METHODS IN ENZYMOLOGY
EDITORS-IN-CHIEF
John N. Abelson Melvin I. Simon
DIVISION OF BIOLOGY
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA
FOUNDING EDITORS
Sidney P. Colowick and Nathan O. Kaplan
Preface
Five years ago, Academic Press published parts A and B of volumes of Methods
in Enzymology devoted to Macromolecular Crystallography, which we had
edited. The editors of the series, in their wisdom, requested that we assemble
the present volumes. We have done so with the same logical style as before,
moving smoothly from methods required to prepare and characterize high
quality crystals and to measure high quality data, in the first volume, to
structure solving, refinement, display, and evaluation in the second. Although
we continue to look forward in these volumes, we also look resolutely back in
time by having recruited three chapters of reminiscence from some of those on
whose shoulders we stand in developing methods in modern times: Brian
Matthews, Michael Rossmann, and Uli Arndt. A spiritually similar contribu-
tion opens the second volume: David Blow’s introduction to our Phases section
has his personal reflections on the impact that Johannes Bijvoet has had on
modern protein crystallography.
In the earlier volumes, we foreshadowed a time when macromolecular
crystallography would become as automated as the technique applied to small
molecules. That time is not quite upon us, but we all feel rattling of the
windows from the heavy tread of high-throughput synchrotron-based macro-
molecular crystallography. As for the previous volumes, we have tried to
provide in this volume sufficient reference that those becoming immersed in
the field might find an explanation of methods they confront, while hopefully
also stimulating others to create the new and better methods that sustain

intellectual vitality. The years since publication of parts A and B have seen
amazing advances in all areas of the discipline. Super high brightness synchro-
tron sources (Advanced Photon Source in the United States, European Syn-
chrotron Radiation Facility in Europe, and Super Photon Ring-8 in Japan) are
producing numerous important results even while the older sources are increas-
ing productivity. Proteomics and structural genomics have appeared in the
lexicon of all biologists and have become vital research programs in many
laboratories. In the spirit of the time, these chapters approach many of the
methods that are pertinent to high-throughput structure determination. These
are now robots for large-scale screening of crystal-growth conditions using
sub-microliter volumes, which were accessible only in a few dedicated research
laboratories a decade ago. Similarly, automation has begun to assume increas-
ing roles in cryogenic specimen changing for data collection; many laboratories
are building and beginning to use robots for this purpose.
xiii
The first and largest section of technical chapters dissects the cutting-edge
methods for thinking about or accomplishing crystal growth, including theoret-
ical aspects, using physical chemistry to understand and improve crystal dif-
fraction quality, robotics, and cryocrystallography. The other large section
addresses phasing. A profound shift has occurred with the growing appreci-
ation that map interpretation and model refinement are inseparable from the
phase problem itself. Various methods of integrating the two processes in
automated algorithms constitute an important step toward realization of
high-throughput. More importantly perhaps, they improve the resulting struc-
tures themselves. New algorithms for representing the variance parameters
have come into wider practice.
The database of solved macromolecular structures has grown to the point
where its statistical properties now afford impressive insight and can be used
to improve the quality of structures. Concurrently, simulation methods have
become more accessible, reliable, and relevant. The validation process is there-

fore one that impacts a widening sphere of activities, including homology
modeling and the presentation and analysis of conformational, packing, and
surface properties. Many of these are reviewed in the concluding chapters.
We take little credit, either for the quality of the volume, which goes to the
chapter authors, or for comprehensive coverage of competing methods. We will
happily accept blame for mistakes and omissions. Academic Press has
remained supportive and helpful throughout the long and trying process of
completing this job, earning our sincere appreciation.
Charles W. Carter
Robert M. Sweet
xiv preface
Contributors to Volume 374
Article numbers are in parentheses and following the names of contributors.
Affiliations listed are current.
Jan Pieter Abrahams (8), Biophysical
Structureal Chemistry, Leiden Institute
of Chemistry, 2300 RA Leiden, The
Netherlands
Paul D. Adams (3), Lawrence Berkeley
Laboratory, 1 Cyclotron Road, Berkeley,
California 94720
Vandim Alexandrov (23), Department of
Biochemistry and Biophysics, Texas A &
M University, College Station, Texas,
77843
W. Bryan Arendall, III. (18), Depart-
ment of Biochemistry, Duke University,
Duke Building, Durham, North Carolina
27708
Nenad Ban (8), Institute for Molecular

Biology and Biophysics, Swiss Federal
Institute of Technology, CH8093 Zurich,
Switzerland
Joel Berendzen (3), Biophysics Group,
Los Alamos National Laboratory, Los
Alamos, New Mexico 87545
Helen M. Berman (17), Research
Callaboratory for Structural Bio-
informatics, Department of Chemistry,
Rutgers The State University of
New York, Piscataway, New Jersey 08854
D. M. Blow (1), 26 Riversmeet, Appledore,
Bideford, Devon EX39 1RE, United
Kingdom
Jose M. Borreguero (25), Department of
Biochemistry and Biophysics, University
of North Carolina at Chapel Hill, Chapel
Hill, North Carolina 27599
Axel T. Brunger (3), The Howard Hughes
Medical Institute and Departments of
Molecular and Cellular Physiology,
Neurology, and Neurological Sciences,
Stanford Radiation Laboratory, Stanford
University, 1201 Welch Road, Stanford,
California 94205
Sergey V. Buldyrev (25), Department of
Biochemistry and Biophysics, University
of North Carolina at Chapel Hill, Chapel
Hill, North Carolina 27599
Kyle Burkhardt (17), Research Calla-

boratory for Structural Bioinformatics,
Department of Chemistry, Rutgers The
State University ofNewYork, Piscataway,
New Jersey 08854
Raul E. Cachau (15), Advanced Bio-
medical Computer Center, Frederic,
Maryland 21703
Stephen Cammer (22), University of
California San Diego Libraries, 9500
Gillman Drive, La Jolla, California 92093
Charles W. Carter,Jr. (7, 22),
Department of Biochemistry and Bio-
physics, University of North Carolina at
Chapel Hill, Chapel Hill, North Carolina
27599
Zbigniew Dauter (5), Synchroton
Radiation Research Section, NCI Broo-
khaven National Laboratory Building,
Upton, New York 11973
Feng Ding (25), Department of Biochemis-
try and Biophysics, University of North
Carolina at Chapel Hill, Chapel Hill,
North Carolina 27599
ix
Eleanor J. Dodson (3), Department of
Chemistry, University of York, Heslington
York YO1 5DD, United Kingdom
Nikolay V. Dokholyn (25), Department
of Biochemistry and Biophysics, Univer-
sity of North Carolina at Chapel Hill,

