Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-24T22:29:29.670Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

Published online by Cambridge University Press:  20 August 2015

M. Holst ,
J.A. McCammon ,
Z. Yu ,
Y.C. Zhou  and
Y. Zhu
M. Holst*
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA Department of Physics, University of California at San Diego, La Jolla, CA 92093, USA Center for Theoretical Biological Physics (CTBP), University of California at San Diego, La Jolla, CA 92093, USA National Biomedical Computation Resource (NBCR), University of California at San Diego, La Jolla, CA 92093, USA
J.A. McCammon
Affiliation:
Department of Chemistry & Biochemistry, University of California at San Diego, La Jolla, CA 92093, USA Center for Theoretical Biological Physics (CTBP), University of California at San Diego, La Jolla, CA 92093, USA National Biomedical Computation Resource (NBCR), University of California at San Diego, La Jolla, CA 92093, USA Howard Hughes Medical Institute, University of California at San Diego, La Jolla, CA 92093, USA
Z. Yu
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA National Biomedical Computation Resource (NBCR), University of California at San Diego, La Jolla, CA 92093, USA
Y.C. Zhou
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA Center for Theoretical Biological Physics (CTBP), University of California at San Diego, La Jolla, CA 92093, USA
Y. Zhu
Affiliation:
Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA Howard Hughes Medical Institute, University of California at San Diego, La Jolla, CA 92093, USA
*
*Corresponding author.Email:mholst@math.ucsd.edu
Get access

Abstract

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a prioriL∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

Keywords

92-0865N0665N3065N3865N5092C05Poisson-Boltzmann equationsemi-linear partial differential equationssupercritical nonlinearitysingularitya priori L∞ estimatesexistenceuniquenesswell-posednessGalerkin methodsdiscrete a priori L∞ estimatesquasi-optimal a priori error estimatesadaptive finite methodscontractionconvergenceoptimalitysurface and volume mesh generationmesh improvement and decimation
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 1 , January 2012 , pp. 179 - 214
Copyright
Copyright © Global Science Press Limited 2012

