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A treecode-accelerated boundary integral Poisson-Boltzmann solver for electrostatics of solvated biomolecules. (English) Zbl 1349.78084
Summary: We present a treecode-accelerated boundary integral (TABI) solver for electrostatics of solvated biomolecules described by the linear Poisson-Boltzmann equation. The method employs a well-conditioned boundary integral formulation for the electrostatic potential and its normal derivative on the molecular surface. The surface is triangulated and the integral equations are discretized by centroid collocation. The linear system is solved by GMRES iteration and the matrix-vector product is carried out by a Cartesian treecode which reduces the cost from \(O(N^2)\) to \(O(N\log N)\), where \(N\) is the number of faces in the triangulation. The TABI solver is applied to compute the electrostatic solvation energy in two cases, the Kirkwood sphere and a solvated protein. We present the error, CPU time, and memory usage, and compare results for the Poisson-Boltzmann and Poisson equations. We show that the treecode approximation error can be made smaller than the discretization error, and we compare two versions of the treecode, one with uniform clusters and one with non-uniform clusters adapted to the molecular surface. For the protein test case, we compare TABI results with those obtained using the grid-based APBS code, and we also present parallel TABI simulations using up to eight processors. We find that the TABI solver exhibits good serial and parallel performance combined with relatively simple implementation, efficient memory usage, and geometric adaptability.

MSC:
78M15 Boundary element methods applied to problems in optics and electromagnetic theory
65N38 Boundary element methods for boundary value problems involving PDEs
78A30 Electro- and magnetostatics
92C30 Physiology (general)
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[1] Honig, B.; Nicholls, A., Classical electrostatics in biology and chemistry, Science, 268, 1144-1149, (1995)
[2] Zhang, Z.; Witham, S.; Alexov, E., On the role of electrostatics in protein-protein interactions, Phys. Biol., 8, 035001, (2011)
[3] Roux, B.; Simonson, T., Implicit solvent models, Biophys. Chem., 78, 1-20, (1999)
[4] Feig, M.; Brooks, C. L., Recent advances in the development and application of implicit solvent models in biomolecule simulations, Curr. Opin. Struct. Biol., 14, 217-224, (2004)
[5] Klapper, I.; Hagstrom, R.; Fine, R.; Sharp, K.; Honig, B., Focusing of electric fields in the active site of cu-zn superoxide dismutase: effects of ionic strength and amino-acid modification, Proteins: Struct. Funct. Genet., 1, 47-59, (1986)
[6] Davis, M. E.; McCammon, J. A., Electrostatics in biomolecular structure and dynamics, Chem. Rev., 90, 509-521, (1990)
[7] Fogolari, F.; Brigo, A.; Molinari, H., The Poisson-Boltzmann equation for biomolecular electrostatics: a tool for structural biology, J. Mol. Recognit., 15, 377-392, (2002)
[8] Baker, N. A., Poisson-Boltzmann methods for biomolecular electrostatics, Methods Enzymol., 383, 94-118, (2004)
[9] Callenberg, K. M.; Choudhary, O. P.; de Forest, G. L.; Gohara, D. W.; Baker, N. A.; Grabe, M., Apbsmem: a graphical interface for electrostatic calculations at the membrane, PLoS ONE, 5, e12722, (2010)
[10] Beard, D. A.; Schlick, T., Modeling salt-mediated electrostatics of macromolecules: the discrete surface charge optimization algorithm and its application to the nucleosome, Biopolymers, 58, 106-115, (2001)
[11] Baker, N. A., Improving implicit solvent simulations: a Poisson-centric view, Curr. Opin. Struct. Biol., 15, 137-143, (2005)
