An envelope-based approach for direct phase determination of macromolecular structures

An initial electron density distribution for a crystal structure may be directly derived from observed diffraction data by maximizing the product of the observed and calculated Patterson functions with respect to the electron density values within an envelope. This maximization problem may be formulated as an eigenvalue equation, in which potential electron density distributions are obtained as eigendensities (eigenvectors) of a symmetric matrix. Elements of this matrix depend only on the indices and intensities of the observed reflections, and on the coordinates of grid points inside the envelope. Eigendensities are calculated for a set of small envelopes (enclosing about 20% of the molecular volume) covering a unique region of the unit cell whose points are unrelated by space-group operations, origin shifts or changes in enantiomorph. On the basis of correlation coefficients between the observed and calculated values for both the Patterson function and structure factor amplitudes, a small set of eigendensities are selected for combination into a final electron density distribution. This electron density distribution may be Fourier transformed to yield calculated structure factors. Test calculations on lysozyme indicate that phase errors of less than 60° may be obtained for strong low-resolution reflections by this procedure. An extension of this approach to handle crystal structures containing non-crystallographic symmetry is described.