Propagation of Conformational Coordinates Across Angular Space in Mapping the Continuum of States from Cryo-EM Data by Manifold Embedding

Recent approaches to the study of biological molecules employ manifold learning to single-particle cryo-EM data sets to map the continuum of states of a molecule into a low-dimensional space spanned by eigenvectors or "conformational coordinates". This is done separately for each projection direction (PD) on an angular grid. One important step in deriving a consolidated map of occupancies, from which the free energy landscape of the molecule can be derived, is to propagate the conformational coordinates from a given choice of "anchor PD" across the entire angular space. Even when one eigenvector dominates, its sign might invert from one PD to the next. The propagation of the second eigenvector is particularly challenging when eigenvalues of the second and third eigenvector are closely matched, leading to occasional inversions in their ranking as we move across the angular grid. In the absence of a computational approach, this propagation across the angular space has been done thus far "by hand" using visual clues, thus greatly limiting the general use of the technique. In this work we have developed a method that is able to solve the propagation problem computationally, by using optical flow and a probabilistic graphical model. We demonstrate its utility by selected examples.