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On preconditioning the treecode-accelerated boundary integral (TABI) Poisson-Boltzmann solver. (English) Zbl 1416.65487
Summary: We recently developed a treecode-accelerated boundary integral (TABI) solver for solving Poisson-Boltzmann (PB) equation [the second author and R. Krasny, ibid. 247, 62–78 (2013; Zbl 1349.78084)]. The solver has combined advantages in accuracy, efficiency, memory, and parallelization as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an \(O(N \log N)\) treecode to accelerate the GMRES iterative solver. However, as observed in our previous work [the second author and F. Jacob, Comput. Phys. Commun. 184, No. 6, 1490–1496 (2013; Zbl 1310.78017)], occasionally when the mesh generator produces low quality triangles, the number of GMRES iterations required to solve the discretized boundary integral equations \(A x = b\) could be large. To address this issue, we design a preconditioning scheme using preconditioner matrix \(M\) such that \(M^{- 1} A\) has much improved condition while \(M^{- 1} z\) can be rapidly computed for any vector \(z\). In this scheme, the matrix \(M\) carries the interactions between boundary elements on the same leaf only in the tree structure thus is block diagonal with many computational advantages. The sizes of the blocks in \(M\) are conveniently controlled by the treecode parameter \(N_0\), the maximum number of particles per leaf. The numerical results show that this new preconditioning scheme improves the TABI solver with significantly reduced iteration numbers and better accuracy, particularly for protein sets on which TABI solver previously converges slowly. In addition, this preconditioning scheme potentially can improve the condition number of various multipole method accelerated boundary elements solvers in scattering, fluids, elasticity, etc.

65N38 Boundary element methods for boundary value problems involving PDEs
92C40 Biochemistry, molecular biology
65F08 Preconditioners for iterative methods
Full Text: DOI
[1] Geng, W.; Krasny, R., A treecode-accelerated boundary integral Poisson-Boltzmann solver for electrostatics of solvated biomolecules, J. Comput. Phys., 247, 62-78, (2013) · Zbl 1349.78084
[2] Geng, W.; Jacob, F., A GPU-accelerated direct-sum boundary integral Poisson-Boltzmann solver, Comput. Phys. Commun., 184, 6, 1490-1496, (2013) · Zbl 1310.78017
[3] Schlick, T., Molecular modeling and simulation: an interdisciplinary guide, (2010), Springer · Zbl 1320.92007
[4] Baker, N. A., Improving implicit solvent simulations: a Poisson-centric view, Curr. Opin. Struct. Biol., 15, 2, 137-143, (2005)
[5] Cherezov, V.; Rosenbaum, D. M.; Hanson, M. A.; Rasmussen, S. G.F.; Thian, F. S.; Kobilka, T. S.; Choi, H.-J.; Kuhn, P.; Weis, W. I.; Kobilka, B. K.; Stevens, R. C., High-resolution crystal structure of an engineered human 2-adrenergic g protein-coupled receptor, Science, 318, 5854, 1258-1265, (2007)
[6] Dong, F.; Vijaykumar, M.; Zhou, H. X., Comparison of calculation and experiment implicates significant electrostatic contributions to the binding stability of barnase and barstar, Biophys. J., 85, 1, 49-60, (2003)
[7] Huang, N.; Chelliah, Y.; Shan, Y.; Taylor, C. A.; Yoo, S.-H.; Partch, C.; Green, C. B.; Zhang, H.; Takahashi, J. S., Crystal structure of the heterodimeric CLOCK:BMAL1 transcriptional activator complex, Science, 337, 6091, 189-194, (2012)
[8] 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)
[9] Alexov, E.; Mehler, E. L.; Baker, N.; Baptista, A. M.; Huang, Y.; Milletti, F.; Nielsen, J. E.; Farrell, D.; Carstensen, T.; Olsson, M. H.M.; Shen, J. K.; Warwicker, J.; Williams, S.; Word, J. M., Progress in the prediction of pka values in proteins, Proteins, 79, 3260-3275, (Dec. 2011)
[10] Antosiewicz, J.; McCammon, J. A.; Gilson, M. K., The determinants of p\(K_a\)s in proteins, Biochemistry, 35, 24, 7819-7833, (1996)
[11] Hu, J.; Zhao, S.; Geng, W., Accurate pka computation using matched interface and boundary (MIB) method based Poisson-Boltzmann solver, Commun. Comput. Phys., 2, 520-539, (2018)
[12] Nielsen, J. E.; McCammon, J. A., Calculating pka values in enzyme active sites, Protein Sci., 12, 9, 1894-1901, (2003)
[13] Zhou, Y. C.; Lu, B.; Gorfe, A. A., Continuum electromechanical modeling of protein-membrane interactions, Phys. Rev. E, 82, (Oct. 2010)
[14] 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, 1-12, (09 2010)
[15] García-García, C.; Draper, D. E., Electrostatic interactions in a peptide-RNA complex, J. Mol. Biol., 331, 1, 75-88, (2003)
[16] Nguyen, D. D.; Wang, B.; Wei, G.-W., Accurate, robust, and reliable calculations of Poisson-Boltzmann binding energies, J. Comput. Chem., 38, 13, 941-948, (2017)
[17] L. Wilson, R. Krasny, A comparison of two molecular surface triangulation codes in a boundary integral Poisson-Boltzmann framework, 2018, in preparation.
