On the direct determination of three-dimensional crystallographic phases at low resolution: crambin at 6 A

Using a pseudo-atom approach, the three-dimensional crystallographic phases for the protein crambin (a = 40.76, b = 18.49, c = 22.33 A, beta = 90.61 degrees, space group P2(1)) were determined to 6 A by direct methods. First, the centrosymmetric h0l set was assigned phases by symbolic addition, and the initial solution was then refined by Fourier methods. Phase values of strong reflections were then permuted, and the decision to change the phase value for two of these was made by consulting a cross-correlation of the experimental density histogram to the theoretical or known histogram for the protein. The two-dimensional basis was then extended by the Sayre equation into three dimensions by assigning a phase to a third allowed hkl origin-defining reflection and an algebraic value to another axial reflection. The correct solution was again identified by the histogram correlation, yielding a solution in which the mean phase error for all 98 reflections was 61.5 degrees or 23.1 degrees for the 21 most intense reflections. A parallel study with another protein indicates this method may have general utility.