Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T20:33:58.725Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Optimization of a Walk-on-Spheres Solver for the Linear Poisson-Boltzmann Equation

Published online by Cambridge University Press:  03 June 2015

Travis Mackoy ,
Robert C. Harris ,
Jesse Johnson ,
Michael Mascagni  and
Marcia O. Fenley
Travis Mackoy*
Affiliation:
Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306, USA
Robert C. Harris*
Affiliation:
Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Jesse Johnson*
Affiliation:
Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA
Michael Mascagni*
Affiliation:
Departments of Computer Science, Mathematics and Scientific Computing, Florida State University, Tallahassee, FL 32306, USA
Marcia O. Fenley*
Affiliation:
Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306, USA
*
Email:mackoy@cs.fsu.edu
Email:rch02c@fsu.edu
Email:johnson@biomaps.rutgers.edu
Email:mascagni@fsu.edu
Corresponding author.Email:mfenley@sb.fsu.edu
Get access

Abstract

Stochastic walk-on-spheres (WOS) algorithms for solving the linearized Poisson-Boltzmann equation (LPBE) provide several attractive features not available in traditional deterministic solvers: Gaussian error bars can be computed easily, the algorithm is readily parallelized and requires minimal memory and multiple solvent environments can be accounted for by reweighting trajectories. However, previously-reported computational times of these Monte Carlo methods were not competitive with existing deterministic numerical methods. The present paper demonstrates a series of numerical optimizations that collectively make the computational time of these Monte Carlo LPBE solvers competitive with deterministic methods. The optimization techniques used are to ensure that each atom’s contribution to the variance of the electrostatic solvation free energy is the same, to optimize the bias-generating parameters in the algorithm and to use an epsilon-approximate rather than exact nearest-neighbor search when determining the size of the next step in the Brownian motion when outside the molecule.

