Rapid calculation of RMSDs using a quaternion-based characteristic polynomial

A common measure of conformational similarity in structural bioinformatics is the minimum root mean square deviation (RMSD) between the coordinates of two macromolecules. In many applications, the rotations relating the structures are not needed. Several common algorithms for calculating RMSDs require the computationally costly procedures of determining either the eigen decomposition or matrix inversion of a 3x3 or 4x4 matrix. Using a quaternion-based method, here a simple algorithm is developed that rapidly and stably determines RMSDs by circumventing the decomposition and inversion problems.