Three-dimensional diffractive imaging for crystalline monolayers with one-dimensional compact support

The use of a compact support constraint along the beam direction is considered as a solution to the phase problem for diffraction by two-dimensional protein crystals. Specifically we apply the iterative Gerchberg-Saxton-Fienup algorithm to simulated three-dimensional transmission electron diffraction data from monolayer organic crystals. We find that oversampling along the reciprocal-lattice rods (relrods) normal to the monolayer alone does not solve the phase problem in this geometry in general. However, based on simulations for a crystalline protein monolayer (lysozyme), we find that convergence is obtained in three dimensions if phases are supplied from a few high resolution electron microscope images recorded at small tilts to the beam direction. In the absence of noise, amplitude-weighted phase residuals of around 5°, and a cross-correlation coefficient of 0.96 between the true and estimated potential are obtained if phases are included from images at tilts of up to 15°. The performance is almost as good in the presence of noise at a level that is comparable to that commonly observed in electron crystallography of proteins. The method should greatly reduce the time and labor needed for data acquisition and analysis in cryo-electron microscopy of organic thin crystals by avoiding the need to record images at high tilt angles.