Many problems in structural biology, require a researcher to understand the shape of a protein. At first glance, this may seem obvious. By opening a molecular visualizer, one can easily see the shape of a protein. But what about calculating the surface area or volume of the protein? What about performing analyses of the surface, such as looking for concave pockets in a protein that might be binding sites for other molecules? What about calculating the volume and shape of those empty binding pockets, in order to find molecules that might fit in them? What about determining whether a particular small molecule can fit in a binding pocket?

All of these problems require some formal notion of the shape of a protein. A protein structure file usually provides no more information than a list of atom locations in space and their types. It will be assumed that for any given application, a radius may be defined for each atom type. This leads to the space filling representation of a protein, in which each atom is treated as an impenetrable sphere.

This representation allows for visualization, but it brings us no closer to being able to computationally decide which parts of which atoms are on the surface of the protein and which are buried inside the structure. Some additional tool is needed to capture notions of interior and exterior and spatial adjacency.

Using the sphere model for atoms, one way to define the shape of a molecule is as the union of (possibly overlapping) balls in R 3 R 3 .

Since proteins inside our cells are in an aqueous environment, considering a protein's interactions with solvent molecules, particularly water, is very important for appropriately modeling them. Recall that one of the phenomena that determines the structure of a protein is the hydrophobic effect: some amino acid residues are stabilized by the presence of water, and others are repelled. The extent of the interaction of a protein with the surrounding water depends on the surface area of the protein that can be reached by water molecules. Therefore, quantitave modeling of the strength of interaction with solvent often involves computing the solvent accessible surface area (SASA). Computing SASA can be done by regarding each solvent molecule as a sphere of set radius. This is of course a simplification, since water molecules are not spherical. When this sphere rolls about the molecule, its center delineates the SASA. One can think of the SASA of a molecule as the result of growing each atom sphere by the radius of the solvent sphere. Instead, by taking what is swept out by the front of the solvent sphere, we obtain the molecular surface (MS) model of the molecule. Alternatively, the MS can be obtained by removing a layer of solvent radius depth from the SASA model. The surface determined by SASA analysis depends on the size of a typical solvent molecule. The larger the solvent, the less contoured the resulting surface will appear, because a larger probe molecules would not be able to fit into some of the interatomic spaces that a smaller one would.

Part of the problem with defining the shape of a protein is that we start with nothing but a point set, and the "shape" of a set of discontinuous points is poorly defined. The problem is, what do we mean by shape? As you saw above, the shape of a molecule depends on what is being used to measure it. To handle this ambiguity, we will introduce a method of shape calculation based on a parameter, α, which will determine the radius of a spherical probe that will define the surface. The method defines a class of shapes, called α-shapes [4] for any given point set. It allows fast, accurate, and efficient calculations of volume and surface area.

α-shapes are a generalization of the convex hull. Consider a point set S. Define an α-ball as a sphere of radius α. An α-ball is empty if it contains no points in S. For any α between zero and infinity, the α-hull of S is the complement of the union of all empty α-balls.

The volume of a molecule can be approximated using the space-filling model, in which each atom is modeled as a ball whose radius is α, where α is selected depending on the model being used: Van der Waals surface, molecular surface, solvent accessible surface, etc. Unfortunately, calculating the volume is not as simple as taking the sum of the ball volumes because they may overlap. Calculating the volume of a complex of overlapping balls is non-trivial because of the overlaps. If two spheres overlap, the volume is the sum of the volumes of the spheres minus the volume of the overlap, which was counted twice. If three overlap, the volume is the sum of the ball volumes, minus the volume of each pairwise overlap, plus the volume of the three-way overlap, which was subtracted one too many times in accounting for the pairwise overlaps. In the general case, all pairwise, three-way, four-way and so on to n-way intersections (assuming there are n atoms) must be considered. Proteins generally have thousands or tens of thousands of atoms, so the general n-way case may be computationally expensive and may introduce numerical error.

α-shapes provide a way around this undesirable combinatorial complexity [2], and this issue has been one of the motivating factors for introducing α-shapes. To calculate the volume of a protein, we take the sum of all ball volumes, then subtract only those pairwise intersections for which a corresponding edge exists in the α-complex. Only those three-way intersections for which the corresponding triangle is in the α-complex must then be added back. Finally, only four-way intersections corresponding to tetrahedra in the α-complex need to be subtracted. No higher-order intersections are necessary, and the number of volume calculations necessary corresponds directly to the complexity of the α-complex, which is O(n log n) in the number of atoms.

An example of how this approach works is given on page 4 of the Liang et al. article in the Recommended Reading section below. A proof of correctness and derivation is also provided in the article. Surface area calculations, such as solvent-accessible surface area, which is often used to estimate the strength of interactions between a protein and the solvent molecules surrounding it, are made by a similar use of the α-complex.

Recommended Reading