Chapel Hill, North Carolina 27599
Zukang Feng (17), Research Callabora-
tory for Structural Bioinformatics, De-
partment of Chemistry, Rutgers The
State University of New York, Piscat-
away, New Jersey 08854
Andra
´
s Fiser (20), Department of Bio-
chemistry and Seaver Foundation Center
for Bioinformatics, Albert Einstein Col-
lege of Medicine, Bronz, New York 10461
Roger Fourme (4), Soleil (CNRS-CEA-
MEN), Batiment 209d, Universite Paris
XI, 91898 Orsay, Cedex France
Mark Gerstein (23), Department of
Molecular Biophysics and Biochemistry,
Yale University, New Haven, Connecticut
06520
Ralf W. Gosse-Kunstleve (3), Lawrence
Berkeley Laboratory, 1 Cyclotron Road,
Berkeley, California 94720
Dorit Hanein (10), The Burnham Institute,
La Jolla, California 92037
Jan Hermans (19), Department of Bio-
chemistry and Biophysics, University of
North Carolina at Chapel Hill, Chapel
Hill, North Carolina 27599
Barry Honig (21), Department of Biochem-
istry and Molecular Biophysics, Columbia

University, New York, New York 10032
Thomas R. Ioerger (12), Texas A & M
University, College Station, Texas 77843
Ronald Jansen (23), Department of
Molecular Biophysics and Biochemistry,
Yale University, New Haven, Connecticut
06520
Andrzej Joachimiak (15), Structural Biol-
ogy Sciences, Biosciences Division, Ar-
gonne National Laboratory, Argonne,
Illinois 60439
Jochen Junker (23), Max Planck Institut
fur Biophysikalische Chemie, D37070
Gottingen, Germany
Michel H. J. Koch (24), European
Molecular Biology Laboratory, Ham-
burg Outstation, D-22603 Hamburg,
Germany
W. G. Krebs (23), San Diego Supercom-
puter Center, University of California
San Diego, La Jolla California 92093
Victor S. Lamzin (11), European Molecu-
lar Biology Laboratory, Hamburg
Outstation, 22603 Hamburg, Germany
Richard J. Morris (11), European Bioin-
formatics Institute, Wellcome Trust
Genome Campus, Cambridge CB10 1SD,
United Kingdom
Garib N. Murshudov (14), Chemistry De-
partment, University of York, Helsington

York, YO1 5DD, United Kingdom
Ronaldo A. P. Nagem (5), CBME
Laboratorio Nacional de Luz Sincrotron
and Instituto de Fisica Gleb Weataghin,
Unicamp Caixa, CEP 13084-971 Campi-
nas SP, Brazil
Tom Oldfield (13), European Bioinfor-
matics Institute, European Molecular
Biology Laboratory, Wellcome Trust
Genome Campus, Cambridge CB10 1SD,
United Kingdom
Miroslav Z. Papiz (14), Daresbury
Laboratory, Daresbury, Warrington,
WA4 4AD, United Kingdom
Anastassis Perrakis (11), Netherlands
Cancer Institute, Department of
Carcinogenesis, 1066 CX Amsterdam,
The Netherlands
x contributors to volume 374
Donald Petrey (21), The Howard Hughes
Medical Institute, Department of Bio-
chemistry and Molecular Biophysics, Col-
umbia University, New York, New York
10032
Alberto Podjarny (15), Structural
Biology Sciences, Biosciences Division,
Argonne National Laboratory, Argonne,
Illinois 60439
Igor Polikarpov (5), Instituto de Fisica de
Sao Carlos, Universidade de Sao Paulo,

Av Trabalhador, Saovarlense, 13560 Sao
Carlos SP, Brazil
Thierry Prange
´
(4), LURE (CNRS-CEA-
MEN), Batiment 209d, Universite Paris
XI, 91898 Orsay, Cedex France
David C. Richardson (18), Department of
Biochemistry, Duke University, Duke
Building, Durham, North Carolina 27708
Jane S. Richardson (18), Department of
Biochemistry, Duke University, Duke
Building, Durham, North Carolina 27708
Jeffrey Roach (6), Department of Chemis-
try and Biophysics, University of North
Carolina at Chapel Hill, Chapel Hill,
North Carolina
Mark A. Rould (7), Department of Physi-
ology, University of Vermont, School of
Medicine, Burlington, Vermont 05405
James C. Sacchettini (12), Texas A & M
University, College Station, Texas 77843
Andrej S
ˇ
ali (20), Mission Bay Genentech
Hall, University of California at San Fran-
cisco, San Francisco, California 94143
Celia Schiffer (19), Department of Bio-
chemistry and Molecular Pharmacology,
University of Massachusetts, Medical

School, Worcesster, Massachusetts 01655
Marc Schiltz (4), LURE (CNRS-CEA-
MEN), Batiment 209d, Universite Paris
XI, 91898 Orsay, Cedex France
Thomas R. Schneider (3, 15), Lehrstuhl
fur Strukturchemie, Gottingen University
D37077 Gottingen, Germany
Eugene L. Shakhnovich (25), Department
of Biochemistry and Biophysics, Univer-
sity of North Carolina at Chapel Hill,
Chapel Hill, North Carolina 27599
George M. Sheldrick (3), Lehrstuhl fur
Strukturchemie, Gottingen University
D37077 Gottingen, Germany
H. Eugene Stanley (25), Department of
Biochemistry and Biophysics, University
of North Carolina at Chapel Hill, Chapel
Hill, North Carolina 27599
Dmitri I. Svergun (24), Institute of
Crystallography Russian Academy of
Sciences, 117333 Moscow, Russia
Lynn F. Ten Eyck (16), National Partner-
ship for Advanced Computational Infra-
structure, San Diego Supercomputer
Center, La Jolla, California 92093
Thomas C. Terwilliger (2, 3), Los Alamos
National Laboratory, Los Alamos, New
Mexico 87545
Alexander Tropsha (22), Department
of Medicinal Chemistry and Natural

