Motivation: We would like to understand general techniques used in computations and visualization of molecular surfaces.

Goal of This Research: To tie together ideas from graphics, physics, and protein behavior and structure in order to understand how to do computations and visualizations of molecules.

Goal of This Paper: To present ideas from graphics, physics, and protein behavior and structure to summarize methods used in computations and visualization of molecules. The topics include:

1. Physical molecular models
   1. Atom and bond models
   2. Regular symmetric models
   3. Orbital models
   4. Folded chain models
2. Electron density fitting

   Molecular graphics is applied to X-ray crystallography.

3. Solvent-accessible surfaces

   Determining which hydrophobic regions of a protein will fold is useful in modeling protein folding.

4. Molecular surface graphics

   Rendering images of molecules, especially proteins.

5. Molecular volume and protein packing

   Computing van der Waals volume of a molecule and packing, to identify pockets and tunnels in proteins.

6. Shapes of small molecules and proteins
7. Designing drugs based on shape
8. Protein-protein interactions
9. Surface biology, chemistry, and physics

   Studying surfaces of molecules to analyze their behaviors.

Results: Techniques for computations and visualizations for molecular surfaces have been presented, with emphasis on solvent-accessibility ideas. Physical models, X-ray crystallography, molecular graphics, and drug design were also covered.

Motivation: We would like a mathematically precise definition of a pocket in a protein. Pockets in a given protein are important to identify because they are the places where another protein can attach (and hence interact) with the given protein. Unfortunately, we cannot use topology alone to define pockets, since a deep pocket and a flat surface are topologically the same.

Goal of This Research: To be able to automatically analyze the geometry of a protein so that we can analyze its interactions with other proteins.

Goal of This Paper: This paper defines a pocket in a protein as follows. Think of a protein as a closed volume. Then take the complement C of this volume. Now if C contains a region of limited accessibility from the outside, then the protein contains a pocket in that region. This paper describes the algorithm based on this idea and shows how it performs on various known proteins.

Results: The algorithm for detecting pockets cannot find shallow pockets.

Motivation: Understanding the structure of proteins is essential to understanding how they behave. We would like to analyze how several proteins attach to each other to form a protein complex. This process of attaching is called docking.

Goal of This Research: Given a set of proteins, how can they or will they dock?

Goal of This Paper: An algorithm proposed in the mid-1980s tries to predict how proteins will dock based on which knobs (protrusions) can fit into which depressions. This paper extends this idea by adding the Morse-Smale decomposition, based on Morse theory, which can be computed with an O(n log n) algorithm. This decomposition creates a partition of a protein surface in a way that elegantly relates local quantities to global ones. The paper applies this decomposition to the output of Connolly's function, which is a function whose extrema are the knobs and depressions on a protein's surface. Then the paper presents experimental results for several protein models.

Results: Decompositions seem to work well. Whether the techniques in this paper can be used for general shape comparison still needs to be investigated.