We consider NURBS based data structures for molecules and their properties, to support synthetic drug design and structural reasoning applications in molecular chemistry. 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.

Surface Animation

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.

From PDB(Protein Data Bank) which contains structural data of biological molcules, we can build molecular shapes based on CPK model and solvent accessible and solvent excluded surfaces. Given the centers of the molecule atoms and the relative van der Walls radii we can build the CPK representation as a union of balls. Its representation is based on the corresponding alpha-shape.
Centers	Van der Waals' radii spheres	Alpha-Shape
The solvent accessible surface can be obtained by increasing the radius of each atom in the molecule by the radius of the probe sphere assumed as solvent. A different Alpha-Shape is associated with the new set of spheres. This Alpha-Shape and its associated Power Diagram provide all the topological and geometrical information necessary to compute the solvent contact surface of the molecule.
Solvent Accessible Surface	Corresponding alpha shape	Clipped power diagram of SAS and original molecule
The Solvent Excluded Surface (rolling ball blend) is obtained by combining parts of the CPK model with concave and toroidal patches which centers lie on the curvilinear wireframe of the solvent accessible surface (the arcs are the intersection circles between spheres of the solvent accessible surface).
CPK model and solvent accessible wireframe	Toroidal and concave patches of the solvent excluded surface	Complete Solvent Excluded Surface