Development of hardware accelerator for molecular dynamics simulations: a computation board that calculates nonbonded interactions in cooperation with fast multipole method

Evaluation of long-range Coulombic interactions still represents a bottleneck in the molecular dynamics (MD) simulations of biological macromolecules. Despite the advent of sophisticated fast algorithms, such as the fast multipole method (FMM), accurate simulations still demand a great amount of computation time due to the accuracy/speed trade-off inherently involved in these algorithms. Unless higher order multipole expansions, which are extremely expensive to evaluate, are employed, a large amount of the execution time is still spent in directly calculating particle-particle interactions within the nearby region of each particle. To reduce this execution time for pair interactions, we developed a computation unit (board), called MD-Engine II, that calculates nonbonded pairwise interactions using a specially designed hardware. Four custom arithmetic-processors and a processor for memory manipulation ("particle processor") are mounted on the computation board. The arithmetic processors are responsible for calculation of the pair interactions. The particle processor plays a central role in realizing efficient cooperation with the FMM. The results of a series of 50-ps MD simulations of a protein-water system (50,764 atoms) indicated that a more stringent setting of accuracy in FMM computation, compared with those previously reported, was required for accurate simulations over long time periods. Such a level of accuracy was efficiently achieved using the cooperative calculations of the FMM and MD-Engine II. On an Alpha 21264 PC, the FMM computation at a moderate but tolerable level of accuracy was accelerated by a factor of 16.0 using three boards. At a high level of accuracy, the cooperative calculation achieved a 22.7-fold acceleration over the corresponding conventional FMM calculation. In the cooperative calculations of the FMM and MD-Engine II, it was possible to achieve more accurate computation at a comparable execution time by incorporating larger nearby regions.     

Design of a reaction field using a linear‐combination‐based isotropic periodic sum method

In our previous study (Takahashi et al., J. Chem. Theory Comput. 2012, 8, 4503), we developed the linear‐combination‐based isotropic periodic sum (LIPS) method. The LIPS method is based on the extended isotropic periodic sum theory that produces a ubiquitous interaction potential function to estimate homogeneous and heterogeneous systems. The LIPS theory also provides the procedure to design a periodic reaction field. To demonstrate this, in the present work, a novel reaction field of the LIPS method was developed. The novel reaction field was labeled LIPS‐SW, because it provides an interaction potential function with a shape that resembles that of the switch function method. To evaluate the ability of the LIPS‐SW method to describe in homogeneous and heterogeneous systems, we carried out molecular dynamics (MD) simulations of bulk water and water–vapor interfacial systems using the LIPS‐SW method. The results of these simulations show that the LIPS‐SW method gives higher accuracy than the conventional interaction potential function of the LIPS method. The accuracy of simulating water–vapor interfacial systems was greatly improved, while that of bulk water systems was maintained using the LIPS‐SW method. We conclude that the LIPS‐SW method shows great potential for high‐accuracy, high‐performance computing to allow large scale MD simulations. © 2014 Wiley Periodicals, Inc.     

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     

Boundary element solution of the linear Poisson-Boltzmann equation and a multipole method for the rapid calculation of forces on macromolecules in solution

The Poisson-Boltzmann equation is widely used to describe the electrostatic potential of molecules in an ionic solution that is treated as a continuous dielectric medium. The linearized form of this equation, applicable to many biologic macromolecules, may be solved using the boundary element method. A single-layer formulation of the boundary element method, which yields simpler integral equations than the direct formulations previously discussed in the literature, is given. It is shown that the electrostatic force and torque on a molecule may be calculated using its boundary element representation and also the polarization charge for two rigid molecules may be rapidly calculated using a noniterative scheme. An algorithm based on a fast adaptive multipole method is introduced to further increase the speed of the calculation. This method is particularly suited for Brownian dynamics or molecular dynamics simulations of large molecules, in which the electrostatic forces must be calculated for many different relative positions and orientations of the molecules. It has been implemented as a set of programs in C++, which are used to study the accuracy and speed of this method for two actin monomers.     

Molecular dynamics with helical periodic boundary conditions

Helical symmetry is often encountered in nature and thus also in molecular dynamics (MD) simulations. In many cases, an approximation based on infinite helical periodicity can save a significant amount of computer time. However, standard simulations with the usual periodic boundary conditions (PBC) are not easily compatible with it. In the present study, we propose and investigate an algorithm comprising infinitely propagated helicity, which is compatible with commonly used MD software. The helical twist is introduced as a parametric geometry constraint, and the translational PBC are modified to allow for the helical symmetry via a transitional solvent volume. The algorithm including a parallel code was implemented within the Tinker software. The viability of the helical periodic boundary conditions (HPBC) was verified in test simulations including α‐helical and polyproline II like peptide structures. For an insulin‐based model, the HPBC dynamics made it possible to simulate a fibrillar structure, otherwise not stable within PBC. © 2014 Wiley Periodicals, Inc.     

