Distance Geometry: Theory, Algorithms, and Chemical Applications

Distance geometry is the mathematical basis for a geometric theory of molecular conformation.1 This theory plays a role in conformational analysis analogous to that played in statistical mechanics by a hard-sphere fluid . . . which can in fact be regarded as the distance geometry description of a monoatomic fluid. More generally, a distance geometry description of a molecular system consists of a list of distance and chirality constraints. These are, respectively, lower and upper bounds on the distances between pairs of atoms, and the chirality of its rigid quadruples of atoms (i.e., R or S relative to some given order). The distance geometry approach is predicated on the assumption that it is possible to adequately define the set of all possible (i.e., significantly populated) conformations, or conformation space, of just about any nonrigid molecular system by means of such purely geometric constraints. By Occam’s razor, we contend that any properties of the system that can be explained by such a simple model should be explained that way. Distance geometry also plays an important role in the development of computational methods for analyzing distance geometry descriptions. The goal of these calculations is to determine the global properties of the entire conformation space, as opposed to the local properties of its individual members. This is done by deriving new geometric facts about the system from those given explicitly by the distance and chirality constraints, a process known more generally as geometric reasoning. Although numerous constraints can be derived from knowledge of the molecular formula, in many cases (e.g., globular proteins) additional noncovalent constraints are needed in order to define precisely the accessible conformation space. These must be obtained from additional experiments, and thus one of the best-known applications of distance geometry is the determination of molecular conformation from experimental data, most notably NMR spectroscopy. Other important applications include enumerating the conformation spaces of small molecules, ligand docking and pharmacophore mapping in drug design, and the homology modeling of protein structure.