[18] Simonson, T.; Archontis, G.; Karplus, M., Free energy simulations come of age: protein-ligand recognition, Acc. Chem. Res., 35, 6, 430-437, (2002), PMID: 12069628
[19] Wagoner, J. A.; Baker, N. A., Assessing implicit models for nonpolar mean solvation forces: the importance of dispersion and volume terms, Proc. Natl. Acad. Sci., 103, 22, 8331-8336, (2006)
[20] Unwin, N., Refined structure of the nicotinic acetylcholine receptor at 4 å resolution, J. Mol. Biol., 346, 4, 967-989, (2005)
[21] Baker, N. A., Poisson-Boltzmann methods for biomolecular electrostatics, Methods Enzymol., 383, 94-118, (2004)
[22] Lu, Q.; Luo, R., A Poisson-Boltzmann dynamics method with nonperiodic boundary condition, J. Chem. Phys., 119, 21, 11035-11047, (2003)
[23] Im, W.; Beglov, D.; Roux, B., Continuum solvation model: electrostatic forces from numerical solutions to the Poisson-Boltzmann equation, Comput. Phys. Commun., 111, 1-3, 59-75, (1998) · Zbl 0935.78019
[24] Honig, B.; Nicholls, A., Classical electrostatics in biology and chemistry, Science, 268, 5214, 1144-1149, (1995)
[25] Geng, W.; Yu, S.; Wei, G. W., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 127, (2007)
[26] Yu, S.; Geng, W.; Wei, G. W., Treatment of geometric singularities in implicit solvent models, J. Chem. Phys., 126, (2007)
[27] Greengard, L. F.; Huang, J., A new version of the fast multipole method for screened Coulomb interactions in three dimensions, J. Comput. Phys., 180, 2, 642-658, (2002) · Zbl 1143.78372
[28] Li, P.; Johnston, H.; Krasny, R., A Cartesian treecode for screened Coulomb interactions, J. Comput. Phys., 228, 10, 3858-3868, (2009) · Zbl 1165.78304
[29] 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
[30] Chen, J.; Geng, W., Parallel computing of the adaptive n-body treecode algorithm for solving boundary integral Poisson-Boltzmann equation, Lect. Notes Comput. Sci., 9576, 1-9, (2015)
[31] Baker, N. A.; Sept, D.; Holst, M. J.; Mccammon, J. A., The adaptive multilevel finite element solution of the Poisson-Boltzmann equation on massively parallel computers, IBM J. Res. Dev., 45, 3-4, 427-438, (2001)
[32] Jurrus, E.; Engel, D.; Star, K.; Monson, K.; Brandi, J.; Felberg, L. E.; Brookes, D. H.; Wilson, L.; Chen, J.; Liles, K.; Chun, M.; Li, P.; Gohara, D. W.; Dolinsky, T.; Konecny, R.; Koes, D. R.; Nielsen, J. E.; Head-Gordon, T.; Geng, W.; Krasny, R.; Wei, G.-W.; Holst, M. J.; McCammon, J. A.; Baker, N. A., Improvements to the APBS biomolecular solvation software suite, Protein Sci., 27, 112-128, (2018)
[33] Sanner, M. F.; Olson, A. J.; Spehner, J. C., Reduced surface: an efficient way to compute molecular surfaces, Biopolymers, 38, 305-320, (1996)
[34] Yoon, B. J.; Lenhoff, A. M., A boundary element method for molecular electrostatics with electrolyte effects, J. Comput. Chem., 11, 9, 1080-1086, (1990)
[35] Liang, J.; Subranmaniam, S., Computation of molecular electrostatics with boundary element methods, Biophys. J., 73, 1830-1841, (1997)
[36] Boschitsch, A. H.; Fenley, M. O.; Zhou, H.-X., Fast boundary element method for the linear Poisson-Boltzmann equation, J. Phys. Chem. B, 106, 10, 2741-2754, (2002)