Products, University of North Carolina
at Chapel Hill, Chapel Hill, North Caro-
lina 27599
J. Tsai (23), Department of Biochemistry
and Biophysics, Texas A & M University,
College Station, Texas, 77843
Maria G. W. Turkenburg (3), Department
of Chemistry, University of York, He-
slington York YO1 5DD, United Kingdom
Isabel Uson (3), Lehrstuhl fur Strukturch-
emie, Gottingen University D37077 Got-
tingen, Germany
Patrice Vachette (24), LURE Bat. 209d,
University Paris-Sud, F-91898 Orsay,
Cedex France
Iosif I. Vaisman (22), School of
Computational Sciences, George Mason
University, Manassas, Virginia 20110
Niels Volkmann (10), The Burnham
Institute, La Jolla, California 92037
contributors to volume 374 xi
Charles M. Weeks (3), Hauptman-Wood-
ward Medical Research Institute, 73 High
Street, Buffalo, New York 14203
John Westbrook (17), Research Calla-
boratory for Structural Bioinformatics,
Department of Chemistry, Rutgers
The State University of New York,
Piscataway, New Jersey 08854
Martyn D. Winn (14), Daresbury

Laboratory, Daresbury, Warrington,
WA4 4AD, United Kingdom
Kam Y. J. Zhang (9), Department of Struc-
tural Biology, Plexxikon, Inc., Berkeley,
California 94710
xii contributors to volume 374
[1] How Bijvoet Made the Difference:
The Growing Power of Anomalous Scattering
By D. M. Blow
History
Johannes Bijvoet (1892–1980) made pioneering contributions to the de-
termination of noncentrosymmetric structures. He was the first to exploit
the isomorphous replacement method to reveal a noncentrosymmetric
structure, using isomorphous sulfate and selenate salts to determine the
structure of strychnine on the basis of two projections.
1–3
In space group
C2, the selenium atoms, one in each asymmetric unit, make a centrosym-
metric array, and the structure factors of the heavy atoms (with appropriate
choice of origin) are all real. The isomorphous difference then determines
the real part of the strychnine structure factor, but the sign of the imaginary
part of the structure factor is undefined. The best estimate of the strychnine
structure factor is its real part.
This leads to an electron density map in which the structure and its in-
verse are superimposed, with symmetry C2/m. The structure of the strych-
nine molecule was deduced by discarding one of each pair of atoms related
by the mirror, using the same principles that Carlisle and Crowfoot
4
had
used in separating the two images of the cholesteryl iodide molecule gener-

ated by the ‘‘heavy atom’’ method. In both cases, the authors deriving their
structure did not know which interpretation was a true representation of
the molecule, and which was its inverted image.
Bijvoet recognized that anomalous scattering could be used to identify
the correct enantiomorph of a noncentrosymmetric structure. He wrote,
5
There is in principle a general way of determining the sign [of a phase
angle]. We can use the abnormal scattering of an atom for a wavelength just
beyond its absorption limit. It also becomes possible to attribute the d or l
structure to an optically active compound on actual grounds and not merely
by a basic convention.
Nishikawa and Matsukawa
6
and Coster et al.
7
had observed departure
from Friedel’s law
8
in diffraction from opposite polar faces of a zinc sulfide
1
C. Bokhoven, J. C. Schoone, and J. M. Bijvoet, Kon. Ned. Akad. Wet. 51, 825 (1948).
2
C. Bokhoven, J. C. Schoone, and J. M. Bijvoet, Kon. Ned. Akad. Wet. 52, 120 (1949).
3
C. Bokhoven, J. C. Schoone, and J. M. Bijvoet, Acta. Crystallogr. 4, 275 (1951).
4
H. C. Carlisle and D. M. Crowfoot, Proc. R. Soc. A 184, 64 (1945).
5
J. M. Bijvoet, Kon. Ned. Akad. Wet. 52, 313 (1949).
[1] how bijvoet made the difference 3

Copyright 2003, Elsevier Inc.
All rights reserved.
METHODS IN ENZYMOLOGY, VOL. 374 0076-6879/03 $35.00
crystal. In a beautifully clear exposition of anomalous scattering effects,
Bijvoet
9
drew on this example:
Normal X-ray reflection does not detect any difference between one side [of
the octahedral faces of a zinc blende crystal], a dull and poorly developed
tetrahedron plane, and the other, a shining well-developed one. In this respect,
it is less sensitive than the human eye. Coster, however, chose a radiation—
L
1
radiation of gold—which just excites the K electrons of zinc. Now
X-ray analysis not only detects a difference, but it concludes—and this is, of
course completely impossible for the human eye—that it is the dull plane that
has the zinc plane facing outwards.
In 1951 Bijvoet and colleagues
10
observed the intensity differences
between Friedel-related pairs of X-ray reflections from sodium rubidium
tartrate crystals. These observed differences showed that the convention
established by Emil Fischer to discuss the configuration of bonds at asym-
metric carbon atoms, especially in sugars, by good chance represents the
true three-dimensional enantiomorph of these molecules.
This was a substantial achievement, but Bijvoet was looking much fur-
ther ahead. His visionary paper in 1954
11
opens by mentioning
the great successes of X-ray analysis that determined structures as compli-

cated as those of sterols and alkaloids and that now approach the domain of
Nature’s most complicated biochemical compounds, the proteins
A flow chart (see Fig. 1) sets the agenda for structure determination for the
next half-century. It shows how isomorphous substitution and anomalous
scattering can determine phases for all noncentrosymmetric reflections.
But there is a question mark. Bijvoet warns,
11
It has not yet been thoroughly investigated whether the small effect of the
anomalous scattering will be measurable for a sufficient part of the reflections
involved in a complete Fourier synthesis.
How thrilled he would have been to know that tunable X-ray sources
could produce such measurable effects that anomalous scattering could
solve protein structures on its own! These methods began to be introduced
in the last year of his life.
6
S. Nishikawa and R. Matsukama, Proc. Imp. Acad. Jpn. 4, 96 (1928).
7
D. Coster, K. S. Knol, and J. A. Prins, Z. Phys. 63, 345 (1930).
8
G. Friedel, Comptes Rendus 157, 1533 (1913).
9
J. M. Bijvoet, Endeavour 14, 71 (1955).
10
J. M. Bijvoet, J. F. Peerdeman, and J. A. van Bommel, Nature 168, 271 (1951).
11
J. M. Bijvoet, Nature 173, 888 (1954).
4 phases [1]
Center of Symmetry
PHASE DETERMINATION IN THE ISOMORPHOUS SUBSTITUTION METHOD
(determination of amplitude sign)

Heavy atom on center
Algebraic amplitude addition (1936).
Heavy atom out of center
(a) Location of the heavy atom (Patterson
analysis).
(b) Algebraic amplitude addition (1939).
No center of symmetry
(determination of phase angle)
(a) Location of the heavy atom.
(b) Determination of the absolute value of phase angle from amplitude addition in
vector diagram (1949).
F
B
-F
A
(c) Synthesis of double Fourier and resolution (c') Determination of all phase signs
by geometrical considerations. by anomalous scattering (19 ?)
(d) Phase shift by anomalous scattering (1930).
Determination of absolute configuration (1951).
∆f "
∆f "
F
B
-F
A

