Symbolic addition in protein electron crystallography--a method for finding projected helices.

The crystal structure of orthorhombic bacteriorhodopsin was determined in projection by direct methods from electron diffraction amplitudes, assuming that, after re-scaling the problem, the Fourier transform of projected alpha-helices could be modeled by atomic scattering factors. A basic set comprising two origin-defining phases, two phase values from sigma 1 triple estimates and an algebraic unknown (resolved early in the phase determination) was extended to a total set of 20 terms, with only two errors. Five helix sites were observed in the first potential map and, after three cycles of Fourier refinement, the rest of the asymmetric unit was found. The overall phase accuracy was 47 degrees or 22 degrees for the 25 most intense reflections.