Error Analysis of Point Set Alignments for Use In Identification of Conserved Polypeptide Segments

A recognition of conserved protien structure frequently requires a three dimensional alignment of some of the common structural features of pairs of polypeptide segmants represented as rigid point sets. The need for an estimation of a "mean" or representative structure composed from a collection of several possibly overlapping polypeptide segments also arises naturally in the context of such studies. There is a number of least squares based solutions which have been exhaustively compared and described. However techniques for computing a propagation of errors have not been developed. Such methods will provide ways of characterizing the variations tolerated by evolutionary conservation and diversity that may be observed in examples of conserved protein segments. Also, they will provide the means for testing whether new conservation candidates produce alignments consistent with the variability of previous observations. This paper presents a computation for the propagation of errors for the x2 estimation of displacements and (orthogonally constrained) rotation matrices that comprise pose transformations. Further, a consistent formulation of knean or average point-sets constructed from an alignment of a group of rigid point sets is developed from the perspective of a minimization of a x2, together with an error analysis of the average point-sets along with the interactions with the translation and rotation errors of each of the rigid point sets to the average point set.