(see an mpeg example)

The difficulty of modeling and visualization of large molecules derives from the high combinatorial complexity of the typical molecule considered (e.g.  proteins or nucleic acids). There are two main modeling approaches.  The first describes the molecule's primary structure and the detailed 3D position of each of its atoms.  The second groups some regions of the molecule into simpler shapes to describe the folding of the molecule into its secondary, tertiary and higher order structures. 

We develop a B-rep data structure for molecular surfaces that aims to be useful both for visualization and modeling purposes.  This requires the ability (a) to exactly represent the shape of the molecule, (b) to directly render such a representation, and (c) to perform modeling operations that correspond to the addition/deletion of residues.  The natural choice to achieve both goals is to use trimmed NURBS (Non Uniform Rational tensor-product B-Spline with rational B-Spline trimming curves).  They are an industry-wide standard as a modeling primitive and graphics libraries for NURBS rendering are available (e.g.  openGL, OpenInventor).  Moreover, the rational parameterization allows for an exact representation of a  spherical surface.  This alone is not sufficient. In order to have an exact representation of a macromolecular structure we also need to represent for each atom, not its entire sphere, only that portion of the sphere which belongs to the external molecule surface.  This means that from the sphere which represents one atom we must cut away the pieces contained in the neighboring atoms.  We  prove that adopting a certain parameterization each trimming curve (a circle) in  the 3D space is mapped back in the parameter domain to a curve that can be in turn represented exactly as a NURBS curve.  In this way we can represent the contribution of each atom to the molecule surface with a trimmed NURBS patch without any approximation. 

The main contributions of the approach are: 

• the definition of a (minimal size) B-rep with standard trimmed NURBS representation;
• parametric B-rep model of the solvent accessible surface useful for animation 
• the classification of the solvent contact surface and computation of its representation as a trimmed NURBS. 

C. Bajaj, H.Y. Lee, R. Merkert, V. Pascucci 
"NURBS based B-rep Models from Macromolecules and their Properties'', (ps.gz)
In Proceedings Fourth Symposium on Solid Modeling and Applications, Atlanta, Georgia, 1997,C. Hoffmann and W. Bronsvort Eds., ACM Press. pp. 217-228

Abstract

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Adaptive and Quality Quadrilateral/Hexahedral Meshing from Volumetric Data

Results

(Each image is linked to a higher resolution image.)

Implicit solvation models of Haloarcula Marismortui large Ribosome 50S (1JJ2) crystal subunit. The light yellow and the pink color show 5S and 23S rRNA respectively, the remaining colors are proteins. (a): the implicit solvation model at the medium resolution level, p1=0.0625, p2=1.0; (b) and (c): triangular meshes (16700 vertices, 33400 triangles); (d): the interior mesh (230025 vertices, 1141575 tets); (e): an exterior mesh within a sphere (234902 vertices, 1162568 tets); (f): an exterior mesh within a bounding box (260858 vertices, 1315112 tets). 1. Implicit solvation models. (a) Implicit solvation models of Thermus Thermophilus small Ribosome 30S (1J5E) crystal subunit for various Gaussian kernel parameters. The pink color shows 16S rRNA and the remaining colors are proteins. (b) Implicit solvation models of Haloarcula Marismortui large Ribosome 50S (1JJ2) crystal subunit. The light yellow and the pink color show 5S and 23S rRNA respectively, the remaining colors are proteins. 2. Surface Smoothing. (a) Comparison of mAChE (9308 vertices, 18612 triangles) before and after surface smoothing. (a) - original; (b) - after smoothing. (b) Comparison of Ribosome 30S (13428 vertices, 26852 triangles) before and after surface smoothing. Left - original; Right - after smoothing. 3. Interior/Exterior Tetrahedral Meshes. Interior and exterior tetra meshes of monomeric mAChE. The left two pictures conform to the SAS with sigma=2, and the right two pictures conform to the surface constructed from Gaussian summation with p1=0.25, p2=1.0. From left to right: (65147 vertices, 323442 tets), (121670 vertices, 656823 tets), (103680 vertices, 509597 tets) and (138967 vertices, 707284 tets). The color shows potential (leftmost) or residues (the right two). Interior and exterior tetra meshes of tetrameric mAChE, p1=0.5, p2=1.0. The left two pictures show the 1st crystal structure 1C2O (133078 vertices, 670950 tets), and the right two pictures show the 2nd one 1C2B, (106463 vertices, 551074 tets). Cavities are shown in red boxes. Interior and exterior tetra meshes of Ribosome 30S, low resolution, p1=0.03125, p2=1.0. From left to right: (33612 vertices, 163327 tets), (37613 vertices, 186496 tets) and (40255 vertices, 201724 tets). The pink color shows 16S rRNA and other colors show proteins.