[37] Lu, B.; Cheng, X.; McCammon, J. A., A new-version-fast-multipole-method accelerated electrostatic calculations in biomolecular systems, J. Comput. Phys., 226, 2, 1348-1366, (2007) · Zbl 1121.92007
[38] Altman, M. D.; Bardhan, J. P.; White, J. K.; Tidor, B., Accurate solution of multi-region continuum biomolecule electrostatic problems using the linearized Poisson-Boltzmann equation with curved boundary elements, J. Comput. Chem., 30, 1, 132-153, (2009)
[39] Greengard, L.; Gueyffier, D.; Martinsson, P.-G.; Rokhlin, V., Fast direct solvers for integral equations in complex three-dimensional domains, Acta Numer., 18, (2009) · Zbl 1176.65141
[40] 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, 2, 826-848, (2011) · Zbl 1227.92005
[41] Geng, W., Parallel higher-order boundary integral electrostatics computation on molecular surfaces with curved triangulation, J. Comput. Phys., 241, 253-265, (2013) · Zbl 1349.78083
[42] Zhang, B.; Lu, B.; Cheng, X.; Huang, J.; Pitsianis, N. P.; Sun, X.; McCammon, J. A., Mathematical and numerical aspects of the adaptive fast multipole Poisson-Boltzmann solver, Commun. Comput. Phys., 13, 1, 107-128, (2013)
[43] Sun, Q.; Klaseboer, E.; Chan, D. Y.C., A robust and accurate formulation of molecular and colloidal electrostatics, J. Chem. Phys., 145, 5, (2016)
[44] Zhong, Y.; Ren, K.; Tsai, R., An implicit boundary integral method for computing electric potential of macromolecules in solvent, J. Comput. Phys., 359, 199-215, (2018) · Zbl 1383.78014
[45] Davis, M. E.; Madura, J. D.; Sines, J.; Luty, B. A.; Allison, S. A.; McCammon, J. A., Diffusion-controlled enzymatic reactions, Methods Enzymol., 202, 473-497, (1991)
[46] 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, 18, 10037-10041, (2001)
[47] Luo, R.; David, L.; Gilson, M. K., Accelerated Poisson-Boltzmann calculations for static and dynamic systems, J. Comput. Chem., 23, 13, 1244-1253, (2002)
[48] Cai, Q.; Wang, J.; Zhao, H.-K.; Luo, R., On removal of charge singularity in Poisson-Boltzmann equation, J. Chem. Phys., 130, 14, (2009)
[49] Chen, D.; Chen, Z.; Chen, C.; Geng, W.; Wei, G. W., MIBPB: a software package for electrostatic analysis, J. Comput. Chem., 32, 657-670, (2011)
[50] Deng, W.; Zhufu, X.; Xu, J.; Zhao, S., A new discontinuous Galerkin method for the nonlinear Poisson-Boltzmann equation, Appl. Math. Lett., 257, 1000-1021, (2015)
[51] Ying, J.; Xie, D., A new finite element and finite difference hybrid method for computing electrostatics of ionic solvated biomolecule, J. Comput. Phys., 298, 636-651, (2015) · Zbl 1349.78103
[52] Zauhar, R. J.; Morgan, R. S., A new method for computing the macromolecular electric potential, J. Mol. Biol., 186, 4, 815-820, (1985)
[53] Lu, B. Z.; Zhou, Y. C.; Holst, M. J.; McCammon, J. A., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Commun. Comput. Phys., 3, 5, 973-1009, (2008) · Zbl 1186.92005
[54] Rocchia, W.; Sridharan, S.; Nicholls, A.; Alexov, E.; Chiabrera, A.; Honig, B., Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: applications to the molecular systems and geometric objects, J. Comput. Chem., 23, 1, 128-137, (2002)