Fig. 1. The Bijvoet presentation of phase determination in the isomorphous substitution
method. Redrawn with permission from Nature 173, 888–891. Copyright 1954 Macmillan
Magazines Limited.
[1] how bijvoet made the difference 5

Anomalous scattering became popular. Pepinsky’s group
12
devised a
type of anomalous difference Patterson function, known as the P
s
function,
which is the sine transform of the intensities:
P
s
ðuÞ¼ð1=VÞ
X
jFðhÞj
2
sinð2h Á uÞ
Because the sine function is odd [sin x ¼Àsin(Àx)], the terms of this sum-
mation are only nonzero to the extent that intensities for h and Àh differ.
The P
s
function is antisymmetric, and its positive and negative peaks repre-
sent vectors between an anomalous scatterer and a normal scatterer. Peer-
deman and Bijvoet
13
and Ramachandran and Raman
14
both discovered a
simple way to use a centrosymmetric array of anomalous scatterers in a
noncentrosymmetric structure to derive the imaginary components of the
structure factors.
Personal Notes
During 1954–1957 I was a student working on ways to apply isomorph-

ous replacement to phase noncentrosymmetric reflections of proteins, and
especially to deal with the ambiguities and errors that appeared to domin-
ate the results in practice.
15,16
I met Johannes Bijvoet at an international
crystallography meeting in Madrid in April 1956, introduced myself, and
outlined my research project to him. I remember him as a strongly built
man with slightly receding hair (Fig. 2), who was gentle and encouraging
to the nervous student talking to him—indeed, he was clearly excited by
the progress in developing methods to exploit isomorphous replacement
in proteins. He spoke excellent English, and his attitude was warm and
friendly. I met him again at an International Crystallography Congress in
Cambridge in August 1960. By that time the subject of protein crystallog-
raphy had become established by three-dimensional electron density maps
for hemoglobin and myoglobin, and anomalous scattering had been used to
help with the phasing of haemoglobin. Bijvoet was enthusiastic about these
developments.
He retired in 1962 and I did not see him again, but I had one other per-
sonal involvement. In 1972 I was elected to the Royal Society, and a few
days later (before the formal admission ceremony) I learned that a vote
was to be taken on the election of Bijvoet as a Foreign Member of the
12
Y. Okaya, Y. Saito, and R. Pepinsky, Phys. Rev. 98, 1857 (1955).
13
A. F. Peerdeman and J. M. Bijvoet, Acta Crystallogr. 9, 1012 (1956).
14
G. N. Ramachandran and S. Raman, Curr. Sci. 25, 348 (1956).
15
D. M. Blow, Proc. R. Soc. A. 247, 302 (1958).
16

D. M. Blow and F. H. C. Crick, Acta Crystallogr. 12, 794 (1959).
6 phases [1]
Royal Society. I was told it would be in order for me to vote, as I had al-
ready been elected. It was a pleasure and an honor to travel to London
to cast my vote for his election.
Anomalous Scattering in Proteins
In 1956, a consistent and obvious difference was observed between the
diffracted intensities of a Friedel pair for a low-order reflection of myoglo-
bin (what is now known as a Bijvoet difference).
17
Wyckoff ruled out that it
was an experimental artifact, or that it was dependent on solvent concen-
tration, and it was recognized to be an anomalous scattering effect. These
17
J. C. Kendrew, G. Bodo, H. M. Dintzis, J. Kraut, and H. W. Wyckoff, unpublished data
(1956).
Fig. 2. Johannes Bijvoet. Photograph courtesy of Han Meijer.
[1] how bijvoet made the difference 7
studies were made with CuK radiation, for which the iron atom of myo-
globin has a significant anomalous scattering component (about 3.4 elec-
trons). We can now recognize that this effect could have given clear
evidence about the position of the iron atom in the myoglobin crystals. It
suggested that anomalous scattering might provide useful phase informa-
tion, even though the effects at accessible wavelengths were much smaller
than those of isomorphous replacement.
Let us refer
18
to the normal part of the heavy atom structure factor as
F
0

H
. This is calculated using the real component of the atomic scattering
factor f
0
þ f
0
. (In practice, f
0
is a negative quantity.) The anomalous part
F
00
H
is calculated using the imaginary part of the atomic scattering factor,
if
00
. As indicated in Fig. 3,
F
0
H
ðhÞ¼F
0
H
ðÀhÞ
*
F
00
H
ðhÞ¼ÀF
00
H

ðÀhÞ
*
In the simple case of a centrosymmetric distribution of heavy atoms
(which always exists for a single heavy atom site in a space group with an
even-fold symmetry axis), the normal structure factor F
H
of the heavy atom
is real. In this case, the isomorphous replacement method estimates the
cosine of the phase angle, but gives no information about its sine; measure-
ment of the Bijvoet difference estimates the sine of the phase angle, but
gives no information about its cosine (Fig. 1). In a general case (Fig. 4),
the isomorphous replacement method and the anomalous scattering
method give orthogonal information about the phases.
18
Notation: When discussing anomalous scattering, the subscript P refers to all the ordered
atoms in the crystal whose atomic scattering factors are real. The subscript H refers to
atoms that exhibit significant anomalous scattering, usually assumed to be all of the same
type. The structure factor F
H
has two components: F
0
H
, which arises from the normal part
f
0
þf of the scattering factor of the H atoms and F
00
H
, which arises from the anomalous part
of their scattering factor f

