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     

Fast summation boundary element method for calculating solvation free energies of macromolecules

We present a boundary element method (BEM) for calculating the reaction field energy of a macromolecule embedded in a high-dielectric medium such as water. In a BEM calculation, the key computational task is the calculation of the induced surface charge distribution at the dielectric boundary. This is obtained by solving a system of linear equations whose dimension can run into the tens of thousands for a macromolecule. In this work, we use a fast summation hierarchical multipole method to solve for the induced surface charge densities. By careful analysis of the levels of approximation required for the various terms in the calculation, we avoid the unnecessary computation of terms that contribute negligibly to the final outcome and, consequently, achieve high computational efficiency. For a protein such as BPTI with 890 atoms, the calculation of the induced surface charge density distribution and the reaction field energy was completed in 7.9 s on an SGI workstation with an R10000 CPU. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1494–1504, 1998     

Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation

A hybrid approach for solving the nonlinear Poisson-Boltzmann equation (PBE) is presented. Under this approach, the electrostatic potential is separated into (1) a linear component satisfying the linear PBE and solved using a fast boundary element method and (2) a correction term accounting for nonlinear effects and optionally, the presence of an ion-exclusion layer. Because the correction potential contains no singularities (in particular, it is smooth at charge sites) it can be accurately and efficiently solved using a finite difference method. The motivation for and formulation of such a decomposition are presented together with the numerical method for calculating the linear and correction potentials. For comparison, we also develop an integral equation representation of the solution to the nonlinear PBE. When implemented upon regular lattice grids, the hybrid scheme is found to outperform the integral equation method when treating nonlinear PBE problems. Results are presented for a spherical cavity containing a central charge, where the objective is to compare computed 1D nonlinear PBE solutions against ones obtained with alternate numerical solution methods. This is followed by examination of the electrostatic properties of nucleic acid structures.     

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     

A boundary element method for molecular electrostatics with electrolyte effects

A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.     

A simple yet accurate boundary element method for continuum dielectric calculations

A simple yet accurate method for calculating electrostatic potentials using the boundary element continuum dielectric method is presented. It is shown that the limiting factor in accuracy is not the evaluation of integrals involving the interaction between boundary elements but rather a proper estimation of the self-polarization of a patch upon itself. We derive a sum rule that allows us to calculate this important self-polarization term in a self-consistent and simple way. Intricate integration schemes used in previous treatments are consequently rendered unnecessary while concurrently achieving at least comparable accuracy over earlier methods. In some model systems for which analytic solutions are available, the computed surface polarization charge and reaction field energy are correct to better than six significant figures. An application of the method to the calculation of hydration free energies is presented. Good agreement with experimental values is obtained.     

SMART: A solvent-accessible triangulated surface generator for molecular graphics and boundary element applications

Summary An algorithm is presented for generating a representation of the solvent-accessible molecular surface as a smooth triangulated manifold. The algorithm, called SMART (SMooth moleculAR surface Triangulator), divides the contact and reentrant portions of the solvent-accessible molecular surface into curvilinear three-sided elements. In contrast to the author's earlier implementation of this general approach [Zauhar, R.J. and Morgan, R.S., J. Comput. Chem., 11 (1990) 603], the SMART algorithm defines elements directly on the appropriate geometric surface types (rather than using interpolation over cubic elements), and has special features to handle highly distorted regions which often appear in deep crevices and internal cavities. While the method is designed for use with boundary element techniques in continuum electrostatics, it can also be applied to the accurate computation of molecular surface areas and volumes, and the generation of shaded surfaces for display with interactive computer graphics. Availability: Programs (in C) for surface generation, area and volume computation are available from the author. Also available is a graphics display program which runs on Silicon Graphics workstations.     

A method for estimation of transport properties of ZSM-5 zeolite

A method for a quick testing of transport properties of samples of a zeolite catalyst has been developed. The method is based on the application of gas-phase chromatography, where the tested sample is used as the stationary phase. Cyclopentene, cyclohexane, and 1,3,5-trimethylcyclohexane were used as the testing molecules. A retention indexes for samples comparison were introduced. The method is sensitive even to small changes in the zeolite structures caused by binding agents and/or alkali metals.     

Ab initio MO study for electron transfer matrix element in hydrogen-bonded system

The electronic coupling matrix element of electron transfer between donor and acceptor connected with hydrogen bonds has been studied in a model system. The calculated matrix element depends largely on the relative rotational conformation of the electron-donor and electron-acceptor sites and a simple orbital analysis has been presented. Along the approximate proton transfer coordinate, the energy potential is a double well and the matrix element has a single maximum at the center of the double well.     

