Fast Boundary Element Method for the Linear Poisson−Boltzmann Equation

Boschitsch, Alexander H.; Fenley, Márcia O.; Zhou, Huan‐Xiang

doi:10.1021/jp013607q

Cited by 157 publications

(176 citation statements)

References 65 publications

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“…By proper coupling of single and double layer potentials as discussed by Rokhlin (7), we derive an integral equation formulation for systems with an arbitrary number of domains (molecules). Similar formulations have been used for single-domain problems by Juffer et al (8), Liang and Subramaniam (9), and Boschitsch et al (10). Compared with ''direct'' formulations, the condition number of our system does not increase with the number of unknowns, hence the number of iterations in the Krylov subspace-based methods is bounded.…”

mentioning

confidence: 89%

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Cheng

Huang

et al. 2006

Proc. Natl. Acad. Sci. U.S.A.

141

158

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Poisson-Boltzmann electrostatics is a well established model in biophysics; however, its application to large-scale biomolecular processes such as protein-protein encounter is still limited by the efficiency and memory constraints of existing numerical techniques. In this article, we present an efficient and accurate scheme that incorporates recently developed numerical techniques to enhance our computational ability. In particular, a boundary integral equation approach is applied to discretize the linearized Poisson-Boltzmann equation; the resulting integral formulas are well conditioned and are extended to systems with arbitrary numbers of biomolecules. The solution process is accelerated by Krylov subspace methods and a new version of the fast multipole method. In addition to the electrostatic energy, fast calculations of the forces and torques are made possible by using an interpolation procedure. Numerical experiments show that the implemented algorithm is asymptotically optimal O(N) in both CPU time and required memory, and application to the acetylcholinesterasefasciculin complex is illustrated. In recent years, because of the rapid advances in biotechnology, both the temporal and spatial scales of biomolecular studies have been increased significantly: from single molecules to interacting molecular networks in a cell, and from the static molecular structures at different resolutions to the dynamical interactions in biophysical processes. In these studies, the electrostatics modeled by the well established Poisson-Boltzmann (PB) equation has been shown to play an important role under physiological solution conditions. Therefore, its accurate and efficient numerical treatment becomes extremely important, especially in the study of large-scale dynamical processes such as protein-protein association and dissociation in which the PB equation has to be solved repetitively during a simulation.Traditional numerical schemes for PB electrostatics include the finite difference methods, where difference approximations are used on structured grids, and finite element methods in which arbitrarily shaped biomolecules are discretized by using elements and the associated basis functions. The resulting algebraic systems for both are commonly solved by using multigrid or domain decomposition accelerations for optimal efficiency. However, as the grid number (and thus the storage, number of operations, and condition number of the system) increases proportionally to the volume size, finite difference and finite element methods become less efficient and accurate for systems with large spatial sizes, e.g., as encountered in protein association and dissociation. Alternative methods include the boundary element method (BEM) and the boundary integral equation (BIE) method. In these methods, only the surfaces of the molecules are discretized; hence, the number of unknowns is greatly reduced. Unfortunately, in earlier versions of BEM, the matrix is stored explicitly and the resulting dense linear system is solved by using Gauss eli...

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mentioning

confidence: 89%

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Cheng

Huang

et al. 2006

Proc. Natl. Acad. Sci. U.S.A.

141

158

View full text Add to dashboard Cite

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“…Recently the boundary element method has been combined with fast multipole methods, resulting in better scaling than finite difference methods with increasing system size. 28,29 Once the electrostatic potential is known, the electrostatic energy is given by…”

Section: Continuum Solvent Description Based On Poisson Theorymentioning

confidence: 99%

Performance comparison of generalized born and Poisson methods in the calculation of electrostatic solvation energies for protein structures

et al. 2003

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This study compares generalized Born (GB) and Poisson (PB) methods for calculating electrostatic solvation energies of proteins. A large set of GB and PB implementations from our own laboratories as well as others is applied to a series of protein structure test sets for evaluating the performance of these methods. The test sets cover a significant range of native protein structures of varying size, fold topology, and amino acid composition as well as nonnative extended and misfolded structures that may be found during structure prediction and folding/unfolding studies. We find that the methods tested here span a wide range from highly accurate and computationally demanding PB-based methods to somewhat less accurate but more affordable GB-based approaches and a few fast, approximate PB solvers. Compared with PB solvation energies, the latest, most accurate GB implementations were found to achieve errors of 1% for relative solvation energies between different proteins and 0.4% between different conformations of the same protein. This compares to accurate PB solvers that produce results with deviations of less than 0.25% between each other for both native and nonnative structures. The performance of the best GB methods is discussed in more detail for the application for force field-based minimizations or molecular dynamics simulations.

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“…[13][14][15][16] A number of algorithms are available to solve the PB equation numerically, including finite-difference, [17][18][19] finiteelement, [20][21][22][23] and boundary-element methods. [24][25][26][27] The main motivation behind the development of these models is to reduce the computational cost. 28,15 With water molecules integrated out, implicit solvents present far fewer degrees of freedom in computer simulations than the explicit solvents, thus bringing down the associated computational cost.…”

Section: Introductionmentioning

confidence: 99%

An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions

Lin

Baumketner

Deng

et al. 2009

The Journal of Chemical Physics

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In this paper, a new solvation model is proposed for simulations of biomolecules in aqueous solutions that combines the strengths of explicit and implicit solvent representations. Solute molecules are placed in a spherical cavity filled with explicit water, thus providing microscopic detail where it is most needed. Solvent outside of the cavity is modeled as a dielectric continuum whose effect on the solute is treated through the reaction field corrections. With this explicit/implicit model, the electrostatic potential represents a solute molecule in an infinite bath of solvent, thus avoiding unphysical interactions between periodic images of the solute commonly used in the lattice-sum explicit solvent simulations. For improved computational efficiency, our model employs an accurate and efficient multiple-image charge method to compute reaction fields together with the fast multipole method for the direct Coulomb interactions. To minimize the surface effects, periodic boundary conditions are employed for nonelectrostatic interactions. The proposed model is applied to study liquid water. The effect of model parameters, which include the size of the cavity, the number of image charges used to compute reaction field, and the thickness of the buffer layer, is investigated in comparison with the particle-mesh Ewald simulations as a reference. An optimal set of parameters is obtained that allows for a faithful representation of many structural, dielectric, and dynamic properties of the simulated water, while maintaining manageable computational cost. With controlled and adjustable accuracy of the multiple-image charge representation of the reaction field, it is concluded that the employed model achieves convergence with only one image charge in the case of pure water. Future applications to pKa calculations, conformational sampling of solvated biomolecules and electrolyte solutions are briefly discussed.

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Fast Boundary Element Method for the Linear Poisson−Boltzmann Equation

Cited by 157 publications

References 65 publications

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Order N algorithm for computation of electrostatic interactions in biomolecular systems

Performance comparison of generalized born and Poisson methods in the calculation of electrostatic solvation energies for protein structures

An image-based reaction field method for electrostatic interactions in molecular dynamics simulations of aqueous solutions

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