00
. When all the anomalous scatterers are of the same type, F
00
H
is in
quadrature with F
0
H
.
Fig. 3. (a) Normal and anomalous components of the heavy atom structure factor for
reflection h, and for reflection Àh. (b) Comparison of the heavy atom structure factor F
H
(h)
with the complex conjugate of its Friedel mate F
H
(Àh)*.
8 phases [1]
Considering the effects with CuK too small, Blow
15
plated a rotating
anode with chromium and made measurements on a mercury derivative of
hemoglobin, using CrK radiation (f
00
for mercury then estimated as 15e
at 2.29 A
˚
). This was a mistake, because the need for large absorption cor-
rections at this wavelength seriously prejudiced precise observation of Bij-
voet differences. It was subsequently concluded that MoK radiation
would have been more suitable, being relatively close to the mercury L ab-

sorption edge, but at a wavelength at which absorption errors are much
smaller.
The Bijvoet differences did give significant information to resolve ambi-
guities in the phases determined by isomorphous replacement, but the
large errors made a quantitative estimate difficult. The results were simply
categorized as Bijvoet difference probably positive, insufficient informa-
tion, or as Bijvoet difference probably negative. Even this information
was useful in estimating phases, when available isomorphous replacements
left a large ambiguity in phase.
15
Using only CuK radiation, similar methods were employed by Cullis
et al.
19
to help resolve ambiguities of phase left by the isomorphous
19
A. F. Cullis, H. Muirhead, M. F. Perutz, M. G. Rossmann, and A. C. T. North, Proc. R.
Soc. A. 265, 15 (1961).
Fig. 4. Harker constructions for (a) isomorphous replacement difference; (b) Bijvoet
amplitude difference, showing how they give orthogonal phase information. In (a) the real
part of the scattering by the heavy atoms is used to calculate the structure factor F
H
0
appropriate for isomorphous replacement. In (b), the Bijvoet difference is due to opposite
effects of the imaginary part of the heavy atom scattering factor on F(h) and on [F(Àh)]*.
[1] how bijvoet made the difference 9
replacement technique, in determining the hemoglobin structure to 5.8-A
˚
resolution. The squares of the Bijvoet amplitude differences provide a set
of Fourier coefficients of an approximate Patterson function of the anom-
alous scatterers (closely analogous to the difference Patterson for an iso-

morphous pair). Using CuK radiation, the iron atoms of hemoglobin are
the only important anomalous scatterers in the molecule and Rossmann
showed how the iron atom positions could be determined directly from
the Bijvoet differences.
19a
Blow and Rossmann
20
showed that a recogniz-
able but more noisy electron density map could be obtained using only
the data from the parent crystal and a single isomorphous derivative, in-
cluding anomalous scattering observations, the method now known as
SIRAS (single isomorphous replacement with anomalous scattering).
Methods of Analysis at Fixed Wavelength
Blow and Rossmann
20
did not simply use the sign of the observed Bij-
voet difference to resolve the ambiguity of phase left by the single iso-
morphous replacement. Instead, they followed a procedure similar to that
of Blow and Crick,
16
in which a probability is assigned to every possible
phase angle depending on how accurately it fits the observations. For a par-
ticular reflection h, the observations of jF
PH
(h)j and jF
PH
(Àh)j were
treated as separate observations. The calculated heavy-atom structure
factors F
H

(h) and F
H
(Àh)
*
were calculated using a complex atomic scatter-
ing factor f
0
þ f
0
þ if
00
(Fig. 1). The analysis was carried out as though the
two members of the Friedel pair were separate isomorphous derivatives.
An improvement was suggested by North,
21
who pointed out that the
errors in exploiting the Bijvoet difference are far smaller than those that
arise in isomorphous replacement, and the Bijvoet difference can be inter-
preted with greater precision. The implications of the isomorphous differ-
ence are confused by departures from ideal isomorphism between
‘‘parent’’ and ‘‘derivative,’’ but there is no corresponding inaccuracy
affecting the Bijvoet difference. Also, because measurements are made
on the same crystal, often under similar geometric conditions, the ampli-
tude difference is measured more accurately. North suggested a different
algorithm for calculation of the phase probabilities, depending on the three
observed structure amplitudes, jF
P
(h)j, jF
PH
(h)j,andjF

PH
(Àh)j, and on
the estimated normal and anomalous components of the scattering by
the heavy atoms, F
0
H
(h) and F
00
H
(h). However, as North recognized, the
19a
M. G. Rossmann, Acta Crystallogr. 14, 383 (1961).
20
D. M. Blow and M. G. Rossmann, Acta Crystallogr. 14, 1195 (1961).
21
A. C. T. North, Acta Crystallogr. 18, 212 (1965).
10 phases [1]
algorithm depended on an approximation and could be used in different
ways.
This was a considerable improvement, but Matthews
22
found a better
formulation. The essence of the method was to change the variables
used in the analysis. Instead of working with the observed quantities
jF
PH
(h)j and jF
PH
(Àh)j, Matthews worked with the mean structure ampli-
tude

1
2
(jF
PH
(h)jþjF
PH
(Àh)j) and the Bijvoet amplitude difference
jF
PH
(h)jÀjF
PH
(Àh)j. The mean structure amplitude estimates the struc-
ture amplitude that would exist if f
00
were zero, designated F
0
PH
ðh). This
is used in the usual way with jF
P
(h)j and with the calculated normal part
of the heavy atom structure factor F
0
H
(h), to obtain a phase probability dis-
tribution by isomorphous replacement. The Bijvoet amplitude difference is
used in a similar way with the calculated anomalous part of the heavy atom
structure factor. This second contribution to the phase probability distribu-
tion is independent of any assumption about isomorphism with the parent
crystal P. The Bijvoet amplitude difference (jF

PH
(h)jÀjF
PH
(Àh)j) is used
to develop a phase probability distribution derived from anomalous effects.
Because there are no errors due to nonisomorphism, the intrinsic errors in
interpreting the Bijvoet difference are much smaller, so the root-mean-
square (RMS) lack of closure E
00
is smaller, leading to a more tightly
defined phase distribution.
This formulation
22
is valid even when different types of anomalous
scatterer exist, but usually one type of anomalous scatterer is assumed. This
method of phase determination was coded by L. P. Ten Eyck
23
and by
J. E. Ladner
24
into a widely used program PHARE (now incorporating a
maximum likelihood refinement procedure and called MLPHARE
25
).
The Ten Eyck phasing algorithm seems to have been used without
significant change.
In that era X-ray analysis of proteins was practicable only when using
characteristic radiations such as CuK. Investigators concentrated on the
effects of f
00

, whose effect causes a Bijvoet difference, and tended to ignore
f
0
, which modifies the magnitude of the isomorphous difference, because
at the given wavelength it is a fixed quantity.
Minor criticisms of the Matthews algorithm
22
can be made.
1. The normal part of the scattering F
0
PH
(h) is the complex quantity
1
2
[F
PH
(h) þ F
*
PH
(Àh)]. But Matthews approximates jF
0
PH
(h)j as
1
2
[jF
PH
(h)jþ
22
B. W. Matthews, Acta Crystallogr. 20, 82 (1966).

