Page 146 - Macromolecular Crystallography

APPLICATION OF DIRECT METHODS 135

SnB uses the parameter-shift optimization of the the atoms. The imposition of physical constraints
minimal function by default whereas SHELXD uses counteracts the tendency of phase refinement to
the tangent formula. The complete direct-methods propagate errors or produce overly consistent phase
phasing process using the Shake-and-Bake algorithm sets. For example, the ability to eliminate chemi-
is illustrated in flow-chart form in Fig. 9.2, and the cally impossible peaks at special positions using a
main parameters are summarized in Table 9.4. symmetry-equivalent cut-off distance prevents the
occurrence of most false minima (Weeks and Miller,
1999).
9.3.4 Peak picking
Peak picking is a simple but powerful way of impos-
9.4 Recognizing solutions
ing a real-space atomicity constraint. In each cycle of
the Shake-and-Bake procedure, the largest peaks are Since direct methods are implemented in the form
used as an up-dated trial structure without regard to of multiple-trial procedures, it is essential to have
chemical constraints other than a minimum allowed some means of distinguishing the refined trial struc-
distance between atoms (e.g. 1 Å for full structures tures that are solutions from those that are not. The
and 3 Å for substructures). By analogy to the com- SnB program computes three figures of merit that
mon practice in macromolecular crystallography of are useful for this purpose. These quantities are the
omitting part of a structure from a Fourier calcula- minimal function or R MIN (Eq. 5) calculated directly
tion in the hope of finding an improved position for from the constrained phases corresponding to the
the deleted fragment, a variant of the peak picking final peak positions for a trial structure, a crystallo-
procedure in which approximately one-third of the graphic R value based on normalized magnitudes
largest peaks are randomly omitted in each cycle can [i.e., R CRYST = ( ||E OBS |−|E CALC ||)/ |E OBS |],
also be used (Schneider and Sheldrick, 2002). and a correlation coefficient,
If markedly unequal atoms are present, appropri- 2 2 2
E
CC = wE OBS CALC · w− wE OBS · w
ate numbers of peaks (atoms) can be weighted by the
proper atomic numbers during transformation back 2
4 2 2
· E wE · w− wE
to reciprocal space in a subsequent structure-factor CALC OBS OBS
calculation. Thus, a priori knowledge concerning the
1/2
2

chemical composition of the crystal is used, but no · wE 4 CALC · w − wE 2 CALC ,
knowledgeofconstitutionisrequiredorusedduring
(6)
peak selection. It is useful to think of peak picking
in this context as simply an extreme form of density between |E OBS | and |E CALC | where the weight w is
modification appropriate when the resolution of the usually unity (Fujinaga and Read, 1987). The min-
data is small compared to the distance separating imal function and the crystallographic R have rela-
tively small values for solutions, but the correlation
coefficient is relatively large.
Table 9.4 Recommended values of the major Shake-and-Bake
parameters for substructure applications expressed in terms of N, the The use of figures of merit to determine whether
expected number of heavy atoms or anomalous scatterers solutions have been found and, if so, which tri-
als were successful can be illustrated with the
Parameter Recommended value 1JC4 Value methylmalonyl-coA epimerase (1JC4) test data. A
manual BnP job for 100 trial structures was run using
Phases 30N 840
Triplet invariants 300N 8400 the normalized anomalous difference magnitudes
Peaks selected N 28 (|E |) for the peak wavelength data. A maximum
Cycles 2 N 56 resolution cut-off of 3 Å was applied, and only reflec-
tions with a signal-to-noise ratio (i.e. |E |/σ(|E |))
The values used for the 1JC4 example are based on the sequence information greater than 3.0 were used. A histogram of the final
that there are seven methionines per chain and that the Matthews coefficient and
solvent fraction are consistent with four chains in the asymmetric unit; therefore, R MIN values for the 100 trials is shown in Fig. 9.3. A
N = 28. clear bimodal distribution of R MIN values is a strong