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Wavelet BEM on molecular surfaces: solvent excluded surfaces

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Abstract

The present paper is concerned with the rapid solution of a boundary integral equation for the apparent surface charge which arises from solvation continuum models. In order to apply the wavelet Galerkin scheme the molecular surface needs to be represented as a parametric surface consisting of smooth four-sided patches. We develop an algorithm which decomposes a solvent excluded surface into a set of globally continuous four-sided NURBS patches. Numerical experiments are carried out to demonstrate the feasibility and scope of the present approach.

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References

  1. Bajaj C, Xu G, Zhang Q (2009) A fast variational method for the construction of adaptive resolution C 2 smooth molecular surfaces. Comput Methods Appl Mech Eng 198: 1684–1690

    Article  Google Scholar 

  2. Beylkin G, Coifman R, Rokhlin V (1991) The fast wavelet transform and numerical algorithms. Commun Pure Appl Math 44: 141–183

    Article  MathSciNet  MATH  Google Scholar 

  3. Brunnett G (1995) Geometric design with trimmed surfaces. Comput Suppl 10: 101–115

    Google Scholar 

  4. Cancés E, Mennucci B (1998) New applications of integral equations methods for solvation continuum models: ionic solutions and liquid crystals. J Math Chem 23: 309–326

    Article  MathSciNet  MATH  Google Scholar 

  5. Cancés E, Mennucci B, Tomasi J (1997) A new integral equation formalism for the polarizable continuum model: theoretical background and applications to isotropic and anisotropic dielectrics. J Chem Phys 107: 3032–3041

    Article  Google Scholar 

  6. Cheng HL, Shi X (2009) Quality mesh generation for molecular skin surfaces using restricted union of balls. Comput Geom 42: 196–206

    Article  MathSciNet  MATH  Google Scholar 

  7. Connolly M (1983) Analytical molecular surface calculation. J Appl Cryst 16: 548–558

    Article  Google Scholar 

  8. Connolly M (2006) Molecular surface guide. http://connolly.best.vwh.net

  9. Dahmen W, Harbrecht H, Schneider R (2006) Compression techniques for boundary integral equations—optimal complexity estimates. SIAM J Numer Anal 43: 2251–2271

    Article  MathSciNet  MATH  Google Scholar 

  10. Dahmen W, Kunoth A (1992) Multilevel preconditioning. Numer Math 63: 315–344

    Article  MathSciNet  MATH  Google Scholar 

  11. Edelsbrunner H, Shah NR (1996) Incremental topological flipping works for regular triangulations. Algorithmica 15: 223–241

    Article  MathSciNet  MATH  Google Scholar 

  12. Elsässer B (1998) Approximation mit rationalen B-spline Kurven und Flächen. PhD thesis, Technische Universität Darmstadt

  13. Frediani L, Cammi R, Corni S, Tomasi J (2004) A polarizable continuum model for molecules at diffuse interfaces. J Chem Phys 120: 3893–3907

    Article  Google Scholar 

  14. Gallier J (2001) Geometric methods and applications for computer science and engineering. Springer, New York

    Book  MATH  Google Scholar 

  15. Greengard L, Rokhlin V (1987) A fast algorithm for particle simulation. J Comput Phys 73: 325–348

    Article  MathSciNet  MATH  Google Scholar 

  16. Hackbusch W, Khoromskij BN (2000) A sparse \({\mathcal{H}}\) -matrix arithmetic. II: application to multi-dimensional problems. Computing 64: 21–47

    MathSciNet  MATH  Google Scholar 

  17. Hackbusch W, Nowak ZP (1989) On the fast matrix multiplication in the boundary element method by panel clustering. Numer Math 54: 463–491

    Article  MathSciNet  MATH  Google Scholar 

  18. Harbrecht H, Randrianarivony M (2010) From computer aided design to wavelet BEM. Comput Vis Sci 13: 69–82

    Article  MathSciNet  MATH  Google Scholar 

  19. Harbrecht H, Randrianarivony M (2009) Wavelet BEM on molecular surfaces: parametrization and implementation. Computing 86: 1–22

