This research focuses on developing new mathematical techniques to capture geometric features of complicated three-dimensional objects. A key application is the protein-protein interaction problem, the problem of modeling and predicting the locations of sites where pairs of proteins interact. This problem is the key to understanding the dynamics of biological activities, and is one of the major current challenges in structural biology. The research will have broad impact since the algorithms developed will have applications to the many problems that involve shape matching and description of geometric objects.
Geometry will likely play a key role in finding a solution to the protein-protein interaction problem, along with physical functions like electrostatics and solvation. Current approaches to shape fitting problems rely on shape description functions that take values on an interface surface, which are used to match surface patches. Such approaches have drawbacks when working with proteins, including instability caused by uncertainty of the surface location and fluctuations of the surface caused by local motions and flexibility. This research develops a new approach based on functions that have 3-dimensional domains supported in a neighborhood of the boundary surfaces. Topological analysis of the level sets associated to such functions will lead to new descriptions of surface features such as depressions and protrusions that are stable under fluctuation and uncertainty in the data. The research involves the development of a rigorous mathematical theory of surface descriptors based on such 3-dimensional functions.
