On updating torsion angles of molecular conformations

A conformation of a molecule is defined by the relative positions of atoms and by the chirality of asymmetric atoms in the molecule. The three main representations for conformations of molecules are Cartesian coordinates, a distance geometry descriptor (which consists of a distance matrix and the signs of the volumes of quadruples of atoms), and internal coordinates. In biochemistry, conformational changes of a molecule are usually described in terms of internal coordinates. However, for many applications, such as molecular docking, the Cartesian coordinates of atoms are needed for computation. Although, for each conformational change, the Cartesian coordinates of atoms can be updated in linear time (which is optimal asymptotically), the constant factor becomes significant if a large number of updates are needed. Zhang and Kavraki (J. Chem. Inf. Comput. Sci. 2002, 42, 64-70) examined three methods: the simple rotations, the DenavitHartenberg local frames, and the atom-group local frames. On the basis of their implementations, they showed that the atom-group local frames are more efficient than the other two. In this paper, by expressing the torsion-angle change as a composition of translations and rotations, we observe that the simple rotations can be implemented in an efficient way by taking advantage of consecutive operations. Both quantitative and experimental comparisons show that the improved simple rotations, in which rotations are expressed in unit quaternions, are as efficient as the atom-group local frames and, thus, have the advantage of avoiding the need of precomputations of a set of local frames and transformations between them.