23
L. F. Ten Eyck, J. Mol. Biol. 100, 3 (1976).
24
J. E. Ladner, personal communication (2002).
25
Z. Otwinowski, ‘‘CCP4 Study Weekend Proceedings’’ (W. Wolf, P. R. Evans, and A. G. W.
Leslie, eds.), p. 80. Daresbury Laboratory, Warrington, UK, 1991.
[1] how bijvoet made the difference 11
jF
PH
(Àh)j]. By straightforward trigonometry (Fig. 5) (see also Burling
et al.
26
),
jF
0
PH
ðhÞj
2
¼
1
2
ðjF
PH
ðhÞj
2
obs
þjF
PH
ðÀhÞj

2
obs
ÞÀjF
00
PH
ðhÞj
2
calc
(1)
Subscripts ‘‘obs’’ and ‘‘calc’’ emphasize that the calculated part of this
expression is a small correction to the value derived from observation. In
practice the error will often be on the order of 1% and it will rarely exceed
3–4% unless F
00
PH
is extraordinarily large.
2. The isomorphous replacement method, as conventionally used, gives
a phase probability distribution for the parent crystal F
P
. The phase
probability distribution derived from the Bijvoet difference applies to the
normal part of the scattering from the derivative crystal F
0
PH
. That means
it is the phase probability distribution for (F
P
þ F
0
H

). This fact was ignored
by Matthews, who estimated the phase probability as if these two
distributions applied to the same quantity.
The errors created by these simplified assumptions were insignificant in
relation to the precision of phase estimation at the time.
A slightly different approach to isomorphous replacement was intro-
duced by Hendrickson and Lattmann.
27
They devised a method to sum-
marize the phase probability distribution by four coefficients A, B, C, and
D, which essentially represent the first two Fourier components of the
phase probability curve. To achieve this, they expressed the lack of closure
error e() as the lack of agreement of observed and calculated intensity,
eðÞ¼jjF
P
jexpðiÞþF
H
j
2
ÀjF
PH
j
2
26
F. T. Burling, W. I. Weis, K. M. Flaherty, and A. T. Bru
¨
nger, Science 271, 72 (1996).
27
W. A. Hendrickson and E. E. Lattman, Acta Crystallogr. B 26, 136 (1970).
Fig. 5. The two triangles shown include the length jF

PH
0
(h)j, which in each triangle can be
calculated from the lengths of the other sides, and from two related (but unknown) angles.
Equating two trigonometric expressions for jF
PH
0
(h)j leads to Eq. (1).
12 phases [1]
(In this expression jF
P
j and jF
PH
j are derived from the observed intensities,
and F
H
is calculated from the heavy atom parameters.) This method has
been incorporated into a number of computer programs. A criticism of it
would be that the observational error in intensity tends to be proportional
to the intensity, so that larger errors e are usually encountered for intense
reflections. In contrast, the observational error in amplitude is fairly con-
stant for weak and medium strength reflections. Moreover, errors due to
nonisomorphism are not correlated with intensity. Therefore the root-
mean-square error E ¼he(
best
)
2
i
1=2
depends on the intensity. Blow and

Crick
16
used the amplitude error
xðÞ¼jjF
P
jexpðiÞþF
H
jÀjF
PH
j
in their analysis, and this justifies using a single value for the root-mean-
square lack of closure E ¼hx(
best
)
2
i
1=2
at a given resolution, independent
of the observed intensity (see also Kumar and Rossmann
28
).
A new era in the use of anomalous scattering began when Hendrickson
and Teeter
29
spectacularly demonstrated the possibilities of using anomal-
ous scattering on its own, by total determination of the crambin structure
using the anomalous scattering of its six sulfur atoms in CuK radiation.
In terms of a Harker diagram (Fig. 4b), this method (now called SAD:
single-wavelength anomalous diffraction) produces a phase ambiguity
equivalent to that of a single isomorphous replacement, but still gives im-

portant information. Because the constellation of six sulfur atoms will
never be centrosymmetric there is no restriction on the indicated phase
angle. The resulting image is not the structure plus its inverse, but a noisy
image of the structure, which can be refined using other information, espe-
cially at high resolution. As is discussed in the final section of this article,
these methods have now become powerful.
Synchrotron Radiation
In the late 1970s synchrotron radiation became more accessible, the first
beamline facilities for macromolecular crystallography were set up, and for
the first time experiments became feasible using any chosen wavelength.
The possibilities of using anomalous scattering at several wavelengths were
recognized early by Phillips et al.
30
28
A. Kumar and M. G. Rossmann, Acta Crystallogr. D 52, 518 (1996).
29
W. A. Hendrickson and M. Teeter, Nature 290, 107 (1981).
30
J. C. Phillips, A. Wlodawer, J. M. Goodfellow, K. D. Watenpaugh, L. C. Sieker, L. H.
Jensen, and K. O. Hodgson, Acta Crystallogr. A 33, 445 (1977).
[1] how bijvoet made the difference 13
It had now become possible to exploit the changes in f
0
at different
wavelengths, as well as the existence of Bijvoet differences. Karle
31
made
a new analysis of the problem that was rigorous and general. It developed
the possibility of determining phase angles using anomalous scattering by a
single crystal at different wavelengths [now known as MAD: multiple-

wavelength anomalous diffraction (or dispersion)]. Karle’s analysis, al-
though presented in a general way, concentrates on the usual case in
practice, in which there is one type of anomalous scatterer and other types
of atom whose anomalous scattering factors f
0
and f
00
can be taken as zero.
To preserve the notation already adopted, this article shall identify the
scattering by these two parts of the structure by subscripts H and P, re-
spectively (Karle uses subscripts 2 and 1). The subscript PH is omitted, be-
cause it refers to the whole structure of the crystal under investigation,
including normal and anomalous scatterers. Following Karle, a structure
factor F(h) without subscript refers to scattering by the whole crystal. Be-
cause the crystal includes anomalous scatterers, this structure factor
depends on the wavelength and is written

F(h).
An important feature of Karle’s analysis
31
lies in the definition of the
‘‘normal’’ and ‘‘anomalous’’ parts of the structure. The normal structure
factors F
n
are calculated as though all the electrons in the structure scatter
normally. The anomalous component

F
a
is the correction that must be

made to give the actual scattering at some wavelength . To emphasize
wavelength dependence, structure factors that are dependent on wave-
length are given a presuperscript :

FðhÞ¼F
n
ðhÞþ

F
a
ðhÞ
(Although

F
a
derives entirely from the heavy atoms that exhibit anomal-
ous scattering, the subscript H is not needed, because there is no other
anomalous scattering.) If the parameters of the anomalous scatterers are
known (including the complex atomic scattering factors at wavelength ),

F
a
may be calculated. In contrast to the earlier approaches discussed
above, the effects of changes to the real parts of the atomic scattering
factors f
0
are now included in the anomalous component

F
a

. This facili-
tates comparison of scattering at different wavelengths. But the anomalous
scattering

F
a
is not in quadrature with the normal part F
n
. The relation-
ship F
00
H
(Àh)* ¼ÀF
00
H
(h) stated above does not apply to

F
a
because F
00
H
is
only one component of

F
a
. The notation has many advantages, but it does
not allow the Bijvoet difference and the Bijvoet amplitude difference to be
interpreted so simply. The two notations are compared in Fig. 6.

