Page 145 - Macromolecular Crystallography

134 MACROMOLECULAR CRYS TALLOGRAPHY

Stanley, 1963), has proved to be quite powerful computer programs such as MULTAN (Main et al.,
and efficient as an optimization method when used 1980) and SHELXS (Sheldrick, 1990), can provide
to reduce the value of the minimal function. For solutions for structures containing less than 100–150
example, a typical phase-refinement stage consists unique non-hydrogen atoms. With few exceptions,
of three iterations or scans through the reflection however, phase refinement alone is not sufficient
list, with each phase being shifted a maximum of to solve larger structures. In such cases, successful
◦
two times by 90 in either the positive or negative applications require a dual-space optimization pro-
direction during each iteration. The refined value cedure that has come to be known as Shake-and-Bake
for each phase is selected, in turn, through a pro- (Weeks et al 1994; Miller et al., 1993). Shake-and-
cess that involves evaluating the minimal function Bake is also a powerful method for smaller struc-
using the original phase and each of its shifted val- tures and substructures, effectively avoiding most
ues (Weeks et al., 1994). The phase value that results cases of false minima. The distinctive feature of
in the lowest minimal-function value is chosen at this procedure is the repeated and unconditional
each step. Refined phases are used immediately in alternation of reciprocal-space phase refinement
the subsequent refinement of other phases. (Shaking) with a complementary real-space process
that seeks to improve phases by applying con-
straints (Baking). The Shake-and-Bake algorithm was
9.3.3 Dual-space optimization
implemented first in SnB and then independently
Conventional reciprocal-space direct methods, in SHELXD. Although both reciprocal-space opti-
implementing tangent-formula refinement in mization methods are available in both programs,

Start

2 1
Generate Normalize
invariants data
3 7
Generate Yes Another No Manual
trial stop
trial?
FFT
4
Refine Pick
phases Yes FFT-1 peaks
Another
cycle?
No
No
No
5 6 Automated
Solution Auto
Compute FOMs stop
found? Yes run? Yes
Figure 9.2 A flow chart for structure solution using the direct-methods algorithm, Shake-and-Bake. Steps 1 and 2 involve computation of the
normalized structure-factor magnitudes and generation of the triplet structure invariants, respectively. Step 3 is the start of the outer or trial
structure loop in which trial structures consisting of randomly positioned atoms are generated and tested. Step 4 is the inner or Shake-and-Bake
cycle which consists of a phase-refinement step involving either tangent-formula refinement or parameter-shift optimization of the minimal
function, a Fourier transform resulting in an electron-density map, selection of the largest peaks on the map, and an inverse Fourier transform to
generate new phases by means of a structure-factor calculation assuming these peaks to be atoms. The number of such cycles and the number of
peaks used in the structure-factor calculation depend on the number of atoms to be located (Table 9.4). Step 5 is the evaluation of the refined
trial structures based on the values of the figures of merit (see Section 9.4) to determine which trials, if any, are solutions. If solutions are being
recognized automatically (Step 6), the substructure determination is terminated as soon as the first solution is found. Otherwise,
processing continues (Step 7) until a predetermined number of trial structures are completed.