Performance of fast multipole methods for calculating electrostatic interactions in biomacromolecular simulations

The fast multipole method proposed by Greengard and Rokhlin (GR) is applied to large biomacromolecular systems. In this method, the system is divided into a hierarchy of cells, and electric field exerted on a particle is decomposed into two parts. The first part is a rapidly varying field due to nearby cells, so that it needs rigorous pairwise calculations. The second part is a slowly varying local field due to distant cells; hence, it allows rapid calculations through a multipole expansion technique. In this work, two additional possibilities for improving the performance are numerically examined. The first is an improvement of the convergence of the expansion by increasing the number of nearby cells, without including higher-order multipole moments. The second is an acceleration of the calculations by the particle–particle and particle–mesh/multipole expansion (PPPM/MPE) method, which uses fast Fourier transform instead of the hierarchy. For this purpose, the PPPM/MPE method originally developed by the authors for a periodic system is extended to a nonperiodic isolated system. The advantages and disadvantages of the GR and PPPM/MPE methods are discussed for both periodic and isolated systems. It is numerically shown that these methods with reasonable costs can reduce the error in potential felt by each particle to 0.1–1 kcal/mol, much smaller than the 30-kcal/mol error involved in conventional simple truncations. © 1994 by John Wiley & Sons, Inc.     

Compact method for calculating intermolecular interactions

The development of efficient methods for calculating intermolecular interactions (which are responsible for the existence of stable molecular associates, solvation shells, etc.) is a pressing problem of quantum chemistry. We propose a new method for correct calculations of intermolecular interactions, which is based on the solution of SCF equations with fractional occupation numbers. Calculating intermolecular interactions by this method does not require the use of exchange potentials in an explicit form. The method is intended primarily to describe the charge transfer between interacting subsystems. The calculations by this method are compact since the dimensions of matrix problems remain unchanged in the course of the numerical procedure. V. I. Vernadskii Institute of Geochemistry and Analytical Chemistry, Russian Academy of Sciences. Translated fromZhurnal Strukturnoi Khimii, Vol. 36, No. 3, pp. 401–405, May–June, 1995. Translated by I. Izvekova     

A finite field method for calculating molecular polarizability tensors for arbitrary multipole rank

A finite field method for calculating spherical tensor molecular polarizability tensors α_lm;l′m′ = ?Δ_lm/??_l′m′* by numerical derivatives of induced molecular multipole Δ_lm with respect to gradients of electrostatic potential ?_l′m′* is described for arbitrary multipole ranks l and l′. Interconversion formulae for transforming multipole moments and polarizability tensors between spherical and traceless Cartesian tensor conventions are derived. As an example, molecular polarizability tensors up to the hexadecapole–hexadecapole level are calculated for water using the following ab initio methods: Hartree–Fock (HF), Becke three‐parameter Lee‐Yang‐Parr exchange‐correlation functional (B3LYP), Møller–Plesset perturbation theory up to second order (MP2), and Coupled Cluster theory with single and double excitations (CCSD). In addition, intermolecular electrostatic and polarization energies calculated by molecular multipoles and polarizability tensors are compared with ab initio reference values calculated by the Reduced Variation Space method for several randomly oriented small molecule dimers separated by a large distance. It is discussed how higher order molecular polarizability tensors can be used as a tool for testing and developing new polarization models for future force fields. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Variational method for calculating molecular multipole polarizabilities using atom centered extended trial functions

An improvement of the variational method for calculating the electronic multipole polarizabilities is proposed. This modification allows the computation of the polarizabilities at any point and the results are compatible with the relations existing for a change of origin. It is applied to H₂, HF, CO and N₂ by using SCF wavefunctions developed on a limited basis. The computed polarizabilities are systematically too large but this discrepancy is attributed to the fact that the ground state is too far from the exact wavefunction.E.R.A. au C.N.R.S. No. 22.     

Electro-osmotic flows in a microchannel with patterned hydrodynamic slip walls

We present an analysis of the electro-osmotic flow of electrolytic solutions in a microchannel with patterned hydrodynamic slippage on channel walls. A set of governing equations is formulated to account for the effects of small variations in hydrodynamic slippage over the microchannel walls on the electro-osmotic flow. These equations are then solved analytically by using the perturbation method. Two frequently encountered surface patterns, (i) cosine wave variation and (ii) square wave variation in slip length, are considered in our analyses. The results show that patterned slippage over microchannel walls can induce complex flow patterns (such as vortical flows) in otherwise plug-like electro-osmotic flows, which suggests potential applications of such flows in microfluidic mixers.     