31
J. Karle, Int. J. Quantum Chem. Quantum Biol. Symp. 7, 357 (1980).
14 phases [1]
Karle’s analysis
31
expands the algebraic expression for each observable
intensity as a linear combination of terms which depend on four variables.
Two variables are jF
n
P
j
2
and jF
n
H
j
2
, and the other two depend on the sine
and cosine of the angle (
n
À


a
), which determines their phase relation-
ship. If

F
a
can be calculated from the parameters of the heavy atoms, this

angle leads to a phase angle for the normal component of the structure
factor F
n
. At each wavelength two independent intensity observations
[

F(h)]
2
and [

F(Àh)]
2
provide two independent quantities that depend
on these four variables. If measurements are made at two wavelengths,
giving four independent observations, the system is determined in
principle. If three wavelengths are used there are six equations in four
unknowns, and standard linear algebra can give a definite least-squares
solution (as in MADLSQ
32
). The precision of the result depends on
the properties of the normal matrix of the equations—on how well
‘‘conditioned’’ they are.
When the equations are solved for each reflection, the Fourier transform
of jF
n
H
j
2
provides a Patterson function of the array of anomalous scatterers.
From this, the parameters of the heavy atoms may be determined.

Compared with the isomorphous replacement method, anomalous scat-
tering analysis is relatively error-free. The experimental errors arise from
inaccuracies of intensity measurement, and from inaccurate estimation of
the anomalous scattering component, arising from errors in the estimated
parameters of the atoms that cause it, at the particular wavelengths
employed. There is also a significant but usually small error that arises from
the assumption that all of the atoms in the P part of the structure are
‘‘normal’’ scatterers, with f
0
and f
00
precisely zero. For the Karle
method,
31
it was assumed at first that no sophisticated error analysis was
32
W. A. Hendrickson, Trans. Am. Crystallogr. Assoc. 21, 245 (1985).
Fig. 6. Comparison of Karle notation using F
n
,

F
a
with ‘‘pseudo-isomorphous’’ notation
using F
P
, F
H
0
, and F

H
00
. F
H
0
and F
H
00
are orthogonal because all the atoms scattering
anomalously are assumed to be of the same type.
[1] how bijvoet made the difference 15
needed, and many structures were solved in this way, using three and some-
times four different wavelengths. These usually include two very close
wavelengths at the maximum anomalous scattering f
00
(‘‘peak’’) and at
the absorption edge (‘‘edge,’’ where the change f
0
to the normal scattering
is maximized), and one or more ‘‘remote’’ wavelengths where f
0
is fairly
small, although f
00
may be significant.
A study of selenobiotinyl streptavidin by MAD undertook a direct an-
alysis of experimental error.
33
The formulas derived from Karle’s analysis
31

were rearranged into a form parallel to Hendrickson and Lattman’s expres-
sions for MIR (multiple isomorphous replacement).
28
The lack of agree-
ment of observed and calculated intensities e() at each wavelength could
be expressed directly in terms of four quantities A, B, C, and D, which
allow a phase probability distribution to be calculated, leading to a best
phase and figure of merit for each structure factor. The method was
reported to give a smaller phase error, and the map using the resulting
Fourier coefficients was significantly enhanced in appearance and ease of
interpretation, compared with results from MADLSQ.
Two Ways with MAD
An alternative to the Karle approach is to apply the method of
Matthews,
22
originally developed to deal with anomalous dispersion at a
single wavelength in conjunction with isomorphous replacement.
Hendrickson and Ogata
34
and Smith and Hendrickson
35
have contrasted
the two approaches. When the Matthews approach is applied to MAD,
it is referred to as ‘‘pseudo-MIR’’ by Smith and Hendrickson. The methods
based on Karle’s approach,
31–33
developed specifically for multiple
wavelength studies, are called the ‘‘explicit’’ approach.
An important practical difference between the methods arises in
identifying the positions of the anomalous scatterers. In pseudo-MIR the

observed Bijvoet amplitude difference directly provides coefficients
[j

F(h)jÀj

F(Àh)j]
2
for an anomalous difference Patterson synthesis as in
Rossmann.
19a
The coefficients from observations at several wavelengths
may be combined. In the explicit approach, the Karle simultaneous equa-
tions generate jF
n
H
j
2
, which provide coefficients for a Patterson function
of the anomalous scatterers. The advantages and disadvantages of these
approaches are discussed briefly by Smith and Hendrickson.
35
33
A. Pa
¨
hler, J. L. Smith, and W. A. Hendrickson, Acta Crystallogr. A 46, 537 (1990).
34
W. A. Hendrickson and C. M. Ogata, Methods Enzymol. 276, 494 (1997).
35
J. L. Smith and W. A. Hendrickson, in ‘‘International Tables for Crystallography’’ (M. G.
Rossmann and E. Arnold, eds.), Vol. F, p. 299. Kluwer Academic Publishers, Dordrecht,

The Netherlands, 2001.
16 phases [1]
Once the positions of the anomalous scatterers have been established
by either of these approaches, an estimate of F
n
H
ð¼ jF
n
H
j exp i
n
H
) can be
calculated, and the parameters of the H atoms can be refined by many
available methods. When this has been done, either the explicit or the
pseudo-isomorphous method may be used to obtain phases.
In the explicit approach, quantities proportional to cosð
n
P
À 
n
H
) and
sinð
n
P
À 
n
H
) have already been calculated, so that 

n
H
leads directly to
the phase angle 
n
P
, which allows calculation of an electron density map
representing the scattering density of the normal scatterers. Alternatively,
the Karle equations are revisited to generate a best phase and figure of
merit using the ABCD algorithm.
An adaptation of the MIRAS (multiple isomorphous replacement with
anomalous scattering) approach
36,37
uses the Bijvoet difference to give
phase information at each wavelength as in the Matthews method.
22
The
changes in

F
0
H
at different wavelengths are used like isomorphous replace-
ment differences. This approach worked well, but it includes an approxima-
tion because it is not defined which phase is being determined. (Each
Bijvoet difference indicates the phase at a different wavelength.) Terwilli-
ger
38
suggested further approximations, but Burling and colleagues carried
out a more precise analysis.

