The phase problem for two-dimensional crystals. I. Theory.

Properties of the phase problem for two-dimensional crystals are examined. This problem is relevant to protein structure determination using diffraction from two-dimensional crystals that has been proposed using new X-ray free-electron laser sources. The problem is shown to be better determined than for conventional three-dimensional crystallography, but there are still a large number of solutions in the absence of additional a priori information. Molecular envelope information reduces the size of the solution set, and for an envelope that deviates sufficiently from the unit cell a unique solution is possible. The effects of various molecular surface features and incomplete data on uniqueness and prospects for ab initio phasing are assessed. Simulations of phase retrieval for two-dimensional crystal data are described in the second paper in this series.