MMM1D: a method for calculating electrostatic interactions in one-dimensional periodic geometries

We present a new method to accurately calculate the electrostatic energy and forces on charges in a system with periodic boundary conditions in one of three spatial dimensions. We transform the Coulomb sum via a convergence factor into a series of fast decaying functions similar to the Lekner method. Rigorous error bounds for the energies and the forces are derived and numerically verified. The method has a computational complexity of O(N(2)), but is faster and easier to use than previously reported methods.     

An approximate method for calculating depletion and structural interactions between colloidal particles

An approximate method is evaluated for calculating the second virial coefficient in a dilute macromolecular solution bounded by two interfaces. The approximation is essentially the superposition of the coefficients calculated independently for each surface. To test this approach, the depletion interaction between two particles in a solution of nonadsorbing, spherical macromolecules is calculated in systems with either hard-wall or long-range electrostatic interactions. In all systems tested, the interparticle interaction energy calculated with the approximation is found to be in good agreement with that calculated using the exact approach (e.g., error less than 2% at the smallest separations). The primary advantage of this approximation is a significant reduction in the computation time required for calculating the depletion interaction, especially in charged systems. The paper also shows that the expressions for predicting the depletion interaction in purely hard-sphere systems can be used in dilute ionic systems, provided the appropriate effective macromolecule size is used. For the attractive depletion interaction, this effective size is determined by the range of the particle-macromolecule interaction (as opposed to the macromolecule-macromolecule interaction).     

Mechanism of hydrodynamic separation of biological objects in microchannel devices

In this study, the separation mechanism employed in hydrodynamic chromatography in microchannel devices is analyzed. The main purpose of this work is to provide a methodology to develop a predictive model for hydrodynamic chromatography for biological macromolecules in microchannels and to assess the importance of various phenomenological coefficients. A theoretical model for the hydrodynamic chromatography of particles in a microchannel is investigated herein. A fully developed concentration profile for non-reactive particles in a microchannel was obtained to elucidate the hydrodynamic chromatography of these particles. The external forces acting on the particles considered in this model include the van der Waals attractive force, double-layer force as well as the gravitational force. The surface forces, such as van der Waals attractive force as well as the double-layer repulsive force, can either enhance or hinder the average velocity of the macromolecular particles. The average velocity of the particles decreases with the molecular radius because the van der Waals attractive force increases the concentration of the particles near the channel surface, which is the low-velocity region. The transport velocity of the particles is dominated by the gravity and the higher density enlarges the effect caused by gravity.     

A method for DNA detection in a microchannel: fluid dynamics phenomena and optimization of microchannel structure

A simple DNA diagnosis method using microfluidics has been developed which requires simple and straightforward procedures such as injection of sample and probe DNA solutions. This method takes advantage of the highly accurate control of fluids in microchannels, and is superior to DNA microarray diagnosis methods due to its simplicity, highly quantitative determination, and high-sensitivity. The method is capable of detecting DNA hybridization for molecules as small as a 20 mer. This suggests the difference in microfluidic behavior between single strand DNA (ssDNA) and double stranded DNA (dsDNA). In this work, influence of both the inertial force exerted on DNA molecules and the diffusion of DNA molecules was investigated. Based on the determination of these parameters for both ssDNA and dsDNA by experiments, a numerical model describing the phenomena in the microchannel was designed. Computational simulation results using this model were in good agreement with previously reported experimental results. The simulation results showed that appropriate selection of the analysis point and the design of microchannel structure are important to bring out the diffusion and inertial force effects suitably and increase the sensitivity of the detection of DNA hybridization, that is, the analytical performance of the microfluidic DNA chip.     

Thermal and hydrodynamic performance of a microchannel heat sink with carbon nanotube nanofluids

Journal of Thermal Analysis and Calorimetry - The method of evaluating the magnetocaloric effect (MCE) and heat capacity of various magnetic samples from calorimetric experiments are presented....     