26
In this scheme, every pair of observed inten-
sities either related as a Bijvoet pair, or related by a wavelength change,
can be treated separately to provide a phase probability curve. These
results were compared with those obtained by Hendrickson’s method
32
and consistent and significant improvement in phasing accuracy was
reported. Similar results are reported by other authors. Some differences
remain, however, about which phase is to be determined. Burling et al.
26
chose diffraction at the ‘‘remote’’ wavelength to represent the ‘‘parent,’’
but comment that this represents a difference from calculating the phase
of F
n
as defined by Karle.
31
A Better Way?
We must be clear about what phase is actually to be determined. There
are two sensible choices: either the phase of F
P
(the phase of the structure
factor corresponding to the normally scattering atoms in the crystal, but
omitting the anomalous scatterers), or the phase of F
n
(the phase of the
structure factor if all the electrons in the crystal scattered normally). The
36
V. Ramakrishnan, J. T. Finch, V. Graziano, P. L. Lee, and R. M. Sweet, Nature 362, 219
(1993).
37

V. Ramakrishnan and V. Biou, Methods Enzymol. 276, 538 (1997).
38
T. C. Terwilliger, Acta Crystallogr. D 50, 17 (1994).
[1] how bijvoet made the difference 17
Fourier transform of F
P
will show the electron density of all the normal
scatterers in the crystal; the Fourier transform of F
n
will show the density
of all the electrons in the crystal. In the MAD technique, neither of these
structure factors is observable. But because the structure factors

F
H
cor-
responding to the scattering by the anomalous scatterers are calculable
from their parameters (equivalently, the structure factor

F
a
caused by
anomalous scattering effects may be calculated), this creates no fundamental
problem.
A straightforward approach was suggested by Bella and Rossmann,
39
who chose to estimate the phase of F
P
. Each experimental observation of
j


F(h)jor j

F(h)j, together with the calculated contribution of the anomalous
scatterers

F
H
(h)or

F
*
H
(Àh) follows the usual relationship

FðhÞ¼F
P
ðhÞþ

F
H
ðhÞ
Using the Harker construction, each observation generates a circle of pos-
sible values for F
P
(h)(Fig. 7), but because of observational and systematic
errors the circles do not all intersect perfectly. In the MAD technique there
is no direct measure of jF
P
j, and Bella and Rossmann exploited a method of

analysis in which the most probable phase is identified as that where the
Harker circles intersect most closely.
19
This method has several advantages. The analysis is done directly in
terms of the observed quantities j

F(h)j. Each observation is treated in an
equivalent way. There is total clarity about which phase is being evaluated.
Compared with the isomorphous replacement method, there is a complica-
tion because there is no direct measure of jF
P
j. The analysis does not need
to be restricted to the method of selecting close intersection.
19
For any
chosen value of F
P
(specifying both amplitude and phase), a lack of closure
for each observation of j

F( h)j is readily calculated. As seen on the
Harker diagram, it is simply the radial distance x
i
of this value of F
P
from
the corresponding Harker circle (Fig. 8).
This approach could be applied equally well to find a ‘‘best’’ value of
the quantity F
n

defined by Karle.
31
A more sophisticated method of analysis is embodied in the program
SHARP (statistical heavy atom refinement and phasing).
40
In SHARP,
all possible values of an ‘‘unperturbed native structure factor’’ F
P*
are
considered, using a maximum-likelihood formulation in which errors in
all observations and parameters of the problem including the ‘‘lack of
closure’’ are retained as variables. In this way the effects of correlations
and feedback in estimated parameters for a particular reflection are
39
J. Bella and M. G. Rossmann, Acta Crystallogr. D 54, 159 (1998).
40
E. de la Fortelle and G. Bricogne, Methods Enzymol. 276, 472 (1997).
18 phases [1]
included in the analysis, and the resulting estimate of the complex quantity
F
P*
is said to be unbiased.
In SHARP every observation is handled equivalently, and all contrib-
ute to the likelihood function for F
P*
(a two-dimensional function repre-
senting its amplitude as well as its phase). The absence of any
observation of jF
P
j does not change the formulation of the problem, so

the MAD technique can be treated in the same way as usual.
Around the turn of the century, MAD became the most frequently used
technique for direct determination of an unknown macromolecular struc-
ture (where the structure cannot be inferred from a homologous structure).
The different approaches remain in competition. In either case, the analysis
proceeds by two steps, the first of which obtains the positions of the
anomalous scatterers by Patterson methods, either using the Karle equa-
tions for jF
n
H
j
2
, or using coefficients derived directly from the Bijvoet
differences. Phase determination (or maximum likelihood analysis) then
uses all the parameters of the anomalous scatterers. Again, two methods
Fig. 7. Harker diagram for the MAD technique. To keep the diagram as simple as possible
only two wavelengths are illustrated, denoted as 1 and 2. Data from reflection h are identified
by a subscript plus symbol, and data from its Friedel mate Àh are denoted by a subscript
negative symbol. A possible value for F
P
is indicated as a dashed line, but because jF
P
j is not
observable, its amplitude is unknown. The open circle represents the structure factor if all
atoms scattered normally.
[1] how bijvoet made the difference 19
are available (explicit, using the Karle equations, or pseudo-isomorphous),
and there is no reason why the second step should use the same formulation
as the first.
The Future Is SAD

It seems likely, however, that the various improvements to analyze
MAD data more correctly are fading into insignificance. The MAD
technique is losing ground to SAD.
SAD has problems similar to those of SIR, because there are only two
measurements, jF(h)j and jF(Àh)j, so that they indicate two possible values
for the phase angle as in Fig. 4b. If the distribution of anomalous scattering
electrons does not have a tendency to centrosymmetry, the phases of the
anomalous scattering contributions will be fairly random, and will not tend
to generate false symmetry of the kind that Bokhoven et al. encountered
with strychnine.
3
Without other information, the ‘‘best’’ value for F(h)is
the mean of the two possible structure factors indicated by the two phase
angles. The use of this mean structure factor allows considerable error,
introducing noise into the electron density map. In the absence of other
Fig. 8. An enlarged view of a small part of Fig. 7. The solid square indicates a point chosen
as a possible origin for the F
P
vector. The radial distance, measured along F
P
, between this
point and any Harker circle represents a lack of closure x(F
P
) for the corresponding
observation of j

F( h)j. One such distance is identified.
20 phases [1]