An intuitive approach to measuring protein surface curvature

A natural way to measure protein surface curvature is to generate the least squares fitted (LSF) sphere to a surface patch and use the radius as the curvature measure. While the concept is simple, the sphere-fitting problem is not trivial and known means of protein surface curvature measurement use alternative schemes that are arguably less straightforward to interpret. We have developed an approach to solve the LSF sphere problem by turning the sphere-fitting problem into a solvable plane-fitting problem using a transformation known as geometric inversion. The approach works on any arbitrary surface patch, and returns a radius of curvature that has direct physical interpretation. Additionally, it is flexible in its ability to find the curvature of an arbitrary surface patch, and the "resolution" can be adjusted to highlight atomic features or larger features such as peptide binding sites. We include examples of applying the method to visualization of peptide recognition pockets and protein conformational change, as well as a comparison with a. commonly used solid-angle curvature method showing that the LSF method produces more pronounced curvature results.