Extension of the fast multipole method for the rectangular cells with an anisotropic partition tree structure

The fast multipole method (FMM) is an order N method for the numerically rigorous calculation of the electrostatic interactions among point charges in a system of interest. The FMM is utilized for massively parallelized software for molecular dynamics (MD) calculations. However, an inconvenient limitation is imposed on the implementation of the FMM: In three-dimensional case, a cubic MD unit cell is hierarchically divided by the octree partitioning under isotropic periodic boundary conditions along three axes. Here, we extended the FMM algorithm adaptive to a rectangular MD unit cell with different periodicity along the axes by applying an anisotropic hierarchical partitioning. The algorithm was implemented into the parallelized general-purpose MD calculation software designed for a system with uniform distribution of point charges in the unit cell. The partition tree can be a mixture of binary and ternary branches, the branches being chosen arbitrarily with respect to the coordinate axes at any levels. Errors in the calculated electrostatic interactions are discussed in detail for a selected partition tree structure. The extension enables us to execute MD calculations under more general conditions for the shape of the unit cell, partition tree, and boundary conditions, keeping the accuracy of the calculated electrostatic interactions as high as that with the conventional FMM. An extension of the present FMM algorithm to other prime number branches, such as 5 and 7, is straightforward.     

Performance evaluation of the zero‐multipole summation method in modern molecular dynamics software

The zero‐multiple summation method (ZMM) is a cutoff‐based method for calculating electrostatic interactions in molecular dynamics simulations, utilizing an electrostatic neutralization principle as a physical basis. Since the accuracies of the ZMM have been revealed to be sufficient in previous studies, it is highly desirable to clarify its practical performance. In this paper, the performance of the ZMM is compared with that of the smooth particle mesh Ewald method (SPME), where the both methods are implemented in molecular dynamics software package GROMACS. Extensive performance comparisons against a highly optimized, parameter‐tuned SPME implementation are performed for various‐sized water systems and two protein–water systems. We analyze in detail the dependence of the performance on the potential parameters and the number of CPU cores. Even though the ZMM uses a larger cutoff distance than the SPME does, the performance of the ZMM is comparable to or better than that of the SPME. This is because the ZMM does not require a time‐consuming electrostatic convolution and because the ZMM gains short neighbor‐list distances due to the smooth damping feature of the pairwise potential function near the cutoff length. We found, in particular, that the ZMM with quadrupole or octupole cancellation and no damping factor is an excellent candidate for the fast calculation of electrostatic interactions. © 2018 Wiley Periodicals, Inc.     

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518–2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097–3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn–Sham matrix formation. © 2019 Wiley Periodicals, Inc.     

Rapid QM/MM approach for biomolecular systems under periodic boundary conditions: Combination of the density‐functional tight‐binding theory and particle mesh Ewald method

A quantum mechanical/molecular mechanical (QM/MM) approach based on the density‐functional tight‐binding (DFTB) theory is a useful tool for analyzing chemical reaction systems in detail. In this study, an efficient QM/MM method is developed by the combination of the DFTB/MM and particle mesh Ewald (PME) methods. Because the Fock matrix, which is required in the DFTB calculation, is analytically obtained by the PME method, the Coulomb energy is accurately and rapidly computed. For assessing the performance of this method, DFTB/MM calculations and molecular dynamics simulation are conducted for a system consisting of two amyloid‐β(1‐16) peptides and a zinc ion in explicit water under periodic boundary conditions. As compared with that of the conventional Ewald summation method, the computational cost of the Coulomb energy by utilizing the present approach is drastically reduced, i.e., 166.5 times faster. Furthermore, the deviation of the electronic energy is less than . © 2016 Wiley Periodicals, Inc.     

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients

A full implementation of analytical energy gradients for molecular and periodic systems is reported in the TURBOMOLE program package within the framework of Kohn–Sham density functional theory using Gaussian‐type orbitals as basis functions. Its key component is a combination of density fitting (DF) approximation and continuous fast multipole method (CFMM) that allows for an efficient calculation of the Coulomb energy gradient. For exchange‐correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various molecular and periodic model systems, with the largest unit cell of hematite containing 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn–Sham matrix formation. © 2016 Wiley Periodicals, Inc.     

