Protein docking problem as combinatorial optimization using beta-complex

Abstract

The completion of the Human Genome Project has launched the post-sequencing phase of genome research. Once a genome information is identified, protein structures become most important in life sciences. Drug design is one immediate example, and docking, a computational method to understand molecular binding between protein and compound, is one of its fundamental implementation methods. This chapter presents a docking problem between a receptor and a ligand in the framework of combinatorial optimization and is an effort to formalize the problem into a mathematically and computationally rigorous theory so that follow-up studies can safely rely on the concepts presented in this chapter. Docking problem is a mathematical and computational problem and a highly complicated, intractable problem. In addition, this problem is very much interdisciplinary encompassing biology, biochemistry, biophysics, medicinal chemistry, and computational disciplines such as optimization, mathematics, and geometric computations. Therefore, it is desirable to develop a common language. To accomplish this objective, this chapter is based on geometric constructs called the beta-complex and beta-shape of molecules. This chapter has three parts. The first part introduces basic concepts and theory of molecular binding/recognition process and tries to formalize the transformation of the docking problem to a combinatorial optimization problem. This part will be particularly useful for people from mathematical and computational disciplines who wants to learn about the fundamentals of docking problem in a nutshell. The second part introduces the topology representation of molecules and the fundamental building blocks for the computational process with a slightly mathematical rigor. The Voronoi diagram, the spatial tessellations (triangulations), and the beta-complex and beta-shape are primary geometric objects introduced. The third part shows how to formulate and solve a protein-ligand docking problem as combinatorial optimization problems using the beta-complex and beta-shape, as it is done in BetaDock - the docking software developed by author's group.