Accurate Structural Correlations from Maximum Likelihood Superpositions

The cores of globular proteins are densely packed, resulting in complicated networks of structural interactions. These interactions in turn give rise to dynamic structural correlations over a wide range of time scales. Accurate analysis of these complex correlations is crucial for understanding biomolecular mechanisms and for relating structure to function. Here we report a highly accurate technique for inferring the major modes of structural correlation in macromolecules using likelihood-based statistical analysis of sets of structures. This method is generally applicable to any ensemble of related molecules, including families of nuclear magnetic resonance (NMR) models, different crystal forms of a protein, and structural alignments of homologous proteins, as well as molecular dynamics trajectories. Dominant modes of structural correlation are determined using principal components analysis (PCA) of the maximum likelihood estimate of the correlation matrix. The correlations we identify are inherently independent of the statistical uncertainty and dynamic heterogeneity associated with the structural coordinates. We additionally present an easily interpretable method (“PCA plots”) for displaying these positional correlations by color-coding them onto a macromolecular structure. Maximum likelihood PCA of structural superpositions, and the structural PCA plots that illustrate the results, will facilitate the accurate determination of dynamic structural correlations analyzed in diverse fields of structural biology.: Biological macromolecules comprise extensive networks of interconnected atoms. These complex coupled networks result in correlated structural dynamics, where atoms and residues move and evolve together as concerted conformational changes. The availability of a wealth of macromolecular structures necessitates the use of robust strategies for analyzing the correlated modes of motion found in molecular ensembles. Current strategies use a combination of least-squares superpositions and statistical analysis of the structural covariance matrix. However, the least-squares treatment implicitly requires that atoms are uncorrelated and that each atom has the same positional uncertainty, two assumptions which are violated in structural ensembles. For example, the atoms in the proteins are connected by chemical bonds, covalent and non-covalent, resulting in strong correlations. Furthermore, different atoms have different variances, because some atoms are known with less precision or have greater mobility. Using maximum likelihood (ML) analysis, we have developed a technique that is markedly more accurate than the classical least-squares approach by accounting for both correlations and heterogeneous variances. The improved ability to accurately analyze the major modes of dynamic structural correlations will benefit a diverse range of biological disciplines, including nuclear magnetic resonance (NMR) spectroscopy, crystallography, molecular dynamics, and molecular evolution.