Finding Solutions of the Inverse Kinematics Problems in Computer-aided Drug Design

Abstract

The efficient computation of low-energy molecular conformations is of critical importance to problems ranging from protein folding to computer-assisted drug design. Despite the growing number of papers on three-dimensional conformational search, several questions remain open. In this paper we investigate one such question relating to molecular inverse kinematics problems. In these problems we are given an initial conformation of a molecule and the target positions of some feature atoms of the molecule. We wish to automatically compute a new conformation of the molecule that brings the feature atoms to their target positions. We first show how to derive a system of polynomial equations from the geometric constraints of the feature atoms. In contrast with previous work, we do not attempt to solve the system of equations directly, which is computationally expensive. Instead, we adopt a technique based on the Groebner basis from algebraic geometry and develop a novel subdivision algorithm to approximate the real solutions. The approximated solutions can then be used as the starting conformations for existing(heuristic) energy minimization procedures that try to satisfy the target positions of feature atoms and reduce the overall energy of the conformation. To our knowledge, this is the first time that a rigorous algebraic methodology has been used to approximate molecular inverse kinematics solutions and the first time that a subdivision algorithm has been developed to efficiently locate the solutions.