Quality meshing of implicit solvation models of biomolecular structures

This paper describes a comprehensive approach to construct quality meshes for implicit solvation models of biomolecular structures starting from atomic resolution data in the Protein Data Bank (PDB). First, a smooth volumetric electron density map is constructed from atomic data using weighted Gaussian isotropic kernel functions and a two-level clustering technique. This enables the selection of a smooth implicit solvation surface approximation to the Lee-Richards molecular surface. Next, a modified dual contouring method is used to extract triangular meshes for the surface, and tetrahedral meshes for the volume inside or outside the molecule within a bounding sphere/box of influence. Finally, geometric flow techniques are used to improve the surface and volume mesh quality. Several examples are presented, including generated meshes for biomolecules that have been successfully used in finite element simulations involving solvation energetics and binding rate constants.

[1]  Fujio Yamaguchi,et al.  Computer-Aided Geometric Design , 2002, Springer Japan.

[2]  Jonathan Richard Shewchuk,et al.  Constrained Delaunay Tetrahedralizations and Provably Good Boundary Recovery , 2002, IMR.

[3]  Chandrajit L. Bajaj,et al.  A subdivision scheme for hexahedral meshes , 2002, The Visual Computer.

[4]  Paresh Parikh,et al.  Generation of three-dimensional unstructured grids by the advancing-front method , 1988 .

[5]  Shang-Hua Teng,et al.  Unstructured Mesh Generation: Theory, Practice, and Perspectives , 2000, Int. J. Comput. Geom. Appl..

[6]  Nathan A. Baker,et al.  Continuum diffusion reaction rate calculations of wild-type and mutant mouse acetylcholinesterase: adaptive finite element analysis. , 2004, Biophysical journal.

[7]  T. N. Bhat,et al.  The Protein Data Bank , 2000, Nucleic Acids Res..

[8]  Herbert Edelsbrunner,et al.  Sliver exudation , 1999, SCG '99.

[9]  Mark Meyer,et al.  Discrete Differential-Geometry Operators for Triangulated 2-Manifolds , 2002, VisMath.

[10]  Valerio Pascucci,et al.  Dynamic maintenance and visualization of molecular surfaces , 2003, Discret. Appl. Math..

[11]  Tao Ju,et al.  Dual contouring of hermite data , 2002, ACM Trans. Graph..

[12]  M. L. Connolly.  Analytical molecular surface calculation , 1983 .

[13]  K. Shimada,et al.  Anisotropic Triangular Meshing of Parametric Surfaces via Close Packing of Ellipsoidal Bubbles , 2007 .

[14]  Houman Borouchaki,et al.  Delaunay Tetrahedralization using an Advancing-Front Approach , 1996 .

[15]  Michael Garland,et al.  Simplifying surfaces with color and texture using quadric error metrics , 1998, IEEE Visualization.

[16]  Yongjie Zhang,et al.  3D Finite Element Meshing from Imaging Data. , 2005, Computer methods in applied mechanics and engineering.

[17]  Guoliang Xu,et al.  Discrete Laplace-Beltrami Operator on Sphere and Optimal Spherical Triangulations , 2006, Int. J. Comput. Geom. Appl..

[18]  G. Sapiro,et al.  Geometric partial differential equations and image analysis [Book Reviews] , 2001, IEEE Transactions on Medical Imaging.

[19]  L. Paul Chew,et al.  Guaranteed-quality Delaunay meshing in 3D (short version) , 1997, SCG '97.

[20]  Tamal K. Dey,et al.  Sampling and meshing a surface with guaranteed topology and geometry , 2004, SCG '04.

[21]  Mario Botsch,et al.  Feature sensitive surface extraction from volume data , 2001, SIGGRAPH.

[22]  Nathan A. Baker,et al.  Finite element solution of the steady-state Smoluchowski equation for rate constant calculations. , 2004, Biophysical journal.

[23]  R. C. Agarwal.  A new least‐squares refinement technique based on the fast Fourier transform algorithm: erratum , 1978 .

[24]  J. A. Grant,et al.  A Gaussian Description of Molecular Shape , 1995 .

[25]  S.F.F. Gibson,et al.  Using distance maps for accurate surface representation in sampled volumes , 1998, IEEE Symposium on Volume Visualization (Cat. No.989EX300).

[26]  Ho-Lun Cheng,et al.  Dynamic Skin Triangulation , 2001, SODA '01.

