Page 128 - Macromolecular Crystallography

MAD PHASING 117

Where α is the difference in the phase angle between more poorly determined phases computed through
the anomalous and non-anomalous structure fac- the OAS phase-ambiguity resolution procedure (Fan
tors, the equivalent of φ in the Hendrickson formu- et al., 1990). The direct methods phase probabilities
lation above. From these isomorphous equivalents for each phase are determined from each of the OAS
from the MAD data can be formulated subsets, through a modified tangent formula, and
an improved phase set is generated through com-
Iso (λ 0 ) ≈ F H (λ 0 ) cos α
bination with the starting MAD phases, weighted
F(λ i ) − F(λ j ) with their associated figures of merit (Gu et al.,

≈[ f 0 + f (λ j )]

f (λ i ) − f (λ j ) 2001). Estimation of phases from MAD data through

computation of the joint probability distribution
From this, one might estimate approximate values
functions for each phase, based on earlier direct
for F 0 and F(λ 0 ),
methodsapproaches, hasalsobeenreported.Aprob-

f 0 + f (λ j ) ability distribution function approach had earlier
F 0 ≈ F(λ j ) − Iso (λ 0 ) and 0
f 0 + f (λ 0 ) been developed for estimation of | F A (h)| values

(Giacovazzo and Siliqi, 2001; Burla et al., 2003). It has
F(λ 0 ) ≈ F 0 + Iso (λ 0 )
been combined with a procedure for estimation of
0
0
Setting up a formal equivalence between MAD the probabilities for | F T (h)| and φ (h), using nor-
T
and MIR data representations is very useful for com- malized structure factors, derived from measured
bining information from two different sources, for MAD data, to provide a complete solution to the
cases in which each by itself is insufficient for struc- MAD phasing problem. The theoretical principles
ture solution. These sources could, for example, be and application to a number of test data sets have
MAD data sets determined from crystals with dif- been described (Giacovazzo and Siliqi, 2004).
ferent types of anomalous scatterers; or a MAD and
a MIR data set. The assumption, that anomalous sig-
nals are small compared to normal scattering, leads 8.3 Experimental considerations
to some errors, primarily in the estimation disper-
The following sections address some of the choices
sive differences. These errors, however, have been
that need to be made in the design and implementa-
estimated to be no larger than about 4% (Terwilliger,
tion of MAD experiments. Examples are taken from
1994a). An alternative formulation, that does not
the work in the author’s laboratory since he is most
depend on either assuming a special ‘error free’ data
familiar with those.
set or that the anomalous signal be small, has also
been described (Bella and Rossmann, 1998). The
method is based on an earlier, theoretical approach 8.3.1 Incorporation of anomalous scatterers
developed for MIR which treats all isomorphous
data sets as equivalent, in terms of error analysis The best atomic types to include as anomalous scat-
(Cullis et al., 1961). It is also easily extensible to mul- terers are those that produce the largest absorption
tiple kinds of anomalous scatterers, as well as MAD and dispersive signals. Electrons in the L and M
experiments made on different crystals. shells of atoms are less strongly held by their nuclei

More recently, efforts have also been made to and generate larger f and f signals. Most reported
develop direct approaches to the calculation of MAD experiments have been performed at either
phases from MAD data. A method that augments a K or L edge, although at least one has used
other MAD phasing approaches, through improv- an M edge (Liu et al., 2001). Absorption curves
ing poorly estimated phases, has been suggested for the selenium K-edge and holmium L-edge are

(Fan et al., 1990). The method decomposes the shown in Fig. 8.1. Note that the f and f magni-
MAD data into a series of one-wavelength anoma- tudes for selenium are in the 4–5 electrons, while
lous scattering (OAS) subsets. A subset of MAD the corresponding values for holmium) are in the
phases with the highest figures of merit are used 18–20 electron range (Merritt, 1998). Techniques
as a starting point, and improved phases for the for the introduction of anomalous scatterers can be