Geometric structures of proteins for understanding folding, discriminating natives and predicting biochemical functions

Proteins are the main working molecules of a cell. Typical protein molecules contain thousands or more atoms and have complex shapes. Understanding how atoms in proteins pack and how they form intricate shape in three dimensional space is an important task, as it helps us to gain insight on how proteins fold and how they carry out biological functions. There exists a large body of work using geometric constructs such as Voronoi diagram and Delau-nay triangulation to study protein packing, folding, and its physical chemical properties. We refer readers to several excellent reviews that discuss these studies [1–4]. Among these, the review by Poupon provides a comprehensive and concise discussion of recent studies [4]. In this chapter, we focus on several important issues in which computation of the geometric structures of proteins improve our understanding of protein molecules. We first briefly describe the underlying geometric models for accurate description of the complex shapes of protein molecules. We then show how improved geometry based on weighted Delaunay triangulation and alpha shape can help to characterize protein folding speed. We further describe how such accurate geometric description can aid in the development of empirical statistical scoring function for protein structure prediction. We then discuss findings on packing defects in the form of voids and pockets in protein structures , their overall distributions, and their scaling properties with protein size. This is followed by a discussion on the origin of packing defects and the roles played by evolution. Finally, we discuss how to extract evolutionary patterns of protein binding pockets and how to predict enzyme binding activities and functions.