On the multiple simultaneous superposition of molecular structures by rigid body transformations

Abstract

A method of optimally superimposing n coordinate sets on each other by rigid body transformations, which minimizes the sum of all n(n - 1)/2 pairwise residuals, is presented. In the solution phase the work load is approximately linear on n, is independent of the size of the structures, is independent of their initial orientations, and terminates in one cycle if n = 2 or if the coordinate sets are exactly superposable, and otherwise takes a number of cycles dependent only on genuine shape differences. Enantiomorphism, if present, is detected, in which case the option exists to reverse or not to reverse the chirality of relevant coordinate sets. The method also offers a rational approach to the problem of multiple minima and has successfully identified four distinct minima in such a case. Source code, which is arranged to enable the study of the disposition of domains in multidomain structures, is available from the author.