×

A new numerical method for nonlocal electrostatics in biomolecular simulations. (English) Zbl 1193.92006

Summary: The electrostatic behavior of biomolecules solved in water can be described by an elliptic system of partial differential equations for the potential. In previous studies, this system has been solved by the Boundary Element Method (BEM). In this paper, we apply the Explicit Jump Immersed Interface Method (EJIIM) as an alternative method to the BEM. Such a finite difference approach allows for a completely automatized software for analyzing biomolecules in their natural surrounding. The new method shows excellent agreement with the BEM results and has good convergence properties and runtimes. In addition, in contrast to the BEM, where the fundamental solutions of operators are necessary, the EJIIM approach can be easily extended to more complex, especially nonlinear models.

MSC:

92C05 Biophysics
78A70 Biological applications of optics and electromagnetic theory
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Mackerell, A. D., Empirical force fields for biological macromolecules: overview and issues, Journal of Computational Chemistry, 25, 13, 1584-1604 (2004)
[2] Azuara, C.; Orland, H.; Bon, M.; Koehl, P.; Delarue, M., Incorporating dipolar solvents with variable density in Poisson-Boltzmann electrostatics, Biophysical Journal, 95, 12, 5587-5605 (2008)
[3] Blossey, Ralf, Computational biology, Mathematical and Computational Biology Series (2006), Chapman & Hall/CRC · Zbl 1162.92014
[4] Cheng, H.-L.; Shi, X., Quality mesh generation for molecular skin surfaces using restricted union of balls, Computational Geometry, 42, 3, 196-206 (2009) · Zbl 1158.65014
[5] H.L. Cheng, H. Edelsbrunner, Skin.exe. http://biogeometry.duke.edu/software/skin/index.html; H.L. Cheng, H. Edelsbrunner, Skin.exe. http://biogeometry.duke.edu/software/skin/index.html
[6] P. Cignoni, Meshlab. http://meshlab.sourceforge.net; P. Cignoni, Meshlab. http://meshlab.sourceforge.net
[7] Dogonadze, R. R.; Kornyshev, A. A.; Kuznetsov, A. M., Phenomenological description of polar media on the basis of an effective hamiltonian, Teoreticheskaya i Matematicheskaya Fizika, 15, 127 (1973)
[8] Dogonadze, Revaz R.; Kálmán, Erika; Kornyshev, Alexei A.; Ulstrup, Jens, The Chemical Physics of Solvation, Part A: Theory of Solvation (1985), Elsevier Science Ltd.
[9] C. Fasel, S. Rjasanow, O. Steinbach, A boundary integral formulation for nonlocal electrostatics, in: Numerical Mathematics and Advanced Applications - Proceedings of ENUMATH 2007, Springer, 2008, pp. 117-124.; C. Fasel, S. Rjasanow, O. Steinbach, A boundary integral formulation for nonlocal electrostatics, in: Numerical Mathematics and Advanced Applications - Proceedings of ENUMATH 2007, Springer, 2008, pp. 117-124. · Zbl 1155.78307
[10] M. Garland and P. Heckbert. Surface simplification using quadric error metrics, in: SIGGRAPH, 1997.; M. Garland and P. Heckbert. Surface simplification using quadric error metrics, in: SIGGRAPH, 1997. · Zbl 0951.68554
[11] Halgren, T. A., Merck molecular force field i-v, Journal of Computational Chemistry, 17, 490-640 (1996)
[12] Hasted, J. B., Aqueous Dielectrics (1973), Chapman and Hall: Chapman and Hall London
[13] Hildebrandt, A., Biomolecules in a Structured Solvent (2005), Rhombos-Verlag
[14] Hildebrandt, A.; Blossey, R.; Rjasanow, S.; Kohlbacher, O.; Lenhof, H.-P., Novel formulation of nonlocal electrostatics, Physical Review Letters, 93, 10, 108104 (2004)
[15] Hildebrandt, A.; Blossey, R.; Rjasanow, S.; Kohlbacher, O.; Lenhof, H.-P., Electrostatic potentials of proteins in water: a structured continuum approach, Bioinformatics, 23, 2, e99-103 (2007)
[16] David Jackson, John, Classical Electrodynamics (1998), John Wiley & Sons, Inc. · Zbl 0997.78500
[17] Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations (1995), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia · Zbl 0832.65046
[18] Koehl, Patrice; Orland, Henri; Delarue, Marc, Beyond the Poisson-Boltzmann model: modeling biomolecule-water and water-water interactions, Physical Review Letters, 102, 8, 087801 (2009)
[19] O. Kohlbacher, H.P. Lenhof, et al., Ball-Biochemical Algorithms Library. http://www.bioinf.uni-sb.de/OK/BALL; O. Kohlbacher, H.P. Lenhof, et al., Ball-Biochemical Algorithms Library. http://www.bioinf.uni-sb.de/OK/BALL · Zbl 1100.92502
[20] Leach, A., Molecular Modelling: Principles and Applications (2001), Pearson Education
[21] Z. Li, The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces. Ph.D. Thesis, University of Washington, 1994.; Z. Li, The Immersed Interface Method - A Numerical Approach for Partial Differential Equations with Interfaces. Ph.D. Thesis, University of Washington, 1994.
