Section 2.2.10.2.  Ab initio direct phasing of proteins

Section 2.2.10.1 suggests that the mere use of the tangent formula or the Sayre equation cannot solve ab initio protein structures of usual size. However, even in an ab initio situation, there is a source of supplementary information which may be used. Good examples are the `peaklist optimization' procedure (Sheldrick & Gould, 1995) and the SIR97 procedure (Altomare et al., 1999) for refining and completing the trial structure offered by the first E map.

In both cases there are reasons to suspect that the correct structure is sometimes extracted from a totally incorrect direct-methods solution. These results suggest that a direct-space procedure can provide some form of structural information complementary to that used in reciprocal space by the tangent or similar formulae. The combination of real- and reciprocal-space techniques could therefore enlarge the size of crystal structures solvable by direct methods. The first program to explicitly propose the combined use of direct and reciprocal space was Shake and Bake (SnB), which inspired a second package, half-bake (HB). A third program, SIR99, uses a different algorithm.

The SnB method (DeTitta et al., 1994; Weeks et al., 1994; Hauptman, 1995) is the heir of the cosine least-squares method described in Section 2.2.8, point (4). The function where is the triplet phase, and .

is expected to have a global minimum, provided the number of phases involved is sufficiently large, when all the phases are equal to their true values for some choice of origin and enantiomorph. Thus the phasing problem reduces to that of finding the global minimum of (the minimum principle).

SnB comprises a shake step (phase refinement) and a bake step (electron-density modification), the second step aiming to impose phase constraints implicit in real space. Accordingly, the program requires two Fourier transforms per cycle, and numerous cycles. Thus it may be very time consuming and it is not competitive with other direct methods for the solution of the crystal structures of small molecules. However, it introduced into the field the tremendous usefulness of intensive computations for the direct solution of complex crystal structures.

Owing to Sheldrick (1997), HB does most of its work in direct space. Random atomic positions are generated, to which a modified peaklist optimization process is applied. A number of peaks are eliminated subject to the condition that remains as large as possible (only reflections with are involved, where ). The phases of a suitable subset of reflections are then used as input for a tangent expansion. Then an E map is calculated from which peaks are selected: these are submitted to the elimination procedure.

Typically 5–20 cycles of this internal loop are performed. Then a correlation coefficient (CC) between and is calculated for all the data. If the CC is good (i.e. larger than a given threshold), then a new loop is performed: a new E map is obtained, from which a list of peaks is selected for submission to the elimination procedure. The criterion now is the value of the CC, which is calculated for all the reflections. Typically two to five cycles of this external loop are performed.

The program works indefinitely, restarting from random atoms until interrupted. It may work either by applying the true space-group symmetry or after having expanded the data to P1.

The SIR99 procedure (Burla et al., 1999) may be divided into two distinct parts: the tangent section (i.e., a double tangent process using triplet and quartet invariants) is followed by a real-space refinement procedure. As in SIR97, the reciprocal-space part is followed by the real-space refinement, but this time this last part is much more complex. It involves three different techniques: EDM (an electron-density modification process), the HAFR part (in which all the peaks are associated with the heaviest atomic species) and the DLSQ procedure (a least-squares Fourier refinement process). The atomicity is gradually introduced into the procedure. The entire process requires, for each trial, several cycles of EDM and HAFR: the real-space part is able to lead to the correct solution even when the tangent formula does not provide favourable phase values.