Analyzing Protein Structure Using Almost-Delaunay Tetrahedra

Delaunay tessellations and Voronoi diagrams capture proximity relationships among sets of points. When applied to points representing protein atoms or residue positions, they are used to compute molecular surfaces and protein volumes, to define cavities and pockets, to analyze and score packing interactions, and to find structural motifs. Since atom and residue coordinates are known imprecisely, we explore the effect of coordinate perturbation on Delaunay-based scoring and motif finding. We define and compute the almost-Delaunay tetrahedra, which are tetrahedra that can become part of a Delaunay tessellation if the point coordinates are perturbed by at most ≥ 0, and the probability that each is Delaunay assuming random Gaussian perturbations of all points. By analyzing these tetrahedra, we show that Delaunay four-body potential functions are robust and derive a new method to detect structural motifs. An implementation in MATLAB is available from http://www.cs.unc.edu/∼debug/papers/AlmDel.