Molecular Geometry: A Challenge and Opportunity for Geometric Computing

Lecture / Panel
For NYU Community

Speaker: Deok-Soo Kim, Hanyang University


Biotechnology is an emerging discipline and molecules are its main target to be studied. While sequences remain important, it is well-known that molecular structure determines molecular function, and geometry is one of the most fundamental aspects of the molecular structure. Hence, the geometric issues in molecular worlds provide a challenge and opportunity for geometers. Despite its importance, the theory for understanding molecular geometry has not been sufficiently developed. In this talk, we will present a unified theory of molecular geometry and show how the theory can be used for accurately, efficiently, and conveniently solving molecular problems.

The molecular geometry theory is based on the beta-complex which is a derived structure from the Voronoi diagram of atoms and its dual structure called the quasi-triangulation. Unlike the well-known ordinary Voronoi diagram of points, the Voronoi diagram of spherical atoms has been known to be difficult to compute. Once computed, however, it nicely defines the proximity among
the atoms in molecules.

This talk will introduce the Voronoi diagram of atoms and its dual structure, the quasi-triangulation, in the three-dimensional space. Based on the quasi-triangulation, we define the beta-complex which concisely yet efficiently represents the correct proximity among all atoms. It turns out that the beta-complex, together with the Voronoi diagram and quasi-triangulation, can be used to accurately, efficiently, and conveniently solve seemingly unrelated shape problems for molecules within a single theoretical and computational framework. The correctness and efficiency of solutions can be easily mathematically guaranteed. Among many application areas, structural molecular biology and noble material design are the most immediate application area with plenty of visualization problems as well.

Application examples include the following: the most efficient/precise computation of the blending surface over an input molecule, the computation of the van der Waals volume (and area); an efficient docking simulation; the recognition of internal voids and their volume computation; the recognition of molecular tunnels, the comparison (or superposition) of the boundary structures of two molecules, shape reasoning such as measuring the sphericity of molecules, the efficient computation of the optimal side-chain placement for proteins, etc. We anticipate many other important applications will be discovered. In this talk, we will also demonstrate our molecular modeling and analysis software, the BetaMol, which is entirely based on the unified, single representation of the beta-complex. Several programs, including the BetaMol, are freely available at the Voronoi Diagram Research Center (VDRC, The engine software will also be available soon so that researchers can easily create application programs for their own problems using this engine.