[12] Chen, J. H.; Brooks, C. L.; Khandogin, J., Recent advances in implicit solvent-based methods for biomolecular simulations, Curr. Opin. Struct. Biol., 18, 140-148, (2008)
[13] Richards, F. M., Areas, volumes, packing, and protein structure, Ann. Rev. Biophys. Bioeng., 6, 151-176, (1977)
[14] Connolly, M. L., Molecular surface triangulation, J. Appl. Crystallogr., 18, 499-505, (1985)
[15] Davis, M. E.; Madura, J. D.; Luty, B. A.; McCammon, J. A., Electrostatics and diffusion of molecules in solution: simulations with the university of Houston Brownian dynamics program, Comput. Phys. Commun., 62, 187-197, (1991)
[16] Im, W.; Beglov, D.; Roux, B., Continuum solvation model: computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation, Comput. Phys. Commun., 111, 59-75, (1998) · Zbl 0935.78019
[17] Baker, N.; Holst, M.; Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equation: II. refinement at solvent-accessible surfaces in biomolecular systems, J. Comput. Chem., 21, 1343-1352, (2000)
[18] Baker, N. A.; Sept, D.; Joseph, S.; Holst, M. J.; McCammon, J. A., Electrostatics of nanosystems: application to microtubules and the ribosome, Proc. Natl. Acad. Sci. USA, 98, 10037-10041, (2001)
[19] Luo, R.; David, L.; Gilson, M. K., Accelerated Poisson-Boltzmann calculations for static and dynamic systems, J. Comput. Chem., 23, 1244-1253, (2002)
[20] Boschitsch, A. H.; Fenley, M. O., A fast and robust Poisson-Boltzmann solver based on adaptive Cartesian grids, J. Chem. Theory Comput., 7, 1524-1540, (2011)
[21] Chen, D.; Chen, Z.; Chen, C. J.; Geng, W. H.; Wei, G. W., MIBPB: a software package for electrostatic analysis, J. Comput. Chem., 32, 756-770, (2011)
[22] Juffer, A.; Botta, E.; van Keulen, B.; van der Ploeg, A.; Berendsen, H., The electric potential of a macromolecule in a solvent: a fundamental approach, J. Comput. Phys., 97, 144-171, (1991) · Zbl 0743.65094
[23] Bharadwaj, R.; Windemuth, A.; Sridharan, S.; Honig, B.; Nicholls, A., The fast multipole boundary element method for molecular electrostatics: an optimal approach for large systems, J. Comput. Chem., 16, 898-913, (1995)
[24] Liang, J.; Subramaniam, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73, 1830-1841, (1997)
[25] Boschitsch, A. H.; Fenley, M. O.; Zhou, H.-X., Fast boundary element method for the linear Poisson-Boltzmann equation, J. Phys. Chem. B, 106, 2741-2754, (2002)
[26] Altman, M. D.; Bardhan, J. P.; Tidor, B.; White, J. K., FFTSVD: a fast multiscale boundary-element method solver suitable for bio-MEMS and biomolecule simulation, IEEE Trans. Comput.-Aid. Des. Integr. Circ. Syst., 25, 274-284, (2006)
[27] Grandison, S.; Penfold, R.; Vanden-Broeck, J.-M., A rapid boundary integral equation technique for protein electrostatics, J. Comput. Phys., 224, 663-680, (2007) · Zbl 1123.78010
[28] Lu, B. Z.; Cheng, X.; McCammon, J. A., “new-version-fast-multipole-method” accelerated electrostatic calculations in biomolecular systems, J. Comput. Phys., 226, 1348-1366, (2007) · Zbl 1121.92007
[29] Bardhan, J. P., Numerical solution of boundary-integral equations for molecular electrostatics, J. Chem. Phys., 130, 094102, (2009)
[30] Greengard, L.; Gueyffier, D.; Martinsson, P.-G.; Rokhlin, V., Fast direct solvers for integral equations in complex three-dimensional domains, Acta Numer., 243-275, (2009) · Zbl 1176.65141
[31] Lu, B. Z.; Cheng, X.; Huang, J. F.; McCammon, J. A., AFMPB: an adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems, Comput. Phys. Commun., 181, 1150-1160, (2010) · Zbl 1220.78002
[32] Bajaj, C.; Chen, S.-C.; Rand, A., An efficient higher-order fast multipole boundary element solution for Poisson-Boltzmann-based molecular electrostatics, SIAM J. Sci. Comput., 33, 826-848, (2011) · Zbl 1227.92005
