavily on a long range electrostatic Coulomb interaction. The Coulomb potential decays inversely with distance, indicating infinite effective range. In practice, molecular simulations do not directly take into account such an infinite interaction. Therefore, the Ewald, fast multipole, and cutoff methods are frequently used. Although cutoff methods are implemented easily and the calculations are fast, it has been pointed out that they produce serious artifacts. Wolf and coworkers recently discovered one source of the artifacts. They found that when the total charge in a cutoff sphere disappeared, the cutoff error is dramatically suppressed. The Wolf method uses the charge-neutral principle combined with a potential damping that is realized using a complementary error function. To date, many molecular simulation studies have demonstrated the accuracy and reliability of the Wolf method. We propose a novel long-range potential that is constructed only from the charge-neutral condition of the Wolf method without potential damping. We also show that three simulation systems, in which involve liquid sodium-chloride, TIP3P water, and a charged protein in explicit waters with neutralized ions using the new potential, provide accurate statistical and dielectric properties when compared with the particle mesh Ewald method.     

Electrostatic energies and forces computed without explicit interparticle interactions: a linear time complexity formulation

A rapid method for the calculation of the electrostatic energy of a system without a cutoff is described in which the computational time grows linearly with the number of particles or charges. The inverse of the distance is approximated as a polynomial, which is then transformed into a function whose terms involve individual particles, instead of particle pairs, by a partitioning of the double sum. In this way, the electrostatic energy that is determined by the interparticle interactions is obtained without explicit calculation of these interactions. For systems of positive charges positioned on a face-centered cubic lattice, the calculation of the energy by the new method is shown to be faster than the calculation of the exact energy, in many cases by an order of magnitude, and to be accurate to within 1-2%. The application of this method to increase the accuracy of conventional truncation-based calculations in condensed-phase systems is also demonstrated by combining the approximated long-range electrostatic interactions with the exact short-range interactions in a "hybrid" calculation. For a 20-A sphere of water molecules, the forces are shown to be six times as accurate using this hybrid method as those calculated with conventional truncation of the electrostatic energy function at 12 A. This is accomplished with a slight increase in speed, and with a sevenfold increase in speed relative to the exact all-pair calculation. Structures minimized with the hybrid function are shown to be closer to structures minimized with an exact all-pair electrostatic energy function than are those minimized with a conventional 13-A cutoff-based electrostatic energy function. Comparison of the energies and forces calculated with the exact method illustrate that the absolute errors obtained with standard truncation can be very large. The extension of the current method to other pairwise functions as well as to multibody functions, is described.     

Interfacial properties of cyclic hydrocarbons: a Monte Carlo study

The Monte Carlo technique is used to study the vapor-liquid interface of cyclopentane, cyclohexane, and benzene. The OPLS and TraPPE potential fields are compared in the temperature range from 298.15 to 348.15 K (273.15-298.15 K for C5H10). A new method for the treatment of the long-range interactions in inhomogeneous simulations is used. When this new method is employed, the obtained values of saturated liquid density and of enthalpy of vaporization are equal to those obtained using the bulk isothermal-isobaric Monte Carlo technique. The values of surface tension become independent of the cutoff distance and they are significantly larger than those when only simple spherical truncation of intermolecular interactions is used.     

Observations concerning the treatment of long-range interactions in molecular dynamics simulations

Molecular dynamics simulations of pure water employing two different empirical water models have been used to study the effects of different methods for truncation of long-range interactions in molecular mechanics calculations. As has been observed previously in integral equation studies, “shifting” these interactions on an atom-by-atom basis was found to produce artificial structuring in the water and affect diffusion rates. In cases where some form of short-range truncation must be used, the ST2 switching function applied on a group-by-group basis was found to be the most realistic procedure. If atom-based shifting must be employed, a cutoff distance greater than or equal to 12.0 Å was found to be required to produce realistic results. © 1993 John Wiley & Sons, Inc.