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Pockets are empty concavities on a protein surface into which solvent (probe sphere 1.4 A) can gain access, i.e., these concavities have mouth openings connecting their interior with the outside bulk solution. Currently, shallow depressions are excluded from the calculation. A technical definition to distinguish a pocket from a shallow depression is the following: among the infinite number of possible cross sections of a pocket, at least one is larger than the mouth opening of the pocket.

A cavity (or void) is an interior empty space that is not accessible to the solvent probe. It has no mouth openings to the outside bulk solution.

CASTp is based on recent theorectical and algorithmic results of Computational Geometry. It has many advantages: 1) pockets and cavities are identified analytically, 2) the boundary between the bulk solvent and the pocket is defined precisely, 3) all calculated parameters are rotationally invariant, and do not involve discretization and they make no use of dot surface or grid points.

The following contains excerpts from Liang et al, 1998.

The basic ingredients of Computational Geometry applied in CASTp are: Delaunay triangulation, alpha shape, and discrete flow (Edelsbrunner & Mucke 1994; Edelsbrunner 1995; Facello 1995; Edelsbrunner & Shah 1996; Edelsbrunner et al, 1998). Consider a highly simplified hypothetical model, a two dimensional molecule formed by atom disks, all of the same radius (Figure 1a). If nails are figuratively hammered into the plane at each atom center, and a rubber band is stretched around the entire collection of nails, the band encloses a

The Voronoi diagram is formed by the collection of Voronoi cells. For the hypothetical molecule in Figure 1a, Voronoi cells include the convex polygons bounded all around by dashed lines, as well as the polygons with edges defined by dashed lines, but extending to infinity. Each cell contains on atom, and those extending to infinity contain boundary atoms of the convex hull. A Voronoi cell consists of the space around one atom so that the distance of every spatial point in the cell to its atom is less than or equal to the distance to any other atom of the molecule. The Delaunay triangulation can be mapped from the Voronoi diagram directly. Across every Voronoi edge seperating two neighboring Voronoi cells, a line segment connecting the corresponding two atom centers is placed. For every Voronoi vertex where three Voronoi cells intersect, a triangle who's vertices are the three atom centers is placed. In this way, the full Delaunay triangulation is obtained by mapping from the Voronoi diagram. That is, both the Delaunay triangulation and the Voronoi diagram contain equivalent information.

To obtain the

For identifying and measuring pockets, the

All the features of the 2-D description have more complex 3-D counterparts. The convex hull in three
dimensions is a convex polytope instead of a polygon, and its Delaunay triangulation is a tessellation
of the polytope with tetrahedra. When atoms have different radii, the
*weighted Delaunay triangulation*
is required, and the corresponding weighted Voronoi cells are also different (Edelsbrunner, 1995).

- Edelsbrunner H, Mucke EP. 1994. Three-dimensional alpha shapes.
*ACM Trans. Graphics*13:43-72. - Edelsbrunner H. 1995. The union of balls and its dual shape.
*Discrete Comput. Geom.*13:415-440. - Edelsbrunner H, Shah NR. 1996. Incremental topological flipping works for regular triangulations.
*Algorithmica*15:223-241. - Edelsbrunner H, Facello M, Fu P, Liang J. 1995. Measuring proteins and voids in proteins. In:
*Proc. 28th Ann. Hawaii Int'l Conf. System Sciences.*Los Alamitos, California: IEEE Computer Society Press. pp. 256-264. - Edelsbrunner H, Facello M, Liang J. 1998. On the definition and the construction of pockets in macromolecules.
*Disc. Appl. Math.*88:83-102. - Facello MA. 1995. Implementation of a randomized algorihtm for Delaunay and regular triangulations
in three dimensions.
*Computer Aided Geometric Design*. 12:349-370. - Liang J, Edelsbrunner H, Woodward C. 1998. Anatomy of protein pockets and caviteis: measurement
of binding site geometry and implications for ligand design.
*Protein Science.*7:1884-1897. - J. Liang, H. Edelsbrunner, P. Fu, P.V. Sudhakar and S. Subramaniam. 1998. Analytical shape
computing of macromolecules I: molecular area and volume through alpha shape .
*Proteins.*33, 1-17. - Liang J, Edelsbrunner H, Fu P, Sudhakar PV, Subramaniam S. 1998b. Analytical shape computation
of macromolecules. II. Identification and computation of inaccessible cavities in proteins.
*Proteins: Struct. Funct. Genet.*33:18-29.