[55] Hou, S.; Wang, W.; Wang, L., Numerical method for solving matrix coefficient elliptic equation with sharp-edged interfaces, J. Comput. Phys., 229, 19, 7162-7179, (2010) · Zbl 1197.65183
[56] Li, Z.; Ji, H.; Chen, X., Accurate solution and gradient computation for elliptic interface problems with variable coefficients, SIAM J. Numer. Anal., 55, 2, 570-597, (2017) · Zbl 1362.76037
[57] Xie, D., New solution decomposition and minimization schemes for Poisson-Boltzmann equation in calculation of biomolecular electrostatics, J. Comput. Phys., 275, 294-309, (2014) · Zbl 1349.78077
[58] Wang, L.; Hou, S.; Shi, L., A numerical method for solving three-dimensional elliptic interface problems with triple junction points, Adv. Comput. Math., 44, 1, 175-193, (2018) · Zbl 1382.65421
[59] Lu, B.; Cheng, X.; Huang, J.; McCammon, J. A., Order N algorithm for computation of electrostatic interactions in biomolecular systems, Proc. Natl. Acad. Sci., 103, 51, 19314-19319, (2006)
[60] Barnes, J.; Hut, P., A hierarchical O(N log N) force-calculation algorithm, Nature, 324, 12, 446-449, (1986)
[61] Saad, Y.; Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[62] Duan, Z.-H.; Krasny, R., An adaptive treecode for computing nonbonded potential energy in classical molecular systems, J. Comput. Chem., 22, 2, 184-195, (2001)
[63] Lindsay, K.; Krasny, R., A particle method and adaptive treecode for vortex sheet motion in three-dimensional flow, J. Comput. Phys., 172, 2, 879-907, (2001) · Zbl 1002.76093
[64] Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D.; Swaminathan, S.; Karplus, M., CHARMM: a program for macromolecular energy, minimization, and dynamics calculations, J. Comput. Chem., 4, 187-217, (1983)
[65] Dolinsky, T. J.; Czodrowski, P.; Li, H.; Nielsen, J. E.; Jensen, J. H.; Klebe, G.; Baker, N. A., PDB2PQR: expanding and upgrading automated preparation of biomolecular structures for molecular simulations, Nucleic Acids Res., 35, (2007)
[66] Holst, M. J., The Poisson-Boltzmann equation: analysis and multilevel numerical solution, (1994), UIUC, PhD thesis
[67] Lehotzky, R. E.; Partch, C. L.; Mukherjee, S.; Cash, H. L.; Goldman, W. E.; Gardner, K. H.; Hooper, L. V., Molecular basis for peptidoglycan recognition by a bactericidal lectin, Proc. Natl. Acad. Sci., 107, 17, 7722-7727, (2010)
[68] Mukherjee, S.; Zheng, H.; Derebe, M. G.; Callenberg, K. M.; Partch, C. L.; Rollins, D.; Propheter, D. C.; Rizo, J.; Grabe, M.; Jiang, Q.-X.; Hooper, L. V., Antibacterial membrane attack by a pore-forming intestinal C-type lectin, Nature, 505, 103-107, (01 2014)
[69] Decherchi, S.; Rocchia, W., A general and robust ray-casting-based algorithm for triangulating surfaces at the nanoscale, PLoS ONE, 8, 1-15, (04 2013)
[70] Chen, M.; Lu, B., Tmsmesh: a robust method for molecular surface mesh generation using a trace technique, J. Chem. Theory Comput., 7, 1, 203-212, (2011), PMID: 26606233
[71] Liu, B.; Wang, B.; Zhao, R.; Tong, Y.; Wei, G.-W., ESES: software for Eulerian solvent excluded surface, J. Comput. Chem., 38, 7, 446-466, (2017)
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