    Article  MathSciNet  MATH  Google Scholar 

  20. Harbrecht H, Schneider R (2004) Biorthogonal wavelet bases for the boundary element method. Math Nachr 269–270: 167–188

    Article  MathSciNet  Google Scholar 

  21. Harbrecht H, Schneider R (2002) Wavelet Galerkin schemes for boundary integral equations – implementation and quadrature. SIAM J Sci Comput 27: 1347–1370

    Article  MathSciNet  Google Scholar 

  22. Hoschek J, Lasser D (1993) Fundamentals of computer aided geometric design. A.K. Peters, Wellesley

    MATH  Google Scholar 

  23. Laug P, Borouchaki H (2002) Molecular surface modeling and meshing. Eng Comp 18: 199–210

    Article  Google Scholar 

  24. Lee B, Richards F (1971) The interpretation of protein structures estimation of static accessibility. J Mol Biol 55: 379–400

    Article  Google Scholar 

  25. Miertuš S, Scrocco E, Tomasi J (1981) Electrostatic interaction of a solute with a continuum. A direct utilization of ab initio molecular potentials for the prevision of solvent effects. Chem Phys 55: 117–129

    Article  Google Scholar 

  26. Piegl L, Tiller W (1995) The NURBS book. Springer, Berlin

    MATH  Google Scholar 

  27. Pascual-ahuir JL, Silla E, Tuñon I (1994) GEPOL: An improved description of molecular surfaces. III. A new algorithm for the computation of a solvent-excluding surface. J Comput Chem 15: 1127–1138

    Article  Google Scholar 

  28. Pomelli C (2004) A tessellationless integration grid for the polarizable continuum model reaction field. J Comput Chem 25: 1532–1541

    Article  Google Scholar 

  29. Randrianarivony M (2006) Geometric processing of CAD data and meshes as input of integral equation solvers. PhD thesis, Technische Universität Chemnitz

  30. Randrianarivony M (2009) On global continuity of Coons mappings in patching CAD surfaces. Computer-Aided Design, vol 41. Springer, Berlin, pp 782–791

  31. Randrianarivony M, Brunnett G (2008) Molecular surface decomposition using graphical modeling. In: Bildverarbeitung für die Medizin: Algorithmen—Systeme—Anwendungen. Proceedings des Workshops vom 6. bis 8. April 2008, Berlin. Springer, Berlin, pp 197–201

  32. Randrianarivony M (2008) Harmonic variation of edge size in meshing CAD geometries from IGES format. Lect Notes Comput Sci 5102: 56–65

    Article  Google Scholar 

  33. Sanner M, Olsen A, Spehner J (1996) Reduced surface: an efficient way to compute molecular surfaces. Biopolymers 38: 305–320

    Article  Google Scholar 

  34. Sauter S, Schwab C (2004) Randelementmethoden: Analyse, Numerik und Implementierung schneller Algorithmen. Teubner, Stuttgart

    MATH  Google Scholar 

  35. Schoenberg L, Whitney A (1953) On Polya frequency functions III. Trans Am Math Soc 74: 246–259

    MathSciNet  MATH  Google Scholar 

  36. Tomasi J, Mennucci B, Cammi R (2005) Quantum mechanical continuum solvation models. Chem Rev 105: 2999–3094

    Article  Google Scholar 

  37. Weijo V, Randrianarivony M, Harbrecht H, Frediani L (2010) Wavelet formulation of the polarizable continuum model. J Comput Chem 31: 1469–1477

    Google Scholar 

  38. Whitley D (1998) Van der Waals graphs and molecular shape. J Math Chem 23: 377–397

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Analysis und Numerische Simulation, Universität Stuttgart, Stuttgart, Germany

    H. Harbrecht

  2. Institut für Numerische Simulation, Universität Bonn, Bonn, Germany

    M. Randrianarivony

Authors
  1. H. Harbrecht
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  2. M. Randrianarivony
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Correspondence to H. Harbrecht.

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Harbrecht, H., Randrianarivony, M. Wavelet BEM on molecular surfaces: solvent excluded surfaces. Computing 92, 335–364 (2011). https://doi.org/10.1007/s00607-011-0147-y

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  • DOI: https://doi.org/10.1007/s00607-011-0147-y

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