Mixture models for spherical data with applications to protein bioinformatics

TL;DR

This paper develops exact maximum likelihood mixture models using Kent distributions for spherical data, applied to analyze hydrogen bond geometries in proteins to understand protein folding mechanisms.

Contribution

It introduces an exact MLE approach for Kent mixture models and applies it to protein hydrogen bond data, providing new insights into secondary structure interactions.

Findings

Kent mixture models effectively capture hydrogen bond geometry

Distinct bond patterns associated with secondary structures

Enhanced understanding of protein folding mechanisms

Abstract

Finite mixture models are fitted to spherical data. Kent distributions are used for the components of the mixture because they allow considerable flexibility. Previous work on such mixtures has used an approximate maximum likelihood estimator for the parameters of a single component. However, the approximation causes problems when using the EM algorithm to estimate the parameters in a mixture model. Hence the exact maximum likelihood estimator is used here for the individual components. This paper is motivated by a challenging prize problem in structural bioinformatics of how proteins fold. It is known that hydrogen bonds play a key role in the folding of a protein. We explore this hydrogen bond geometry using a data set describing bonds between two amino acids in proteins. An appropriate coordinate system to represent the hydrogen bond geometry is proposed, with each bond represented as a point on a sphere. We fit mixtures of Kent distributions to different subsets of the hydrogen bond data to gain insight into how the secondary structure elements bond together, since the distribution of hydrogen bonds depends on which secondary structure elements are involved.