A fast and Space-efficient boundary element method for computing electrostatic and hydration effects in large molecules

At present, there are two widely used approaches for computing molecular hydration and electrostatic effects within the continuum approximation: the finite difference method, in which the electric potential is directly computed on a cubic grid, and the induced polarization charge or boundary element method, in which an induced charge distribution is first computed on the molecular surface and in which solvation effects are then calculated by reference to the reaction field arising from this induced surface charge. While the induced surface charge approach has a number of advantages over finite differences, especially in the computation of hydration forces and solvent stabilization, the applications of this technique have been largely restricted to small molecules. This is primarily due to the very large system of equations that results when the surface of a macromolecule is discretized into elements small enough to ensure an acceptable level of numerical accuracy within the continuum model. This article describes a new algorithm for implementing boundary element calculations within the continuum model. The essence of our approach is only to compute explicitly those interactions between surface elements that are relatively close together and to approximate long-range interactions by grid-based multipole expansion. The resulting system of equations has a relatively sparse coefficient matrix and requires disk storage that increases linearly with molecular surface area. The technique has numerous applications in the analysis of solvation effects in large molecules, especially in the area of conformational analysis, where it is critical to accurately estimate the global hydration energy for the entire structure. © 1996 by John Wiley & Sons, Inc.     

Effects of T-tubules on dielectric spectra of skeletal muscle simulated by boundary element method with two-dimensional models

In order to investigate the origin of large intensity the alpha-relaxation in skeletal muscles observed in dielectric measurements with extracellular electrode methods, effects of the interfacial polarization in the T-tubules on dielectric spectra were evaluated with the boundary-element method using two-dimensional models in which the structure of the T-tubules were represented explicitly. Each model consisted of a circular inclusion surrounded by a thin shell corresponding to the sarcolemma. The T-tubules were represented by simplified two types of invagination of the shell: straight invagination along the radial directions, and branched one. Each of the models was subjected to two kinds of calculations relevant to experiments with the extracellular and the intracellular electrode methods. Electrical interactions between the cells were omitted in the calculations. Both calculations showed that the dielectric spectra of the models contained two relaxation terms. The low-frequency relaxation term assigned to the alpha-relaxation depended on the structure of the T-tubules. Values of the relaxation frequency of the alpha-relaxation obtained from the two types of calculations agreed with each other. At the low-frequency limit, the permittivity obtained from the extracellular-electrode-type calculations varied in proportion to the capacitance obtained from the intracellular-electrode-type ones. These results were consistent with conventional lumped and distributed circuit models for the T-tubules. This confirms that the interfacial polarization in the T-tubules in a single muscle cell is not sufficient to explain the experimental results in which the intensity of the alpha-relaxation in the extracellular-electrode-type experiments exceeded the intensity expected from the results of the intracellular-electrode-type experiments. The high-frequency relaxation term that was assigned to the beta-relaxation was also affected by the T-tubule structure in the calculations relevant to the extracellular-electrode-type experiments.     

A method to correct defocused element distribution maps in electron probe microanalysis

Element distribution maps obtained on electron microprobes via the beam scan method with wavelength-dispersive spectrometers reveal a defocusing effect if they are taken at sufficiently small magnification. This effect, which occurs where the Bragg condition of the spectrometer is not adequately met, can be avoided or corrected by various methods. A method is presented here to correct defocused element distribution maps with the help of corresponding maps obtained on homogeneous standards.Dedicated to Professor Dr. rer. nat. Dr. h.c. Hubertus Nickel on the occasion of his 65th birthday     

A simple and precise linear integral method for isoconversional data

A simple and precise linear integral method to evaluate the activation energy dependence on the extent of conversion has been proposed. The method leads to consistent results with those from differential and integral non-linear procedure (Vyazovkin method). Moreover, the new procedure yields the pre-exponential factor and the kinetic model. The method was evaluated from isothermal, non-isothermal and non-linear non-isothermal data (CRTA).     

A new parallel method for molecular dynamics simulation of macromolecular systems

Short-range molecular dynamics simulations of molecular systems are commonly parallelized by replicated-data methods, in which each processor stores a copy of all atom positions. This enables computation of bonded 2-, 3-, and 4-body forces within the molecular topology to be partitioned among processors straightforwardly. A drawback to such methods is that the interprocessor communication scales as N (the number of atoms) independent of P (the number of processors). Thus, their parallel efficiency falls off rapidly when large numbers of processors are used. In this article a new parallel method for simulating macromolecular or small-molecule systems is presented, called force-decomposition. Its memory and communication costs scale as N/√P, allowing larger problems to be run faster on greater numbers of processors. Like replicated-data techniques, and in contrast to spatial-decomposition approaches, the new method can be simply load balanced and performs well even for irregular simulation geometries. The implementation of the algorithm in a prototypical macromolecular simulation code ParBond is also discussed. On a 1024-processor Intel Paragon, ParBond runs a standard benchmark simulation of solvated myoglobin with a parallel efficiency of 61% and at 40 times the speed of a vectorized version of CHARMM running on a single Cray Y-MP processor. © 1996 by John Wiley & Sons, Inc.     