Point atomic multipole moments for simulation of electrostatic potential and field in all-siliceous zeolites

Calibration method of atomic multipole moments (AMMs) is presented with respect to geometries of all-siliceous zeolite models obtained with X-ray diffraction (XRD) methods. Mulliken atomic charges and AMMs are calculated for all-siliceous types possessing small size elementary unit cells at the hybrid density functional theory (DFT) (B3LYP) and general gradient approximation (GGA) Perdew-Burke-Ernzerhof (PBE) levels and then used to fit the dependences versus geometry variables for the Mulliken charges and versus special coordinate for the AMMs. Fitted and exact charges and AMMs are used to compute electrostatic potential (EP) and electric field (EF) for all-siliceous zeolites with CRYSTAL. A possibility of application of the point AMMs to quantum mechanical/molecular mechanics computations or classic simulation of physical adsorption is evaluated. The considered models expand over wide range of structural parameters and could be applied even to amorphous all-siliceous systems.     

A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent

A fast multigrid boundary element (MBE) method for solving the Poisson equation for macromolecular electrostatic calculations in a solvent is developed. To convert the integral equation of the BE method into a numerical linear equation of low dimensions, the MBE method uses an adaptive tesselation of the molecular surface by BEs with nonregular size. The size of the BEs increases in three successive levels as the uniformity of the electrostatic field on the molecular surface increases. The MBE method provides a high degree of consistency, good accuracy, and stability when the sizes of the BEs are varied. The computational complexity of the unrestricted MBE method scales as O(N_at), where N_at is the number of atoms in the macromolecule. The MBE method is ideally suited for parallel computations and for an integrated algorithm for calculations of solvation free energy and free energy of ionization, which are coupled with the conformation of a solute molecule. The current version of the 3-level MBE method is used to calculate the free energy of transfer from a vacuum to an aqueous solution and the free energy of the equilibrium state of ionization of a 17-residue peptide in a given conformation at a given pH in ∼ 400 s of CPU time on one node of the IBM SP2 supercomputer. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 569–583, 1997     

The effect of box shape on the dynamic properties of proteins simulated under periodic boundary conditions

The effect of the box shape on the dynamic behavior of proteins simulated under periodic boundary conditions is evaluated. In particular, the influence of simulation boxes defined by the near-densest lattice packing (NDLP) in conjunction with rotational constraints is compared to that of standard box types without these constraints. Three different proteins of varying size, shape, and secondary structure content were examined in the study. The statistical significance of differences in RMSD, radius of gyration, solvent-accessible surface, number of hydrogen bonds, and secondary structure content between proteins, box types, and the application or not of rotational constraints has been assessed. Furthermore, the differences in the collective modes for each protein between different boxes and the application or not of rotational constraints have been examined. In total 105 simulations were performed, and the results compared using a three-way multivariate analysis of variance (MANOVA) for properties derived from the trajectories and a three-way univariate analysis of variance (ANOVA) for collective modes. It is shown that application of roto-translational constraints does not have a statistically significant effect on the results obtained from the different simulations. However, the choice of simulation box was found to have a small (5-10%), but statistically significant effect on the behavior of two of the three proteins included in the study.     

Efficient implementation of periodic boundary conditions in Monte Carlo simulation

Determination of the shortest distances between particles is one of the most time-consuming parts of molecular simulation by the Monte Carlo method. In this work, we demonstrate that the use of signed-integer storage of coordinates in a scaled box allows one to skip multiple conditional statements in realization of periodic boundary conditions in cubic and rectangular boxes, which, in turn, increases the performance. Performance of the improved procedure was tested in NVT Monte Carlo simulations for liquid krypton and water. © 2018 Wiley Periodicals, Inc.     

