Representation of protein 3D structures in spherical (ρ, ϕ, θ) coordinates and two of its potential applications

Three-dimensional objects can be represented using Cartesian, spherical or cylindrical coordinate systems, among many others. Currently all protein 3D structures in the PDB are in Cartesian coordinates. We wanted to explore the possibility that protein 3D structures, especially the globular type (spheroproteins), when represented in spherical coordinates might find useful novel applications. A Fortran program was written to transform protein 3D structure files in Cartesian coordinates (x,y,z) to spherical coordinates (ρ, ϕ, θ), with the centroid of the protein molecule as origin. We present here two applications, namely, (1) separation of the protein outer layer (OL) from the inner core (IC); and (2) identifying protrusions and invaginations on the protein surface. In the first application, ϕ and θ were partitioned into suitable intervals and the point with maximum ρ in each such ‘ϕ-θ bin’ was determined. A suitable cutoff value for ρ is adopted, and for each ϕ-θ bin, all points with ρ values less than the cutoff are considered part of the IC, and those with ρ values equal to or greater than the cutoff are considered part of the OL. We show that this separation procedure is successful as it gives rise to an OL that is significantly more enriched in hydrophilic amino acid residues, and an IC that is significantly more enriched in hydrophobic amino acid residues, as expected. In the second application, the point with maximum ρ in each ϕ-θ bin are sequestered and their frequency distribution constructed (i.e., maximum ρ’s sorted from lowest to highest, collected into 1.50Å-intervals, and the frequency in each interval plotted). We show in such plots that invaginations on the protein surface give rise to subpeaks or shoulders on the lagging side of the main peak, while protrusions give rise to similar subpeaks or shoulders, but on the leading side of the main peak. We used the dataset of Laskowski et al. (1996) to demonstrate both applications.