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]Adams, R. A. and Fornier, J. F., Sobolev Spaces, Academic Press, San Diego, CA, second edition, 2003.Google Scholar
[2]Aksoylu, B., Bond, S. and Holst, M., An odyssey into local refinement and multilevel preconditioning III: implementation and numerical experiments, SIAM J. Sci. Comput., 25(2) (2003), 478–498.Google Scholar
[3]Aksoylu, B. and Holst, M., Optimality of multilevel preconditioners for local mesh refinement in three dimensions, SIAM J. Numer. Anal., 44(3) (2006), 1005–1025.Google Scholar
[4]Babuška, I., The finite element method for elliptic equations with discontinuous coefficients, Computing, 5(3) (1970), 207–213.Google Scholar
[5]Berman, H., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T., Weissig, H., Shindyalov, I. and Bourne, P., The protein data bank, Nucleic Acids Res., 28 (2000), 235–242.CrossRefGoogle ScholarPubMed
[6]Blinn, J., A generalization of algegraic surface drawing, ACM Trans. Graphics, 1(3) (1982), 235–256.Google Scholar
[7]Boschitsch, A. and Fenley, M., A new outer boundary formulation and energy corrections for the nonlinear Poisson-Boltzmann equation, J. Comput. Chem., 28(5) (2007), 909–921.Google Scholar
[8]Boschitsch, A. H. and Fenley, M. O., Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation, J. Comput. Chem., 25(7) (2004), 935–955.Google Scholar
[9]Bramble, J. and King, J., A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math., 6(1) (1996), 109–138.Google Scholar
[10]Cascon, J. M., Kreuzer, C., Nochetto, R. H. and Siebert, K. G., Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. Anal., 46(5) (2008), 2524–2550.Google Scholar
[11]Chen, L., Holst, M. and Xu, J., The finite element approximation of the nonlinear Poisson-Boltzmann equation, SIAM J. Numer. Anal., 45(6) (2007), 2298–2320.Google Scholar
[12]Chen, Z. and Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79(2) (1998), 175–202.Google Scholar
[13]Chena, C.-Y. and Cheng, K.-Y., A sharpness dependent filter for mesh smoothing, Comput. Aided Geom. Des., 22(5) (2005), 376–391.Google Scholar
[14]Chern, I.-L., Liu, J.-G. and Wan, W.-C., Accurate evaluation of electrostatics for macro-molecules in solution, Methods Appl. Anal., 10 (2003), 309–328.Google Scholar
[15]Connolly, M., Analytical molecular surface calculation, J. Appl. Cryst., 16(5) (1983), 548–558.Google Scholar
[16]Dolinsky, T., Nielsen, J., McCammon, J. and Baker, N., PDB2PQR: an automated pipeline for the setup, execution and analysis of Poisson-Boltzmann electrostatics calculations, Nucleic Acids Res., 32 (2004), 665–667.Google Scholar
[17]Duncan, B. and Olson, A., Shape analysis of molecular surfaces, Biopolymers, 33 (1993), 231–238.Google Scholar
[18]Fernandez, J.-J. and Li, S., An improved algorithm for anisotropic nonlinear diffusion for denoising cryo-tomograms, J. Struct. Biol., 144 (2003), 152–161.Google Scholar
[19]Freitag, L. and Ollivier-Gooch, C., Tetrahedral mesh improvement using swapping and smoothing, Int. J. Numer. Methods Eng., 40 (1997), 3979–4002.Google Scholar
[20]Geng, W., Yu, S. and Wei, G., Treatment of charge singularities in implicit solvent models, J. Chem. Phys., 2007.Google Scholar
[21]Gilson, M. K., Davis, M. E., Luty, B. A. and McCammon, J. A., Computationn of electrostatic forces on solvated molecules using the Poisson-Boltzmann equation, J. Phys. Chem., 97 (1993), 3591–3600.Google Scholar
[22]Grant, J. and Pickup, B., A gaussian description of molecular shape, J. Phys. Chem., 99 (1995), 3503–3510.Google Scholar
[23]Holst, M., The Poisson-Boltzmann equation: analysis and multilevel numerical solution (Monograph based on the Ph.D. thesis: Multilevel Methods for the Poisson-Boltzmann Equation), Technical report, Applied Mathematics and CRPC, California Institute of Technology, 1994.Google Scholar
[24]Holst, M., Adaptive numerical treatment of elliptic systems on manifolds, Adv. Comput. Math., 15(1-4) (2001), 139-191.Google Scholar
[25]Holst, M., Baker, N. and Wang, F., Adaptive multilevel finite element solution of the Poisson-Boltzmann equation I: algorithms and examples, J. Comput. Chem., 21 (2000), 1319–1342.Google Scholar
[26]Holst, M. and Saied, F., Numerical solution of the nonlinear Poisson-Boltzmann equation: developing more robust and efficient methods, J. Comput. Chem., 16(3) (1995), 337–364.Google Scholar
[27]Holst, M., Tsogtgerel, G. and Zhu, Y., Local convergence of adaptive methods for nonlinear partial differential equations, submitted.Google Scholar
[28]Lee, B. and Richards, F., The interpretation of protein structures: estimation of static accessibility, J. Mol. Biol., 55(3) (1971), 379–400.Google Scholar
[29]Li, Z., An overview of the immersed interface method and its applications, Taiwanese J. Math., 1 (2003), 1–49.Google Scholar
[30]Li, Z. and Ito, K., Maximum principle preserving schemes for interface problems with discontinuous coefficients, SIAM J. Sci. Comput., 23(1) (2001), 339–361.Google Scholar
[31]Lorensen, W. and Cline, H. E., Marching cubes: a high resolution 3D surface construction algorithm, Comp. Graph., 21(4) (1987), 163–169.CrossRefGoogle Scholar
[32]Lu, B., Cheng, X., Hou, T. and McCammon, J. A., Calculation of the Maxwell stress tensor and the Poisson-Boltzmann force on a solvated molecular surface using hypersingular boundary integrals, J. Chem. Phys., 123 (2005), 084904.Google Scholar
[33]Lu, B., Cheng, X., Huang, J. and McCammon, J. A., Order n algorithm for computation of electrostatic interactions in biomolecular systems, Proc. Natl. Acad. Sci. USA, 103(51) (2006), 19314–19319.Google Scholar
