Common lines Ab initio reconstruction of D₂-symmetric molecules in cryo-electron microscopy^*

Cryo-electron microscopy is a state-of-the-art method for determining high-resolution three-dimensional models of molecules from their two-dimensional projection images. A key step in this method is estimating a low-resolution model of the investigated molecule using only its projection images without any other prior information. Robust algorithms for this step for molecules without symmetry or with cyclic symmetry have been proposed, typically based on common lines between pairs of im-ages. Deriving a common lines algorithm for molecules with D₂ symmetry is more challenging, since such molecules are invariant to an arbitrary orientation-preserving permutation of their coordinate system. This invariance also renders previous common lines algorithms inapplicable to D₂ symmetric molecules. In this work, we present a common lines algorithm for determining the structure of molecules with D₂ symmetry. We derive the geometry that the D₂ symmetry group induces on the common lines between pairs of images, develop a procedure for estimating the relative orientation of pairs of images, characterize the ambiguities inherent in these orientations due to the D₂ sym-metry, and describe a robust method for combining all relative orientations into a single consistent assignment of orientations to all images. In contrast to local-search based methods, our procedure is unbiased, reference free, and guaranteed to find a globally consistent solution for the orientation assignment problem. We demonstrate the applicability of our algorithm using experimental cryo-electron microscopy data.