title: Using quaternions to calculate RMSD.
abstract: A widely used way to compare the structures of biomolecules or solid bodies is to translate and rotate one structure with respect to the other to minimize the root-mean-square deviation (RMSD). We present a simple derivation, based on quaternions, for the optimal solid body transformation (rotation-translation) that minimizes the RMSD between two sets of vectors. We prove that the quaternion method is equivalent to the well-known formula due to Kabsch. We analyze the various cases that may arise, and give a complete enumeration of the special cases in terms of the arrangement of the eigenvalues of a traceless, 4 x 4 symmetric matrix. A key result here is an expression for the gradient of the RMSD as a function of model parameters. This can be useful, for example, in finding the minimum energy path of a reaction using the elastic band methods or in optimizing model parameters to best fit a target structure.
authors.affiliation: Department of Mathematics and Statistics, University of New Mexico, Albuquerque 87131, USA.
authors.forename: Evangelos A
authors.initials: EA
authors.lastname: Coutsias
authors.forename: Chaok
authors.initials: C
authors.lastname: Seok
authors.forename: Ken A
authors.initials: KA
authors.lastname: Dill
mesh_terms: D000465:Algorithms; D008962:Models, Theoretical; D011506:Proteins / Q000737:chemistry*; D013816:Thermodynamics
chemical_list: D011506:Proteins