[34]Lu, B., Zhang, D. and McCammon, J. A., Computation of electrostatic forces between sol-vated molecules determined by the Poisson-Boltzmann equation using a boundary element method, J. Chem. Phys., 122(21) (2005), 214102.Google Scholar
[35]Lu, B., Zhou, Y., Holst, M. and McCammon, J., Recent progress in numerical methods for the Poisson-Boltzmann equation in biophysical applications, Commun. Comput. Phys., 3(5) (2008), 973–1009.Google Scholar
[36]McQuarrie, D. A., Statistical Mechanics, Harper and Row, New York, NY, 1973.Google Scholar
[37]Mekchay, K., Convergence of Adaptive Finite Element Methods, Dissertation, University of Maryland, College Park, Md., Dec. 2005.Google Scholar
[38]Mekchay, K. and Nochetto, R., Convergence of adaptive finite element methods for general second order linear elliptic PDE, SIAM J. Numer. Anal., 43(5) (2005), 1803–1827.Google Scholar
[39]Oevermann, M. and Klein, R., A cartesian grid finite volume method for elliptic equations with variable coefficients and embedded interfaces, J. Comput. Phys., 219(2) (2006), 749–769.Google Scholar
[40]Perona, P. and Malik, J., Scale-space and edge detection using anisotropic diffusion, IEEE T. Pattern Anal., 12(7) (1990), 629–639.Google Scholar
[41]Richards, F., Areas, volumes, packing and protein structure, Ann. Rev. Biophys. Bioeng., 6 (1977), 151–156.Google Scholar
[42]Sanner, M., Olson, A. and Spehner, J., Reduced surface: an efficient way to compute molecular surfaces, Biopolymers, 38 (1996), 305–320.Google Scholar
[43]Savare, G., Regularity results for elliptic equations in lipschitz domains, J. Funct. Anal., 152(1) (1998), 176–201.Google Scholar
[44]Si, H., Tetgen: a quality tetrahedral mesh generator and three-dimensional Delaunay triangulator, Technical Report 9, Weierstrass Institute for Applied Analysis and Stochastics, 2004. (software download: http://tetgen.berlios.de).Google Scholar
[45]Si, H. and Gartner, K., Meshing piecewise linear complexes by constrained Delaunay tetrahedralizations, in Proceedings of the 14th International Meshing Roundtable, 2005.Google Scholar
[46]Stakgold, I. and Holst, M., Green’s Functions and Boundary Value Problems, John Wiley & Sons, Inc., New York, NY, third edition, 2011.Google Scholar
[47]Stevenson, R., An optimal adaptive finite element method, SIAM J. Numer. Anal., 42(5) (2005), 2188–2217.Google Scholar
[48]Tanford, C., Physical Chemistry of Macromolecules, John Wiley & Sons, New York, NY, 1961.Google Scholar
[49]Totrov, M. and Abagyan, R., The contour-buildup algorithm to calculate the analytical molecular surface, J. Struct. Biol., 116 (1996), 138–143.Google Scholar
[50]Verfürth, R., A posteriori error estimates for nonlinear problems, finite element discretizations of elliptic equations, Math. Comput., 62(206) (1994), 445–475.Google Scholar
[51]Wan, Z., Xu, B., Huang, K., Chu, Y.-C., Li, B., Nakagawa, S. H.,Y. Qu, Hu, S.-Q., Katsoyannis, P. G. and Weiss, M. A., Enhancing the activity of insulin at the receptor interface: crystal structure and photo-cross-linking of a8 analogues, Biochemistry, 43 (2004), 16119–12133.Google Scholar
[52]Wang, W.-C., A jump condition capturing finite difference scheme for elliptic interface problems, SIAM J. Sci. Comput., 25(5) (2004), 1479–1496.Google Scholar
[53]Weickert, J., Anisotropic Diffusion in Image Processing, ECMI Series, Teubner-Verlag, Stuttgart, 1998.Google Scholar
[54]Xie, D. and Zhou, S., A new minimization protocol for solving nonlinear Poisson-Boltzmann mortar finite element equation, BIT Numer. Math., 47(4) (2007), 853–871.Google Scholar
[55]Xu, H. and Newman, T., 2d fe quad mesh smoothing via angle-based optimization, in Proceeding of 5th Int’l Conference on Computational Science, pages 9–16, 2005.Google Scholar
[56]Yu, Z. and Bajaj, C., A segmentation-free approach for skeletonizationof gray-scale images via anisotropic vector diffusion, in Proc. Int’l Conf. Computer Vision and Pattern Recognition, pages 415–420, 2004.Google Scholar
[57]Yu, Z. and Bajaj, C., Computational approaches for automatic structural analysis of large biomolecular complexes, IEEE/ACM Trans. Comput. Bioinform., 5(4) (2008), 568–582.Google Scholar
[58]Yu, Z., Holst, M., Cheng, Y. and McCammon, J., Feature-preserving adaptive mesh generation for molecular shape modeling and simulation, J. Mol. Graph. Model., 26(8) (2008), 1370–1380.Google Scholar
[59]Yu, Z., Holst, M. and McCammon, J., High-fidelity geometric modeling for biomedical applications, Finite Elem. Anal. Des., 44(11) (2008), 715–723.Google Scholar
[60]Zhang, Y., Xu, G. and Bajaj, C., Quality meshing of implicit solvation models of biomolecular structures, The special issue of Computer Aided Geometric Design (CAGD) on Applications of Geometric Modeling in the Life Sciences, 23(6) (2006), 510–530.Google Scholar
[61]Zhou, T. and Shimada, K., An angle-based approach to two-dimensional mesh smoothing, in Proceedings of the Ninth International Meshing Roundtable, pages 373–384, 2000.Google Scholar
[62]Zhou, Y. C., Feig, M. and Wei, G. W., Highly accurate biomolecular electrostatics in continuum dielectric environments, J. Comput. Chem., 29(1) (2008), 87–97.Google Scholar
[63]Zhou, Y. C., Zhao, S., Feig, M. and Wei, G. W., High order matched interface and boundary (mib) schemes for elliptic equations with discontinuous coefficients and singular sources, J. Comput. Phys., 213 (2006), 1–30.Google Scholar
[64]Zhou, Z., Payne, P., Vasquez, M., Kuhn, N. and Levitt, M., Finite-difference solution of the Poisson-Boltzmann equation: complete elimination of self-energy, J. Comput. Chem., 17 (1996), 1344–1351.Google Scholar