[22] Lu, B. Z.; Cheng, X.; Huang, J.; McCammon, J. A., An adaptive fast multipole boundary element method for Poisson-Boltzmann electrostatics, Journal of Chemical Theory and Computation, 5, 6, 1692-1699 (2009)
[23] 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, Communications in Computational Physics, 3, 5, 973-1009 (2008) · Zbl 1186.92005
[24] Marquart, M.; Walter, J.; Deisenhofer, J.; Bode, W.; Huber, R., The geometry of the reactive site and of the peptide groups in trypsin, trypsinogen and its complexes with inhibitors, Acta Crystallographica Section B, 39, 4, 480-490 (1983)
[25] Mmff94 validation suite, Merck and Inc., Co., http://ftp.ccl.net/cca/data/MMFF94/index.shtml; Mmff94 validation suite, Merck and Inc., Co., http://ftp.ccl.net/cca/data/MMFF94/index.shtml
[26] Pearlman, D. A.; Case, D. A.; Caldwell, J. W.; Ross, W. S.; Cheatham, T. E.; DeBolt, S. E.; Ferguson, D. M.; Seibel, G. L.; Kollman, P. A., Amber, a package of computer programs for applying molecular mechanics, normal mode analysis, molecular dynamics and free energy calculations to simulate the structural and energetic properties of molecules, Computation Physics Letters (1995) · Zbl 0886.92035
[27] Rocchi, Claudia; Bizzarri, Anna Rita; Cannistraro, Salvatore, Water dynamical anomalies evidenced by molecular-dynamics simulations at the solvent-protein interface, Physical Review E, 57, 3, 3315-3325 (1998)
[28] V. Rutka, Immersed Interface Methods for Elliptic Boundary Value Problems. Ph.D. Thesis, TU Kaiserslautern, 2005.; V. Rutka, Immersed Interface Methods for Elliptic Boundary Value Problems. Ph.D. Thesis, TU Kaiserslautern, 2005.
