Likelihood and Empirical Bayes Superposition of Multiple Macromolecular Structures
Superpositioning plays a fundamental role in current macromolecular structural analysis. By orienting structures so that their atoms match closely, superpositioning enables the direct analysis of conformational similarities and differences in three-dimensional Euclidean space. Superpositioning is a special case of Procrustes problems, in which coordinate vector sets are optimally oriented via rigid body rotations and translations. Optimal transformations are conventionally determined by minimizing the sum of the squared distances between corresponding atoms in the structures. However, the ordinary unweighted least-squares (OLS) criterion can produce inaccurate results when the atoms have heterogeneous variances (heteroscedasticity) or the atomic positions are correlated, both of which are common features of real data. In contrast, model-based probabilistic methods can easily allow for heterogeneous variances and correlations. Our likelihood treatment of the superposition problem results in more accurate superpositions and provides a framework for a full Bayesian analysis.