Automatic recovery of missing amplitudes and phases in tilt-limited electron crystallography of two-dimensional crystals

Electron crystallography of 2D protein crystals provides a powerful tool for the determination of membrane protein structure. In this method, data is acquired in the Fourier domain as randomly sampled, uncoupled, amplitudes and phases. Due to physical constraints on specimen tilting, those Fourier data show a vast un-sampled "missing cone" of information, producing resolution loss in the direction perpendicular to the membrane plane. Based on the flexible language of projection onto sets, we provide a full solution for these problems with a projective constraint optimization algorithm that, for sufficiently oversampled data, produces complete recovery of unmeasured data in the missing cone. We apply this method to an experimental data set of Bacteriorhodopsin and show that, in addition to producing superior results compared to traditional reconstruction methods, full, reproducible, recovery of the missing cone from noisy data is possible. Finally, we present an automatic implementation of the refinement routine as open source, freely distributed, software that will be included in our 2dx software package.