[29] Rutka, V., A staggered grid based explicit jump immersed interface method for two-dimensional stokes flows, International Journal for Numerical Methods in Fluids, 57, 1527-1543 (2008) · Zbl 1140.76028
[30] Rutka, V.; Wiegmann, A., Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials, Numerical Algorithm, 43, 309-330 (2006) · Zbl 1150.74095
[31] Sandberg, L.; Casemyr, R.; Edholm, O., Calculated hydration free energies of small organic molecules using a nonlinear dielectric continuum model, The Journal of Physical Chemistry B, 106, 32, 7889-7897 (2002)
[32] J. Schöberl, H. Gerstmayr, R. Gaisbauer, Netgen – automatic mesh generator. http://www.hpfem.jku.at/netgen/; J. Schöberl, H. Gerstmayr, R. Gaisbauer, Netgen – automatic mesh generator. http://www.hpfem.jku.at/netgen/
[33] Sethian, J. A.; Wiegmann, A., Structural boundary design via level set and explicit jump immersed interface methods, Journal of Computation Physics, 163, 2, 489-528 (2000) · Zbl 0994.74082
[34] Shi, X.; Koehl, P., The geometry behind numerical solvers of the Poisson-Boltzmann equation, Communications in Computational Physics, 3, 5, 1032-1050 (2008) · Zbl 1199.82022
[35] Sitkoff, Doree; Sharp, Kim A.; Honig, Barry, Accurate calculation of hydration free energies using macroscopic solvent models, Journal of Physical Chemistry, 98, 1978-1988 (1994)
[36] Stein, P. E.; Leslie, A. G.; Finch, J. T.; Carrell, R. W., Crystal structure of uncleaved ovalbumin at 1.95A resolution, Journal of Molecular Biology, 222, 3, 941-959 (1991)
[37] Steinbach, Olaf, Numerische Näherungsverfahren für elliptische Randwertprobleme - Finite Elemente und Randelemente, Advances in Numerical Mathematics (2003), Teubner Verlag/GWV Fachverlage GmbH: Teubner Verlag/GWV Fachverlage GmbH Wiesbaden, (in German) · Zbl 1130.65001
[38] Godehard Sutmann, Die nichtlokale dielektrische Funktion von Wasser. Ph.D. Thesis, Forschungszentrum Jülich, Institut für Werkstoffe und Verfahren der Energietechnik, 1999.; Godehard Sutmann, Die nichtlokale dielektrische Funktion von Wasser. Ph.D. Thesis, Forschungszentrum Jülich, Institut für Werkstoffe und Verfahren der Energietechnik, 1999.
[39] Swanson, J.; Adcock, S.; McCammon, J. A., Optimized radii for Poisson-Boltzmann calculations with the amber force field, Journal of Chemical Theory and Computation, 1, 3, 484-493 (2005)
[40] Vorotynsev, M. A.; Kornyshev, A. A., Electrostatics of Media with Spatial Dispersion (1993), Nauka: Nauka Moscow · Zbl 0851.00007
[41] Wang, J.; Tan, C.; Tan, Y.-H.; Lu, Q.; Luo, R., Poisson-Boltzmann solvents in molecular dynamics simulations, Communications in Computational Physics, 3, 5, 1010-1031 (2008)
[42] Clint Whaley, R.; Petitet, Antoine; Dongarra, Jack J., Automated empirical optimization of software and the ATLAS project, Parallel Computing, 27, 1-2, 3-35 (2001), Also available as University of Tennessee LAPACK Working Note#147, UT-CS-00-448, 2000 · Zbl 0971.68033
[43] A. Wiegmann, The Explicit-Jump Immersed Interface Method and Interface Problems for Differential Equations. Ph.D. Thesis, University of Washington, 1998.; A. Wiegmann, The Explicit-Jump Immersed Interface Method and Interface Problems for Differential Equations. Ph.D. Thesis, University of Washington, 1998.
[44] A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds. Technical Report LBNL-43565, Lawrence Berkeley National Laboratory, MS 50A-1148, One Cyclotron Rd, Berkeley CA 94720, 1999.; A. Wiegmann, Fast Poisson, Fast Helmholtz and Fast Linear Elastostatic Solvers on Rectangular Parallelepipeds. Technical Report LBNL-43565, Lawrence Berkeley National Laboratory, MS 50A-1148, One Cyclotron Rd, Berkeley CA 94720, 1999.
[45] Wiegmann, A.; Bube, K. P., The immersed interface method for nonlinear differential equations with discontinuous coefficients and singular sources, SIAM Journal on Numerical Analasys, 35, 1, 177-200 (1998) · Zbl 0913.65076
[46] Wiegmann, A.; Bube, K. P., The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM Journal on Numerical Analasys, 37, 3, 827-862 (2000) · Zbl 0948.65107
[47] Zauhar, R. J., Smart: a solvent-accessible triangulated surface generator for molecular graphics and boundary element applications, Journal of Computer Aided Molecular Design, 9, 149-159 (1995)
[48] Zhou, Ruhong, Free energy landscape of protein folding in water: explicit vs. implicit solvent, Proteins: Structure Function and Genetics, 53, 2, 148-161 (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.