Defining and Computing Optimum RMSD for Gapped and Weighted Multiple-Structure Alignment
Pairwise structure alignment commonly uses root-mean-square deviation (RMSD) to measure the structural similarity, and methods for optimizing RMSD are well established. We extend RMSD to weighted RMSD for multiple structures. By using multiplicative weights, we show that weighted RMSD for all pairs is the same as weighted RMSD to an average of the structures. Thus, using RMSD or weighted RMSD implies that the average is a consensus structure. Although we show that in general, the two tasks of finding the optimal translations and rotations for minimizing weighted RMSD cannot be separated for multiple structures like they can for pairs, an inherent difficulty and a fact ignored by previous work, we develop an iterative algorithm, in which each iteration takes linear time and the number of iterations is small, to converge weighted RMSD to a local minimum. The 10,000 experiments of gapped alignment done on each of 23 protein families from HOMSTRAD (where each structure starts with a random translation and rotation) converge rapidly to the same minimum. Comparisons to other multiple-structure alignment programs show that our algorithm achieves better RMSDs. Finally, we propose a heuristic method to iteratively remove the effect of outliers and find well-aligned positions that determine the structural conserved region by modeling B-factors and deviations from the average positions as weights and iteratively assigning higher weights to better aligned atoms.
multiple-structure alignment weighted rmsd structural conserved region optimization methods