This example is designed to follow step by step our procedure for computing the surface and volume of a protein, as well as for identifying pockets in that protein.

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A simple protein chain: 11 residues of Protein G (PDB code: 1PGB)

Let us consider a small protein chain of 11 residues, shown in figure 1. By selecting "Mainchain" and/or "Sidechain" with the labeled buttons, figure 1 shows either a trace of the mainchain of the proteins, and/or the sidechains attached to this mainchain. Both are shown as "wireframe", i.e. with the chemical bonds shown as line segments.

Another common representation of protein is to show each atom as a ball, whose radius is the vdW radius of that atom. This "ball" representation is referred to as the Corey-Pauling-Koltun or CPK or spacefilling representation of the protein. The CPK picture gives a graphical view of the volume occupied by the protein. The surface area and the volume of the union of these balls are the vdW surface and volume of the proteins.

The exposure of protein atoms to solvent can be obtained by computing their surface area in contact with solvent molecules. An atom is defined as accessible if a solvent molecule of specified size can be brought into van der Waals contact. van der Waals contact is the position of the center of a solvent molecule as it rolls along the protein. In practice, the accessible surface is computed as the surface of the union of balls obtained by expanding the vdW radius od each atom by the radius of the solvent sphere. This union of "solvated" balls can be visualized in Figure 1 by turning on the "CPK_H2O" button.

The geometry of the protein chain: Delaunay, -complex and Pockets

Our calculation of volume, surface and pockets of proteins is based on the Alpha-Shape theory. A complete description of this theory can be found here.

The procedure starts by computing the weighted 3D Delaunay tetrahedrization of the set of atoms of the protein of interest, whose weights are the vdW radii augmented by the radius of the probe considered. The Delaunay complex is a convex partition of the space occupied by the set of atoms. Its edges are shown in blue in Figure 2.

Not all simplices of the Delaunay complex represent the volume occupied by the protein. The alpha-complex for alpha=0 is a subset of the Delaunay that include vertices, edges, triangles and tetrahedra that exactly represent the protein. The inclusion-exclusion formula is then applied to all simplices of the (alpha=0)-complex , providing the exact analytical surface and volume of the protein. In figure 2, when a tetahedron belongs to the alpha-complex, all its edges are shown in red. Singular triangles of the alpha-complex (i.e. triangles whose parent tetrahedra do not belong to the alpha-complex) are shown in cyan, and singular edges are shown in magenta.

Among the tetrahedra of the Delaunay complex that do not belong to the alpha-complex, some correspond to shallow regions in contact with the solvent, while others fit in concave regions of proteins that can be identified as "pockets". These pockets are formally characterized using the concept of flow. The 11 residue fragment considered here contains one pocket.

Vizualizing Pockets in Proteins

In the section above, pockets have been defined as a collection of tetrahedra that do not belong to the alpha-complex, and form a concave region within the protein structures. The vertices of these tetrahedra correspond to the atoms of the protein which border the pocket. Figure 3 shows a A CPK model of the atoms bordering the pocket detected in figure 2. The radii of the balls are chosen to be the vdW radii of the atoms.