[33] Yokota, R.; Bardhan, J. P.; Knepley, M. G.; Barba, L. A.; Hamada, T., Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUS and a billion unknowns, Comput. Phys. Commun., 182, 1272-1283, (2011) · Zbl 1259.78044
[34] Zhang, B.; Lu, B. Z.; Cheng, X.; Huang, J. F.; Pitsianis, N. P.; Sun, X.; McCammon, J. A., Mathematical and numerical aspects of the adaptive fast multipole Poisson-Boltzmann solver, Commun. Comput. Phys., 13, 107-128, (2013) · Zbl 1373.78002
[35] Qiao, Z. H.; Li, Z. L.; Tang, T., A finite difference scheme for solving the nonlinear Poisson-Boltzmann equation modeling charged spheres, J. Comput. Math., 24, 252-264, (2006) · Zbl 1105.78015
[36] Yu, S. N.; Geng, W. H.; Wei, G. W., Treatment of geometric singularities in implicit solvent models, J. Chem. Phys., 126, 244108, (2007)
[37] Zhou, Y. C.; Feig, M.; Wei, G. W., Highly accurate biomolecular electrostatics in continuum dielectric environments, J. Comput. Chem., 29, 87-97, (2008)
[38] Geng, W. H.; Wei, G. W., Multiscale molecular dynamics using the matched interface and boundary method, J. Comput. Phys., 230, 435-457, (2011) · Zbl 1246.82028
[39] Geng, W. H.; Yu, S. N.; Wei, G. W., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127, 114106, (2007)
[40] Cai, Q.; Wang, J.; Zhao, H.-K.; Luo, R., On removal of charge singularity in Poisson-Boltzmann equation, J. Chem. Phys., 130, 145101, (2009)
[41] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73, 325-348, (1987) · Zbl 0629.65005
[42] Cheng, H.; Greengard, L.; Rokhlin, V., A fast adaptive multipole algorithm in three dimensions, J. Comput. Phys., 155, 468-498, (1999) · Zbl 0937.65126
[43] Greengard, L.; Huang, J. F., A new version of the fast multipole method for screened Coulomb interactions in three dimensions, J. Comput. Phys., 180, 642-658, (2002) · Zbl 1143.78372
[44] Saad, Y.; Schultz, M. H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-859, (1986) · Zbl 0599.65018
[45] Bardhan, J. P.; Altman, M. D.; Willis, D. J.; Lippow, S. M.; Tidor, B.; White, J. K., Numerical integration techniques for curved-element discretizations of molecule-solvent interfaces, J. Chem. Phys., 127, 014701, (2007)
[46] Golberg, M. A.; Chen, C. S., Discrete projection methods for integral equations, (1997), Computational Mechanics Publications Southampton, UK · Zbl 0900.65384
[47] Barnes, J.; Hut, P., A hierarchical \(O(N \log N)\) force-calculation algorithm, Nature, 324, 446-449, (1986)
[48] Duan, Z.-H.; Krasny, R., An adaptive treecode for computing nonbonded potential energy in classical molecular systems, J. Comput. Chem., 22, 184-195, (2001)
[49] Lindsay, K.; Krasny, R., A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, J. Comput. Phys., 172, 879-907, (2001) · Zbl 1002.76093
[50] Li, P.; Johnston, H.; Krasny, R., A Cartesian treecode for screened Coulomb interactions, J. Comput. Phys., 228, 3858-3868, (2009) · Zbl 1165.78304
[51] Sanner, M. F.; Olson, A. J.; Spehner, J. C., Reduced surface: an efficient way to compute molecular surfaces, Biopolymers, 38, 305-320, (1996)
[52] MSMS website. <mgl.scripps.edu/people/sanner/html/msms_home.html>.
[53] Netlib repository. <http://www.netlib.org/>.
[54] Kirkwood, J. G., Theory of solution of molecules containing widely separated charges with special application to zwitterions, J. Chem. Phys., 7, 351-361, (1934) · Zbl 0009.27504
[55] Briercheck, D. M.; Wood, T. C.; Allison, T. J.; Richardson, J. P.; Rule, G. S., The NMR structure of the RNA binding domain of E. coli rho factor suggests possible RNA-protein interactions, Nat. Struct. Biol., 5, 393-399, (1998)
[56] Protein Data Bank. <www.rcsb.org/pdb/home/home.do>.
