# α-shapes and Proteins

 Pawel Winter Henrik Sterner Peter Sterner Dept. of Computer Science, Univ. of Copenhagen, Universitetsparken 1, 2100 Copenhagen O, Denmark, e-mail: pawel@diku.dk E-mail: henrik.sterner@gmail.com E-mail: psterner@gmail.com

## Abstract

We provide a unified description of (weighted) α-shapes, $\beta$-shapes and the corresponding simplicial complexes. We discuss their applicability to various protein-related problems. We also discuss filtrations of α-shapes and touch upon related persistence issues. We claim that the full potential of α-shapes and related geometrical constructs in protein-related problems yet remains to be realized and verified. We suggest parallel algorithms for (weighted) α-shapes, and we argue that future use of filtrations and
kinetic variants for larger proteins will need such implementations.

Article is available in ps, dvi, pdf.

## Software

Limited space does not permit us to list various types of open source libraries, software and applets in one way or another way related to α-shapes. However, this list can be found at
http://www.diku.dk/~pawel/alphasoft.html. We do not claim the list be complete and we did not try to make any kind of the evaluation.

## Applets

Voronoi diagrams and Delaunay triangulations applets in 2d can be found at
α-shape applet in 2d can be found at

## Computational Geometry Libraries

CGAL is an open source library of reliable and efficient computational geometry algorithms written in \textit{C++}. It supports construction and visualization of Delaunay triangulations,
regular triangulations, Voronoi diagrams, power diagrams, unweighted and weighted α-shapes, and filtrations of the latter in 2d and 3d. A kinetic data structure
framework is integrated supporting Delaunay triangulations in 2d and 3d and regular triangulations in 3d. It can be found at

LEDA

QHULL supports computation of Delaunay triangulations and Voronoi diagrams in 2d and 3d. Available at

http://www.qhull.org

Hull is an ANSI C program that computes the convex hull of a point set in general (but small!) dimension. The program can also compute Delaunay triangulations and α-shapes,
and volumes of Voronoi regions. Available at

## α-Shape Software

CAVE and Alvis - Cave Automatic Virtual Environment (CAVE); one of its applications is modeling 3D protein structures, generating both outside-in and inside-out views of geometric models. The successors of CAVE, called ALpha Shapes VISualizer (ALVIS) and Virtual ALVIS (VALVIS) available at

BioGeometry 3d α-shape software is available at

3d α-Shapes software is available at

## Biomolecular Software

CASTp server uses the weighted Delaunay triangulation and the α-complex for shape measurements. It provides identification and measurements of surface accessible
pockets as well as interior inaccessible cavities, for proteins and other molecules. It measures analytically the area and volume of each pocket and cavity, both in the SA and MS
models. It also measures the number of mouth openings, area of the openings, circumference of mouth lips for eaxh pocket (in both SA and MS models). Available at

Progeom is a program for computing protein surface, volume, and their derivatives. Available at

APROPOS is a program for the detection of pockets in proteins. APROPOS was written in C and was available for Sun and SGI machines on request. See

Pocket is a program for detecting protein pockets available at

MolAxis is available on line and as a standalone program. Available at

CIEL a software package for generating protein-protein interface surfaces from protein structural data. Available at