A strict solution for the optimal superimposition of protein structures

Existing methods for the optimal superimposition of one vector set on another in the comparison of parts or the whole of related protein molecules are based on the precondition that the centroids of the two sets are coincident. As a result, the translation components of the transformation are artificially removed from the superimposition process. This is obviously not strict in the mathematical sense. The theorem presented in this paper is a strict solution for the optimal superimposition of two vector sets, which is in fact the problem of the weighted optimal rigid superimposition of two vector sets. Examples show its advantages compared with the method of simply coinciding the centroids of the two vector sets for the translation transformation.