[57] MacKerell, A. D.; Bashford, D.; Bellott, M.; Dunbrack, J. D.; Evanseck, M. J.; Field, M. J.; Fischer, S.; Gao, J.; Guo, H.; Ha, S.; Joseph-McCarthy, D.; Kuchnir, L.; Kuczera, K.; Lau, F. T.K.; Mattos, C.; Michnick, S.; Ngo, T.; Nguyen, D. T.; Prodhom, B.; Reiher, W. E.; Roux, B.; Schlenkrich, M.; Smith, J. C.; Stote, R.; Straub, J.; Watanabe, M.; Wiorkiewicz-Kuczera, J.; Yin, D.; Karplus, M., All-atom empirical potential for molecular modeling and dynamics studies of proteins, J. Phys. Chem. B, 102, 3586-3616, (1998)
[58] APBS website. <www.poissonboltzmann.org/apbs>.
[59] Humphrey, W.; Dalke, A.; Schulten, K., VMD-visual molecular dynamics, J. Mol. Graph., 14, 33-38, (1996)
[60] Warren, M. S.; Salmon, J. K., A portable parallel particle program, Comput. Phys. Commun., 87, 266-290, (1995) · Zbl 0923.76191
[61] Grama, A.; Kumar, V.; Sameh, A., Scalable parallel formulations of the Barnes-hut method for n-body simulations, Parallel Comput., 24, 797-822, (1998) · Zbl 0909.68219
[62] Marzouk, Y. M.; Ghoniem, A. F., K-means clustering for optimal partitioning and dynamic load balancing of parallel hierarchical N-body simulations, J. Comput. Phys., 207, 493-528, (2005) · Zbl 1176.70005
[63] Winkel, M.; Speck, R.; Hübner, H.; Arnold, L.; Krause, R.; Gibbon, P., A massively parallel, multi-disciplinary Barnes-hut tree code for extreme-scale N-body simulations, Comput. Phys. Commun., 183, 880-889, (2012)
[64] Smith, W., Molecular dynamics on hypercube parallel computers, Comput. Phys. Commun., 62, 229-248, (1991)
[65] Liu, D.; Duan, Z.-H.; Krasny, R.; Zhu, J., Parallel implementation of the treecode ewald method, (Proceedings of the 18th International Parallel and Distributed Processing Symposium, (2004), IEEE Computer Society Press Santa Fe, New Mexico)
[66] Gropp, W.; Lusk, E.; Skjellum, A., Using MPI: portable parallel programming with the message-passing interface, (1994), The MIT Press Cambridge, Massachusetts
[67] Geng, W. H., Parallel higher-order boundary integral electrostatics computation on molecular surfaces with curved triangulation, J. Comput. Phys., 241, 253-265, (2013) · Zbl 1349.78083
[68] Bates, P. W.; Wei, G. W.; Zhao, S., Minimal molecular surfaces and their applications, J. Comput. Chem., 29, 380-391, (2008)
[69] Yu, Z.; Holst, M. J.; Cheng, Y.; McCammon, J. A., Feature-preserving adaptive mesh generation for molecular shape modeling and simulation, J. Mol. Graph. Model., 26, 1370-1380, (2008)
[70] Bajaj, C. L.; Xu, G.; Zhang, Q., A fast variational method for the construction of resolution adaptive \(C^2\)-smooth molecular surfaces, Comput. Methods Appl. Mech. Eng., 198, 1684-1690, (2009) · Zbl 1227.74106
[71] D. Xu, Y. Zhang, Generating triangulated macromolecular surfaces by Euclidean Distance Transform, PLoS ONE 4(12) (2009) e8140, doi:10.1371/journal.pone.0008140
[72] Chen, M. X.; Lu, B. Z., Tmsmesh: a robust method for molecular surface mesh generation using a trace technique, J. Chem. Theory Comput., 7, 203-212, (2011)
[73] Cramer, C. J.; Truhlar, D. G., Implicit solvation models: equilibria, structure, spectra, and dynamics, Chem. Rev., 99, 2161-2200, (1999)
[74] Chipman, D. M., Solution of the linearized Poisson-Boltzmann equation, J. Chem. Phys., 120, 5566-5575, (2004)
[75] Tomasi, J.; Mennucci, B.; Cammi, R., Quantum mechanical continuum solvation models, Chem. Rev., 105, 2999-3093, (2005)
[76] Lange, A. W.; Herbert, J. M., A simple polarizable continuum solvation model for electrolyte solutions, J. Chem. Phys., 134, 204110, (2011)
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