Keywords

02.70.Uu41.20Cv05.40.FbPoisson-Boltzmann equationwalk-on-spheressolvation
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 1 , January 2013 , pp. 195 - 206
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Lu, B. Z., Zhou, Y. C., Holst, M. J. and McCammon, J. A., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Commun. Comput. Phys., 3(2008), 973–1009.Google Scholar
[2]Lamm, G., The Poisson-Boltzmann equation, Rev. Comput. Chem., 19 (2003), 147–365.Google Scholar
[3]Baker, N. A., Sept, D., Joseph, S., Holst, M. J. and McCammon, J. A., Electrostatics of nanosystems: application to microtubules and the ribosome, Proc. Natl. Acad. Sci. USA, 98 (2001), 10037–10041.Google Scholar
[4]Fogolari, F., Zuccato, P., Esposito, G. and Viglino, P., Biomolecular electrostatics with the linearized Poisson-Boltzmann equation, Biophys. J., 76 (1999), 1–16.Google Scholar
[5]Davis, M. E. and McCammon, J. A., Solving the finite difference linearized Poisson-Boltzmann equation: a comparison of relaxation and conjugate gradient methods, J. Comput. Chem., 10 (1989), 386–391.Google Scholar
[6]Nicholls, A. and Honig, B., A rapid finite difference algorithm, utilizing successive overrelaxation to solve the Poisson-Boltzmann equation, J. Comput. Chem., 12 (1991), 435–445.Google Scholar
[7]Geng, W., Yu, S. and Wei, G., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127 (2007), 114106.Google Scholar
[8]Rocchia, W., Sridharan, S., Nicholls, A., Alexov, E., Chiabrera, A. and Honig, B., Rapid grid-based construction of the molecular surface for both Molecules and geometric objects: applications to the finite difference Poisson-Boltzmann method, J. Comput. Chem., 23 (2002), 128–137.Google Scholar
[9]Davis, M. E., Madura, J. D., Luty, B. A. and McCammon, J. A., Electrostatics and diffusion of molecules in solution: simulations with the University of Houston Brownian dynamics program, Comput. Phys. Commun., 62 (1991), 187–197.Google Scholar
[10]Bashford, D., Scientific Computing in Object-Oriented Parallel Environments, Ishikawa, Y., Oldehoeft, R. R., Reynders, J. and Tholburn, V. W. M., Springer Berlin Heidelberg, 1343 (1997), 233–240.Google Scholar
[11]Holst, M. and Saied, F., Multigrid solution of the Poisson-Boltzmann equation, J. Comput. Chem., 14 (1993), 105–113.Google Scholar
[12]Holst, M., Baker, N. and Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equations I: algorithms and examples, J. Comput. Chem., 21 (2000), 1319–1342.Google Scholar
[13]Cortis, C. M. and Friesner, R. A., Numerical solution of the Poisson-Boltzmann equation using tetrahedral finite-element meshes, J. Comput. Chem., 18 (1997), 1591–1608.Google Scholar
[14]Yoon, B. J. and Lenhoff, A. M., A boundary element method for molecular electrostatics with electrolyte effects, J. Comput. Chem., 11 (1990), 1080–1086.Google Scholar
[15]Boschitsch, A. H., Fenley, M. O. and Zhou, H.-X., Fast boundary element method for the linear Poisson-Boltzmann equation, J. Phys. Chem. B, 106 (2002), 2741–2754.Google Scholar
[16]Chipman, D. M., Solution of the linearized Poisson-Boltzmann equation, J. Chem. Phys., 120 (2004), 5566–5575.CrossRefGoogle ScholarPubMed
[17]Altman, M. D., Bardhan, J. P., White, J. K. and Tidor, B., Accurate solution of multi-region continuum electrostatic problems using the linearized Poisson-Boltzmann equation and curved boundary elements, J. Comput. Chem., 30 (2009), 132–153.Google Scholar
[18]Yap, E.-H. and Head-Gordon, T., A new and efficient Poisson-Boltzmann solver for interaction of multiple proteins, J. Chem. Theory Comput., 6 (2010), 2214–2224.CrossRefGoogle ScholarPubMed
[19]Juffer, A., F, E. F.Botta, B.van Keulen, A. M., van der Ploeg, A. and Berendsen, H. J. C., The electric potential of a macromolecule in a solvent: a fundamental approach, J. Comput. Phys., 97 (1991), 144–171.Google Scholar
[20]Simonov, N. A., Doklady Mathematics, Springer, 656–659.Google Scholar
[21]Simonov, N. A., Mascagni, M. and Fenley, M. O., Monte Carlo-based linear Poisson-Boltzmann approach makes accurate salt-dependent solvation free energy predictions possible, J. Chem. Phys., 127 (2007), 185105.Google Scholar
[22]Bossy, M., Champagnat, N., Maire, S. and Talay, D., Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in molecular dynamics, ESAIM. Math. Model. Numer. Anal., 44 (2010), 997–1048.Google Scholar
[23]Fenley, M. O., Mascagni, M., McClain, J., Silalahi, A. R. J. and Simonov, N. A., Using correlated Monte Carlo sampling for efficiently solving the linearized Poisson-Boltzmann equation over a broad range of salt concentration, J. Chem. Theory Comput., 6 (2009), 300–314.Google Scholar
[24]Mascagni, M. and Simonov, N. A., Monte Carlo methods for calculating some physical properties of large molecules, SIAM J. Sci. Comput., 26 (2005), 339.CrossRefGoogle Scholar
[25]Tjong, H. and Zhou, H.-X., GBr6: a parameterization-free, accurate, analytical generalized Born method, J. Phys. Chem. B, 111 (2007), 3055–3061.Google Scholar
[26]Berman, H. M., Bhat, T. N., Bourne, P. E., Feng, Z., Gilliland, G., Weissig, H. and Westbrook, J., The protein data bank and the challenge of structural genomics, Nat. Struct. Mol. Biol., 7 (2000), 957–959.Google Scholar
[27]Cornell, W. D., Cieplak, P., Bayly, C. I., Gould, I. R., Merz, K. M. Jr., Ferguson, D. M., Spellmeyer, D. C., Fox, T., Caldwell, J. W. and Kollman, P. A., A second generation force field for the simulation of proteins, nucleic acids, and organic molecules, J. Am. Chem. Soc., 117 (1995), 5179–5197.CrossRefGoogle Scholar
[28]Bondi, A., van der Waals Volumes and Radii, J. Phys. Chem., 68 (1964), 441–451.CrossRefGoogle Scholar
[29]Boschitsch, A. H. and Fenley, M. O., A fast and robust Poisson-Boltzmann solver based on adaptive cartesian grids, J. Chem. Theory Comput., 7 (2011), 1524–1540.Google Scholar
[30]Rasulov, A., Karaivanova, A. and Mascagni, M., Quasirandom sequences in branching random walks, Monte Carlo Methods Appl., 10 (2004), 551–558.Google Scholar
[31]Jackson, J. D., Classical Electrodynamics Third Edition, Wiley, 1998.Google Scholar
[32]Qin, S. and Zhou, H.-X., Do electrostatic interactions destabilize protein-nucleic acid binding?, Biopolymers, 86 (2007), 112–118.Google Scholar
[33]Connolly, M. L., Solvent-accessible surfaces of proteins and nucleic acids, Science, 221 (1983), 709–713.Google Scholar
[34]Mount, D. M. and Arya, S., ANN: A library for approximate nearest neighbor searching, Proc. Center for Geometric Computing Second Ann. Fall Workshop Computational Geometry, 1997.Google Scholar