Sphericity of a protein via the beta-complex.

Molecular shape is a fundamental factor in determining the function of a molecule. As proteins tend to fold into globular shapes, the shape descriptor for protein sphericity is important in understanding molecular functions. In this paper, a definition of protein sphericity is introduced based on the recently developed geometric constructs of the beta-complex and beta-shape of a protein. The beta-complex represents the Euclidean proximity among all the atoms in a protein, and the beta-shape is the polyhedron contained within the boundary of the corresponding beta-complex. Hence, the beta-shape determines the proximity among the atoms on the boundary of a protein. Given the volume of a beta-shape, the ratio between the surface area of a sphere with this volume and the surface area of the beta-shape itself is a good measure to classify the sphericity of a protein, especially when the radius of a probe is 3.0 A. The presented measure is invariant to translation and rotation.