FAMUSAMM: An algorithm for rapid evaluation of electrostatic interactions in molecular dynamics simulations

Within molecular dynamics simulations of protein–solvent systems the exact evaluation of long-range Coulomb interactions is computationally demanding and becomes prohibitive for large systems. Conventional truncation methods circumvent that computational problem, but are hampered by serious artifacts concerning structure and dynamics of the simulated systems. To avoid these artifacts we have developed an efficient and yet sufficiently accurate approximation scheme which combines the structure-adapted multipole method (SAMM) [C. Niedermeier and P. Tavan, J. Chem. Phys., 101 , 734 (1994)] with a multiple-time-step method. The computational effort for MD simulations required within our fast multiple-time-step structure-adapted multipole method (FAMUSAMM) scales linearly with the number of particles. For a system with 36,000 atoms we achieve a computational speed-up by a factor of 60 as compared with the exact evaluation of the Coulomb forces. Extended test simulations show that the applied approximations do not seriously affect structural or dynamical properties of the simulated systems. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1729–1749, 1997     

Computationally efficient canonical molecular dynamics simulations by using a multiple time‐step integrator algorithm combined with the particle mesh Ewald method and with the fast multipole method

An efficient implementation of the canonical molecular dynamics simulation using the reversible reference system propagator algorithm (r‐RESPA) combined with the particle mesh Ewald method (PMEM) and with the macroscopic expansion of the fast multipole method (MEFMM) was examined. The performance of the calculations was evaluated for systems with 3000, 9999, 30,000, 60,000, and 99,840 particles. For a given accuracy, the optimal conditions for minimizing the CPU time for the implementation of the Ewald method, the PMEM, and the MEFMM were first analyzed. Using the optimal conditions, we evaluated the performance and the reliability of the integrated methods. For all the systems examined, the r‐RESPA with the PMEM was about twice as fast as the r‐RESPA with the MEFMM. The difference arose from the difference in the numerical complexities of the fast Fourier transform in the PMEM and from the transformation of the multipole moments into the coefficients of the local field expansion in the MEFMM. Compared with conventional methods, such as the velocity‐verlet algorithm with the Ewald method, significant speedups were obtained by the integrated methods; the speedup of the calculation was a function of system size, and was a factor of 100 for a system with 3000 particles and increased to a factor of 700 for a system with 99,840 particles. These integrated calculations are, therefore, promising for realizing large‐scale molecular dynamics simulations for complex systems. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 201–217, 2000     

A new program for optimizing periodic boundary models of solvated biomolecules (PBCAID)

Simulations of solvated macromolecules often use periodic lattices to account for long-range electrostatics and to approximate the surface effects of bulk solvent. The large percentage of solvent molecules in such models (compared to macromolecular atoms) makes these procedures computationally expensive. The cost can be reduced by using periodic cells containing an optimized number of solvent molecules (subject to a minimal distance between the solute and the periodic images). We introduce an easy-to-use program "PBCAID" to initialize and optimize a periodic lattice specified as one of several known space-filling polyhedra. PBCAID reduces the volume of the periodic cell by finding the solute rotation that yields the smallest periodic cell dimensions. The algorithm examines rotations by using only a subset of surface atoms to measure solute/image distances, and by optimizing the distance between the solute and the periodic cell surface. Once the cell dimension is optimized, PBCAID incorporates a procedure for solvating the domain with water by filling the cell with a water lattice derived from an ice structure scaled to the bulk density of water. Results show that PBCAID can optimize system volumes by 20 to 70% and lead to computational savings in the nonbonded computations from reduced solvent sizes. Copyright 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1843-1850, 2001     

Stochastic dynamics simulation of alanine dipeptide: Including solvation interaction determined by boundary element method

A merger of the Poisson–Boltzmann equation and stochastic dynamics simulation is examined using illustrative calculations of alanine dipeptide. The boundary element method (BEM) is used to calculate the hydration forces acting on the solute molecule based on the surroundings. Computational efficiency is achieved by the use of a simple hydration model and a coarse boundary element. Nonetheless, the conformational distribution obtained from this new method is reasonable compared with other theoretical and computational results. Detailed analysis has been accomplished in terms of the hydration interactions and solvation energies. The results indicate that the new simulation method provides an obvious improvement over the conventional stochastic dynamics simulation technique. The further improvement of the hydration model and future application to large molecules are also discussed. © 1997 John Wiley & Sons, Inc. J Comput Chem 18 : 1440–1449, 1997     

Local response dispersion method in periodic systems: Implementation and assessment

We report the implementation of the local response dispersion (LRD) method in an electronic structure program package aimed at periodic systems and an assessment combined with the Perdew–Burke–Ernzerhof (PBE) functional and its revised version (revPBE). The real‐space numerical integration was implemented and performed exploiting the electron distribution given by the plane‐wave basis set. The dispersion‐corrected density functionals revPBE+LRD was found to be suitable for reproducing energetics, structures, and electron distributions in simple substances, molecular crystals, and physical adsorptions. © 2014 Wiley Periodicals, Inc.