Page 127 - Macromolecular Crystallography

116 MACROMOLECULAR CRYS TALLOGRAPHY

There are three unknowns in these equations: small, compared to the total diffraction from a unit
cell. This approach permits ready generalization to
0 0 0 0
| F T (h)|, | F A (h)| and φ (h) − φ (h) = φ
A
T
the simultaneous presence of many kinds of anoma-
The quantities, a(λ), b(λ), and c(λ) are, by def- lous scatterers in the asymmetric unit (Terwilliger,
inition, scalar functions of f and f and can be 1994a). Briefly, the pseudo SIRAS formalism may be

estimated through direct measurements of these summarized as below.
−
+
quantities, without knowledge of the structure. His- Denoting by F (λ j ) and F (λ j ), the two compo-
torically, in early work on MAD phasing, these nents of a Bijvoet pair, at wavelength λ j , the average
three unknowns were determined through an alge- structure factor amplitude at that wavelength is
braic method due to Karle (1980). Measurements given by
at several wavelengths provided the over deter- 1 + −
mination needed for a least squares determination F(λ j ) = 2 [F (λ j ) + F (λ j )]
of these unknown parameters (Hendrickson, 1985). and the Bijvoet difference by
Although the above relations treat the case of a
+
−
Ano (λ j ) = F (λ j ) − F (λ j )
single anomalous scatterer, the methodology can,
To achieve analogy with MIR, the normal and
in principle, be generalized to several anomalous
anomalous diffraction parts for a reflection are
scatterers.
expressed as separate quantities,
Most of the later applications of MAD phasing
have, however, followed the treatment of MAD F(λ j ) =|F 0 + F H (λ j )|
phasing as a special case of the MIR phasing
where F 0 and F H (λ j ) are the structure factor for
approach. Following a suggestion by Hendrickson
all the non-anomalously scattering atoms and that
(Hendrickson, 1991), the first practical application of
for just the anomalous scatterers, respectively. The
this approach was in the determination of the struc-
method then goes on to derive estimates for F 0 ,
tureofafragmentofhistoneH5(Ramakrishnan etal.,
Ano (λ 0 ) and F(λ 0 ), where the subscript ‘0’ indi-
1993). This approach has the advantage that the sig-
cates that one of the wavelengths that the data have
nificant body of theoretical knowledge accumulated
been measured at can be arbitrarily chosen for this
in the application of the MIR procedure, as well
evaluation. An estimate for F H (λ j ), the structure
as the numerous programs that implement it, can
factor at any of the wavelengths other than the
directly be used in analysis of MAD data. The basic
chosen one, can be derived from the observation
tenet of the MAD as a special case of MIR approach is
that anomalous scattering changes in magnitude
to treat the data measured at one of the wavelengths
with wavelength although the phase is wavelength
as ‘native’. Data measured at the other wavelengths
independent,
are then treated as a set of derivatives; dispersive

f 0 + f (λ j )
differences play the role of isomorphous differences F H (λ j ) = F H (λ 0 )

while the Bijvoet differences provide the orthogo- f 0 + f (λ 0 )

nal, anomalous information; their traditional role where the terms containing the f s represent the
in MIR. The theoretical basis of this approach is real part of the scattering factor for the anomalous
detailed in a review by Ramakrishnan and Biou scatterers.
(1997). However, a limitation of this approach is that An approximation to Ano (λ 0 ) is also estimated
it cannot be easily generalized to the case of multiple from similar arguments (Terwilliger and Eisenberg,
types of anomalous scatterers within a unit cell or to 1987),

multiple MAD data sets. In addition, one set of data, f (λ j )
Ano (λ j ) ≈ Ano (λ 0 )
the ‘native’, is arbitrarily treated as not containing f (λ 0 )

an anomalous component. This latter assumption
If one assumes that the anomalous component is
is, in most MAD data measurement regimes, quite
small relative to the normal scattering from the unit
unjustified. A modification, which approximates a
cell contents, then
pseudo SIRAS situation, is based on the sugges-
tion that the magnitude of anomalous diffraction is F(λ j ) ≈ F 0 + F H (λ j ) cos α