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Journal of Chemical Physics / Volume 127 / Issue 1 / ARTICLES / Surfaces, Interfaces, and Materials

J. Chem. Phys. 127, 014701 (2007); http://dx.doi.org/10.1063/1.2743423 (18 pages)

Numerical integration techniques for curved-element discretizations of molecule-solvent interfaces

Jaydeep P. Bardhan1, Michael D. Altman2, David J. Willis3, Shaun M. Lippow4, Bruce Tidor5, and Jacob K. White6

1Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439
2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
3Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
4Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
5Department of Electrical Engineering and Computer Science and Department of Biological Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
6Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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(Received 29 September 2006; accepted 30 April 2007; published online 2 July 2007)

Surface formulations of biophysical modeling problems offer attractive theoretical and computational properties. Numerical simulations based on these formulations usually begin with discretization of the surface under consideration; often, the surface is curved, possessing complicated structure and possibly singularities. Numerical simulations commonly are based on approximate, rather than exact, discretizations of these surfaces. To assess the strength of the dependence of simulation accuracy on the fidelity of surface representation, here methods were developed to model several important surface formulations using exact surface discretizations. Following and refining Zauhar’s work [J. Comput.-Aided Mol. Des. 9, 149 (1995) ], two classes of curved elements were defined that can exactly discretize the van der Waals, solvent-accessible, and solvent-excluded (molecular) surfaces. Numerical integration techniques are presented that can accurately evaluate nonsingular and singular integrals over these curved surfaces. After validating the exactness of the surface discretizations and demonstrating the correctness of the presented integration methods, a set of calculations are presented that compare the accuracy of approximate, planar-triangle-based discretizations and exact, curved-element-based simulations of surface-generalized-Born (sGB), surface-continuum van der Waals (scvdW), and boundary-element method (BEM) electrostatics problems. Results demonstrate that continuum electrostatic calculations with BEM using curved elements, piecewise-constant basis functions, and centroid collocation are nearly ten times more accurate than planar-triangle BEM for basis sets of comparable size. The sGB and scvdW calculations give exceptional accuracy even for the coarsest obtainable discretized surfaces. The extra accuracy is attributed to the exact representation of the solute-solvent interface; in contrast, commonly used planar-triangle discretizations can only offer improved approximations with increasing discretization and associated increases in computational resources. The results clearly demonstrate that the methods for approximate integration on an exact geometry are far more accurate than exact integration on an approximate geometry. A MATLAB implementation of the presented integration methods and sample data files containing curved-element discretizations of several small molecules are available online as supplemental material.

© 2007 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. BACKGROUND
    1. Surface formulations of biophysical problems
      1. Molecular electrostatics
      2. Surface-generalized Born
      3. Continuum van der Waals
    2. Defining molecule-solvent interfaces
  3. SURFACE DISCRETIZATION
    1. Toroidal element definition
    2. Spherical element definition
  4. CURVED-ELEMENT INTEGRATION METHODS
    1. Far-field quadrature
      1. Generalized spherical triangle coordinate transformation
      2. Toroidal element coordinate transformation
    2. Near-field integration techniques
      1. Single-layer potential
        1. Spherical element single layer.
        2. Toroidal element single layer.
      2. Double-layer potential
  5. RESULTS
    1. Problem geometries
      1. Alanine tripeptide
      2. Alanine dipeptide
      3. Barnase-barstar complex
    2. Validating the surface discretization
    3. Validating curved boundary-element integration
    4. Surface-generalized-Born calculations
    5. Continuum van der Waals calculations
    6. Poisson-Boltzmann electrostatics problems
      1. Spherical geometry
      2. Alanine dipeptide
    7. Performance
  6. DISCUSSION

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KEYWORDS and PACS

Keywords

molecular biophysics, integration, boundary-elements methods, biology computing, electrostatics, biochemistry, bioelectric phenomena

PACS

  • 87.15.A-

    Theory, modeling, and computer simulation

  • 87.15.H-

    Dynamics of biomolecules

  • 87.15.N-

    Properties of solutions of macromolecules

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.2743423

PUBLICATION DATA

ISSN

0021-9606 (print)  
1089-7690 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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