A new method of evaluation of element pollutant mobility in sediments

The application of a strong chelating agent for the screening test of element mobility in sedimentary systems was investigated. Single-step and sequential extraction procedures were applied to four sediment samples collected from an industrially polluted region of Eastern Slovakia. A sequential extraction procedure (SEP), recommended by the Institute for Reference Materials and Measurements (IRMM), was applied and used as a reference extraction method. A single-step extraction with 0.05 mol dm⁻³ ethylenediaminetetraacetic acid (EDTA) was adapted for sediments when extraction conditions were optimised. The extraction efficiency of the single-step procedure was compared with that of SEP. The contents of elements extracted by Na2EDTA were in good agreement with the sum of the first three steps of the SEP for Fe, Mn, and Co. Na₂EDTA can therefore be considered capable to extract the majority of elements associated with the reducible sedimentary phase — bound to Fe and Mn oxides in the regional geological conditions of the monitored region. Thus, Na₂EDTA extraction of Fe and Mn can serve as an economical, time-saving supplementary test for the IRMM procedure. This paper is dedicated to the memory of Erika Krakovská Presented at the XVIIIth Slovak Spectroscopic Conference, Spišská Nová Ves, 15–18 October 2006.     

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.     

A flexible triangulation method to describe the solvent-accessible surface of biopolymers

A relatively simple protein solvent-accessible surface triangulation method for continuum electrostatics applications employing the boundary element method is presented. First, the protein is placed onto a three-dimensional lattice with a specified lattice spacing. To each lattice point, a box is assigned. Boxes located in the solvent region and in the interior of the protein are removed from the set. Improper connections between boxes and the possible existence of cavities in the interior of the protein which would destroy the proper connectivity of the triangulated surface are taken care of. The remaining set of boxes define the outer contour of the protein. Each free face exposed to the solvent of the remaining set of boxes is triangulated after the surface defined by the free faces has been smoothed to follow the shape of the macromolecule more accurately. The final step consists of a mapping of triangle vertices onto a set of surface points which define the solvent-accessible surface. Normal vectors at triangle vertices are obtained also from the free faces which define the orientation of the surface. The algorithm was tested for six molecules including four proteins; a dipeptide, a helical peptide consisting of 25 residues, calbindin, lysozyme, calmodulin and cutinase. For each molecule, total areas have been calculated and compared with the result computed from a dotted solvent-accessible surface. Since the boundary element method requires a low number of vertices and triangles to reduce the number of unknowns for reasons of efficiency, the number of triangles should not be too high. Nevertheless, credible results are obtained for the total area with relative errors not exceeding 12% for a large lattice spacing (0.30 nm) while close to zero for a smaller lattice spacing (down to 0.16 nm). The output of the triangulation computer program (written in C++) is rather simple so that it can be easily converted to any format acceptable for any molecular graphics programs.     

Intrinsic thermal stability and quenching recovery of thin-film superconductors with thermal boundary resistance

The stability behaviour of a thin-film superconductor under a localized release of thermal disturbance is investigated. Two-dimensional conjugate film/substrate conduction equation with anisotropic thermal conductivity of the film, and Joule heat are employed to investigate effects of substrate and thermal properties on the intrinsic stability and quenching recovery. To consider the thermal boundary resistance between film and substrate, an interfacial-layer model (ILM) with very low diffusivity and an acoustic mismatch model (AMM) are employed. Results show that the thermal boundary resistance influences strongly the intrinsic stability. Thermal boundary resistance increases intrinsic stability if the thermal conductivity of the substrate or the disturbance energy is large. Higher Biot numbers and thermal conductivity ratios of film to substrate in longitudinal direction influence stability favorably. We demonstrate also that operation of a film/substrate system, such as YBCO/MgO, is either intrinsically stable or irrecoverably unstable.The authors wish to express their sincere appreciation to Dr. R. C. Chen for his invaluable advice and suggestions during the course of this paper. This research was supported by the National Science Council of the R. O. C. through grant NSC 83-0401-E-009-006. The computations were performed on the IBM ES/9000 at the National Center For High-Performance Computing.     

Modification of the direct moving boundary method for the determination of transference numbers for concentrated solutions

The upper concentration limit for the direct moving boundary method for measuring transference numbers can be appreciably extended by introducing three new concepts. First, the closed side of the cell must contain an electrolyte whose apparent molar volume changes relatively little with concentration. Second, the closed electrode chamber must be stirred to avoid the build-up of regions of differing concentration. Third, the solution in this chamber should be pre-saturated with the insoluble salt involved in the electrode reaction. These three modifications allow the large volume correction for concentrated solutions to be calculated much more exactly. The new technique was tested with fair success for 3M KCl in water at 25°C using the KCl LiCl system. The high current dependence of the transference number was mainly attributed to the Soret effect at the boundary.     

The fast multipole boundary element method for molecular electrostatics: An optimal approach for large systems

We propose a fast implementation of the boundary element method for solving the Poisson equation, which approximately determines the electrostatic field around solvated molecules of arbitrary shape. The method presented uses computational resources of order O(N) only, where N is the number of elements representing the dielectric boundary at the molecular surface. The method is based on the Fast Multipole Algorithm by Rokhlin and Greengard, which is used to calculate the Coulombic interaction between surface elements in linear time. We calculate the solvation energies of a sphere, a small polar molecule, and a moderately sized protein. The values obtained by the boundary element method agree well with results from finite difference calculations and show a higher degree of consistency due to the absence of grid dependencies. The boundary element method can be taken to a much higher accuracy than is possible with finite difference methods and can therefore be used to verify their validity. © 1995 by John Wiley & Sons, Inc.