A convergent multipole expansion for 1,3 and 1,4 Coulomb interactions

Traditionally force fields express 1,3 and 1,4 interactions as bonded terms via potentials that involve valence and torsion angles, respectively. These interactions are not modeled by point charge terms, which are confined to electrostatic interactions between more distant atoms (1,n where n>4). Here we show that both 1,3 and 1,4 interactions can be described on the same footing as 1,n (n>4) interactions by a convergent multipole expansion of the Coulomb energy of the participating atom pairs. The atomic multipole moments are generated by the theory of quantum chemical topology. The procedure to make the multipole expansion convergent is based on a "shift procedure" described in earlier work [L. Joubert and P. L. A. Popelier, Molec. Phys. 100, 3357 (2002)].     

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.     

Outer shell ionization potentials for ethane by a many-body Green's function method

A calculation based upon the many-body Green's function method is employed to obtain the outer shell vertical ionization potentials of the ethane molecule. An extended basis set is employed to represent the approximate optical potential, derived by the functional derivative approach, as well as the one-electron Green's function. The results obtained confirm a²E_g state for the ion arising from the first ionization process. Supported by a fellowship of the Scuola Normale Superiore.     

Force constants and many-body electrostatic interactions in two-dimensional colloidal crystal

A model of a two-dimensional colloidal crystal with a hexagonal lattice, the electrostatic interactions in which are described by the nonlinear Poisson-Boltzmann equation, is considered. The calculation procedure for force constants of this crystal is treated in detail. Properties of system symmetry, which make it possible to significantly decrease the volume of calculations and to classify force constants, are analyzed. Numerical data for force constants of a crystal as functions of lattice parameters at different particle sizes are reported. A method that allows us to disclose the presence of many-body interactions in a system by the behavior of force constants at some interval of the values of lattice parameters is proposed. The application of this method to the system under consideration demonstrated that electrostatic interparticle interactions in the system cannot be reduced to simply a pair interaction of any kind; the introduction of many-body potentials is required for the adequate representation of the elastic properties of a crystal.     

Krylov subspace methods for computing hydrodynamic interactions in Brownian dynamics simulations

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studies methods based on Krylov subspaces for computing Brownian noise vectors. These methods are related to Chebyshev polynomial approximations, but do not require eigenvalue estimates. We show that only low accuracy is required in the Brownian noise vectors to accurately compute values of dynamic and static properties of polymer and monodisperse suspension models. With this level of accuracy, the computational time of Krylov subspace methods scales very nearly as O(N(2)) for the number of particles N up to 10 000, which was the limit tested. The performance of the Krylov subspace methods, especially the "block" version, is slightly better than that of the Chebyshev method, even without taking into account the additional cost of eigenvalue estimates required by the latter. Furthermore, at N = 10 000, the Krylov subspace method is 13 times faster than the exact Cholesky method. Thus, Krylov subspace methods are recommended for performing large-scale Brownian dynamics simulations with hydrodynamic interactions.     

Towards a mesoscopic model of water-like fluids with hydrodynamic interactions

We present a mesoscopic lattice model for non-ideal fluid flows with directional interactions, mimicking the effects of hydrogen bonds in water. The model supports a rich and complex structural dynamics of the orientational order parameter, and exhibits the formation of disordered domains whose size and shape depend on the relative strength of directional order and thermal diffusivity. By letting the directional forces carry an inverse density dependence, the model is able to display a correlation between ordered domains and low density regions, reflecting the idea of water as a denser liquid in the disordered state than in the ordered one.     

Parallel implementation of a direct method for calculating electrostatic potentials

The authors present a method for calculating the electrostatic potential directly in a straightforward manner. While traditional methods for calculating the electrostatic potential usually involve solving the Poisson equation iteratively, the authors obtain the electrostatic interaction potential by performing direct numerical integration of the Coulomb-law expression using finite-element functions defined on a grid. The singularity of the Coulomb operator is circumvented by an integral transformation and the resulting auxiliary integral is obtained using Gaussian quadrature. The three-dimensional finite-element basis is constructed as a tensor (outer) product of one-dimensional functions, yielding a partial factorization of the expressions. The resulting algorithm has, without using any prescreening or other computational tricks, a formal computational scaling of Omicron(N4/3), where N is the size of the grid. The authors show here how to implement the method for efficiently running on parallel computers. The matrix multiplications of the innermost loops are completely independent, yielding a parallel algorithm with the computational costs scaling practically linearly with the number of processors.