Introduction | Representation | Function | Dynamic Structure | Hierarchical Model | Protein Docking | Simulations | Meshing | Visualization |

**2. Molecular Shape Generation**

### Links

Interactive Visualization of Bio-molecules and their Properties (pdf).

## 2.1 CPK Model, Solvent Accessible and Solvent Excluded surfaces

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.

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’s 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.

### Papers

- C. Bajaj, H.Y. Lee, R. Merkert, V. Pascucci

NURBS based B-rep Models from Macromolecules and their Properties (ps),

In Proceedings Fourth Symposium on Solid Modeling and Applications, Atlanta, Georgia, 1997,

C. Hoffmann and W. Bronsvort Eds., ACM Press. pp. 217-228 - C. L. Bajaj, V. Pascucci, A. Shamir, R. Holt, A. Netravali

Multiresolution Molecular Shapes,

TICAM Report No. 99-42 (pdf)

### Links

Molecular Surfaces

Molecular Visualization Presentation

## 2.2 Isosurfaces of Molecular Electron Density

Comments | Images | |

Fos-Jun complex (binds DNA) | ||

Anthrax protective antigen (half of a dimer) | ||

HIV-1 protease | ||

MAP kinase P38/Sb203580 | ||

Prion Protein Domain Prp (121-231) | ||

RNA polymerase |

### Papers

- C. Bajaj, V. Pascucci, D. Schikore.

Fast Isocontouring for Improved Interactivity

Proceedings: ACM Siggraph/IEEE Symposium on Volume Visualization, ACM Press, (1996), San Francisco, CA (ps.gz) - C. Bajaj, V. Pascucci, R. Holt, A. Netravali.

Dynamic Maintenance and Visualization of Molecular Surfaces

Fourth issue in the special series of Discrete Applied Mathematics on Computational Molecular Biology (pdf), (ps.gz pages 1-17), (ps.gz pages 19-31), Color Plate (page 18) (pdf),

(ps.gz), (Abstract) - C. L. Bajaj, V. Pascucci, A. Shamir, R. Holt, A. Netravali.

Multiresolution Molecular Shapes

TICAM report # 99-42 (ps.gz) (pdf) - C. Bajaj, H.Y. Lee, R. Merkert, V. Pascucci

NURBS based B-rep Models from Macromolecules and their Properties,

In Proceedings Fourth Symposium on Solid Modeling and Applications, Atlanta,Georgia, 1997,C. Hoffmann and W. Bronsvort Eds., ACM Press. pp. 217-228 (ps.gz) - C. Bajaj, F. Bernardini, K. Sugihara

A Geometric Approach to Molecular Docking and Similarity

Technical Report 94-017 (20 pages), Purdue University, Computer Sciences (3/94) (pdf) (ps.gz) - X. Zhang, C. Bajaj, V. Ramachandra

Parallel and Out-of-core View-dependent Isocontour Visualization Using Random Data Distribution

Joint Eurographics-IEEE TCVG Symposium on Visualization 2002, pages 9-18. (pdf) - C. Bajaj, V. Pascucci,D.Thompson, X.Y. Zhang

Parallel Accelerated Isocontouring for Out-Of-Core Visualization,

i In Proceedings of IEEE Parallel Visualization and Graphics Symposium, October 24-29,1999 San Francisco, CA, pages 97 – 104. (ps) (pdf) - C. Bajaj, V. Pascucci

Progressive IsoContouring,

TICAM report #99-36i (ps.gz) (pdf) - C. Bajaj, V. Pascucci, D.Schikore

Accelerated IsoContouring of Scalar Fields, Data Visualization Techniques, edited by C. Bajaj, John Wiley and Sons (ps.gz) (pdf) - C. Bajaj, V. Pascucci, D. Schikore

Fast Isocontouring for Improved Interactivity, Proceedings: ACM Siggraph/IEEE Symposium on Volume Visualization, ACM Press, (1996), San Francisco, CA (ps.gz) - C. Bajaj, V. Pascucci, D. Schikore

Fast Isocontouring for Improved Interactivity, Technical Report 96-024, Purdue University, Computer Sciences(1996) (ps.gz)

## 2.3 Cryo-Electron Microscopy

From electron density field, we compute iso-surfaces of molecules and using properties of molecules such as static electricity, we can also compute electro static field from which we can compute the color values used to color iso-surfaces of molecular shapes.