Bayesian alignment using hierarchical models, with applications in protein bioinformatics

Abstract

SUMMARYAn important problem in shape analysis is to match configurations of points in space filtering out some geometrical transformation. In this paper we introduce hierarchical models for such tasks, in which the points in the configurations are either unlabelled, or have at most a partial labelling constraining the matching, and in which some points may only appear in one of the configurations. We derive procedures for simultaneous inference about the matching and the transformation, using a Bayesian approach. Our model is based on a Poisson process for hidden true point locations; this leads to considerable mathematical simplification and efficiency of implementation. We find a novel use for classical distributions from directional statistics in a conditionally conjugate specification for the case where the geometrical transformation includes an unknown rotation. Throughout, we focus on the case of affine or rigid motion transformations. Under a broad parametric family of loss functions, an optimal Bayesian point estimate of the matching matrix can be constructed, that depends only on a single parameter of the family.Our methods are illustrated by two applications from bioinformatics. The first problem is of matching protein gels in two dimensions, and the second consists of aligning active sites of proteins in three dimensions. In the latter case, we also use information related to the grouping of the amino acids. We discuss some open problems and suggest directions for future work.

Generation of three-dimensional random rotations in fitting and matching problems

“…When superimposing two rigid bodies or point sets by successive rotation and translation, the posterior probability density function of the rotation matrix has the above form. Recently, Green and Mardia derived such a posterior distribution in the context of protein structure alignment (Green and Mardia 2006). The same distribution occurs in a probabilistic solution of the Procrustes problem (Theobald and Wuttke 2006).…”

“…First we test our algorithm for a simple case where A is chosen randomly: We generated 10,000 rotation matrices using our algorithm, a random walk Metropolis scheme and the algorithm outlined in Green and Mardia (2006). In the Metropolis scheme and in the algorithm of Green and Mardia, the rotation matrix R is parameterized in Euler angles without transforming it into the optimal basis.…”

“…In their article, Green and Mardia (2006) present an algorithm for sampling threedimensional rotation matrices from the above distribution (1). However, their sampling algorithm appears to be suboptimal in two respects: the parameters (Euler angles) are coupled, and the polar angle cannot be drawn from a known distribution but has to be generated using a random walk Metropolis algorithm.…”

Alignment of Multiple Configurations Using Hierarchical Models

“…We focus specifically on generalizing the approach of Green and Mardia (2006), who described a Bayesian methodology for aligning two point configurations. We wish to extend this methodology to deal with an arbitrary number of configurations.…”

“…Specifically, we extend the two-configuration matching approach of Green and Mardia (2006) to the multiple configuration setting. Our approach is based on the introduction of a set of hidden locations underlying the observed configuration points.…”