[27]  R C Wade,et al.  Analytically defined surfaces to analyze molecular interaction properties. , 1996, Journal of molecular graphics.

[28]  Jonathan Richard Shewchuk,et al.  Tetrahedral mesh generation by Delaunay refinement , 1998, SCG '98.

[29]  J. Grassi,et al.  Conformational Flexibility of the Acetylcholinesterase Tetramer Suggested by X-ray Crystallography* , 1999, The Journal of Biological Chemistry.

[30]  D. A. Field.  Laplacian smoothing and Delaunay triangulations , 1988 .

[31]  Guoliang Xu.  Discrete Laplace-Beltrami operators and their convergence , 2004, Comput. Aided Geom. Des..

[32]  Gregory M. Nielson,et al.  Interval volume tetrahedrization , 1997, Proceedings. Visualization '97 (Cat. No. 97CB36155).

[33]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[34]  Valerio Pascucci,et al.  NURBS based B-rep models for macromolecules and their properties , 1997, SMA '97.

[35]  I. Fujishiro,et al.  Volumetric Data Exploration Using Interval Volume , 1996, IEEE Trans. Vis. Comput. Graph..

[36]  Mark S. Shephard,et al.  Automatic three‐dimensional mesh generation by the finite octree technique , 1984 .

[37]  Qing Pan,et al.  Discrete surface modelling using partial differential equations , 2006, Comput. Aided Geom. Des..

[38]  Vinay Siddavanahalli,et al.  Compressed representations of macromolecular structures and properties. , 2005, Structure.

[39]  P. Taylor,et al.  Acetylcholinesterase inhibition by fasciculin: Crystal structure of the complex , 1995, Cell.

[40]  M. Sanner,et al.  Reduced surface: an efficient way to compute molecular surfaces. , 1996, Biopolymers.

[41]  Ivo Babuška,et al.  Accuracy estimates and adaptive refinements in finite element computations , 1986 .

[42]  Herbert Edelsbrunner,et al.  Sliver exudation , 2000, J. ACM.

[43]  Matthew L. Staten,et al.  An Approach to Combined Laplacian and Optimization-Based Smoothing for Triangular, Quadrilateral, and Quad-Dominant Meshes , 1998, IMR.

[44]  Michael J. Holst,et al.  Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples , 2001 .

[45]  Herbert Edelsbrunner,et al.  Triangulating the Surface of a Molecule , 1996, Discret. Appl. Math..

[46]  Lori A. Freitag,et al.  On combining Laplacian and optimization-based mesh smoothing techniques , 1997 .

[47]  Paul G. Mezey,et al.  Shape in Chemistry: An Introduction to Molecular Shape and Topology , 1993 .

[48]  Nathan A. Baker,et al.  Electrostatics of nanosystems: Application to microtubules and the ribosome , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[49]  Nathan A. Baker,et al.  Tetrameric mouse acetylcholinesterase: continuum diffusion rate calculations by solving the steady-state Smoluchowski equation using finite element methods. , 2005, Biophysical journal.

[50]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[51]  Herbert Edelsbrunner,et al.  Deformable Smooth Surface Design , 1999, Discret. Comput. Geom..

[52]  Rainald Löhner,et al.  Three-dimensional grid generation by the advancing front method , 1988 .

[53]  Charles L. Brooks,et al.  New analytic approximation to the standard molecular volume definition and its application to generalized Born calculations , 2003, J. Comput. Chem..

[54]  B. Lee,et al.  The interpretation of protein structures: estimation of static accessibility. , 1971, Journal of molecular biology.

[55]  Houman Borouchaki,et al.  Molecular Surface Modeling and Meshing , 2002, Engineering with Computers.

[56]  T. Steitz,et al.  The structural basis of ribosome activity in peptide bond synthesis. , 2000, Science.

[57]  NelsonL. Max.  Approximating molecular surfaces by spherical harmonics , 1988 .

[58]  David E. Breen,et al.  Semi-regular mesh extraction from volumes , 2000 .

[59]  Chandrajit L. Bajaj,et al.  Adaptive and quality 3D meshing from imaging data , 2003, SM '03.

[60]  James F. Blinn,et al.  A Generalization of Algebraic Surface Drawing , 1982, TOGS.

[61]  Yutaka Ohtake,et al.  Mesh regularization and adaptive smoothing , 2001, Comput. Aided Des..

[62]  Steven J. Owen,et al.  A Survey of Unstructured Mesh Generation Technology , 1998, IMR.