Molecular shape analysis based upon the morse-smale complex and the connolly function

Abstract

Docking is the process by which two or several molecules form a complex. Docking involves the geometry of the molecular surfaces, as well as chemical and energetical considerations. In the mid-eighties, Connolly proposed a docking algorithm matching surface knobs with surface depressions. Knobs and depressions refer to the extrema of the Connolly function, which is defined as follows. Given a surface M bounding a three-dimensional domain X, and a sphere S centered at a point p of M, the Connolly function is equal to the solid angle of the portion of S containing within X.We recast the notions of knobs and depressions in the framework of Morse theory for functions defined over two-dimensional manifolds. First, we study the critical points of the Connolly function for smooth surfaces. Second, we provide an efficient algorithm for computing the Connolly function over a triangulated surface. Third, we introduce a Morse-Smale decomposition based on Forman's discrete Morse theory, and provide an O(n log n) algorithm to construct it. This decomposition induces a partition of the surface into regions of homogeneous flow, and provides an elegant way to relate local quantities to global ones--from critical points to Euler's characteristic of the surface. Fourth, we apply this Morse-Smale decomposition to the discrete gradient vector field induced by Connolly's function, and present experimental results for several mesh models.