Sphericity of a protein via the β-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 β-complex and β-shape of a protein. The β-complex represents the Euclidean proximity among all the atoms in a protein, and the β-shape is the polyhedron contained within the boundary of the corresponding β-complex. Hence, the β-shape determines the proximity among the atoms on the boundary of a protein. Given the volume of a β-shape, the ratio between the surface area of a sphere with this volume and the surface area of the β-shape itself is a good measure to classify the sphericity of a protein, especially when the radius of a probe is 3.0 Å. The presented measure is invariant to translation and rotation.