Lecture 6: The Phase Problem and Methods of
Solution
6.1 The Phase problem
6.2 Isomorphous Replacement
6.3 Anomalous Scattering
6.4 Molecular Replacement
6.4.1 The Patterson Function
6.4.2 The Problem of Phase Bias
6.5 Methods for improving poorly phased maps. (“Density modification”)
6.5.1 Solvent flattening
6.5.2 Histogram matching
6.5.3 Non-crystallographic symmetry averaging.
6.5.4 Mask definition
Tuesday, 25 March 14
1
Okay, so we have scaled and merged our data
… are we ready to go ?
No … Because the computation of electron density requires not just
.
estimates for the structure factor amplitude of each reflection
|F(hkl)| , but also the phase α(hkl)
To remind you, here’s the equation for Fourier synthesis, written in its
“cosine” form
The electron density at a point x,y,z in a unit cell of volume V is
1
ρ (xyz) = ∑ ∑ ∑ F(hkl) cos[2π (hx + ky + lz) − α (hkl)]
V hk l
What we want
Tuesday, 25 March 14
What we have
What we still need
2
From Rupp (2010)
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3
The phase problem and some methods to
overcome it.
We’ll
briefly discuss three methods for estimating the phases, required for
.
computation of electron density maps, and look at the two of these in a little
more detail
1. Multiple isomorphous replacement (MIR)
2. Multiple-wavelength anomolous diffraction (MAD)
3. Molecular replacement (MR)
Tuesday, 25 March 14
4
Need one native crystal and
several derivatized crystals..
.
The method of multipleisomorphous replacement
(MIR).
In this method, crystals are derivatized with heavy
atoms, which cause measurable changes in the
diffracted intensities. These differences are exploited to
calculate phases.
Advantages.
•Experiments can be done on a laboratory X-ray source
•Large changes to diffracted intensities (easy to measure)
Disadvantages
•Can be very difficult to prepare truly isomorphous derivatives in
which the only change in the crystal is at the site of heavy atom
binding. Soaking crystals in heavy atom compounds often causes
them to shrink or expand, making the derivative useless for phase
calculations.
Must collect X-ray diffraction
data from each crystal
Tuesday, 25 March 14
•Many useful “heavy atoms” (eg. Mercury) are toxic and extremely
hazardous
5
Isomorphous replacement ...
Perutz’s description of his
critical realization.
From Judson (1979)
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6
Why Isomorphous replacement works
Let’s say there’s 15 light atoms (e.g. C,
N,O,S) in the unit cell and one heavy atom
(e.g. Hg)
Note the number of electrons associated
with each of these atoms
(C,6; N,7; O, 8; S,16; Hg, 80)
F(s)
real
imaginary
If the light atoms are randomly distributed
throughout the unit cell (approximately true
for protein crystals), then the summed
scattering from the light atoms corresponds
to a short-stepped random walk in the
complex plane
(This probabilistic way of looking at the of
scattering from a group of atoms was
developed by A.J.C Wilson, and is generally
known as “Wilson statistics”)
Then we add the Heavy atom
The key result is that the contribution from the heavy atom “dominates” the
resultant structure factor (even though the light atoms, collectively, have more electrons)
Tuesday, 25 March 14
7
Isomorphous replacement ... a real world example
Close inspection of the following precession photographs will reveal that the relative intensities of the
spots have changed due to derivitization of the protein with a heavy atom compound
From Blundell and Johnson (1976)
Native phosphorylase crystals
h0l zone
Tuesday, 25 March 14
Phosphorylase crystals derivatized with ethyl
mercuri thiosalicylate (EMTS)
h0l zone
8
How we get phase information from Isomorphous
replacement.
Conceptually, we split the total
scattering factor from the derivatized
crystal, FPH, into two parts - one
representing the contribution of the
protein FP and one representing the
contribution of the heavy atoms FH.
FPH(s)
FH(s)
FP(s)
real
FPH = FP + FH
imaginary
Tuesday, 25 March 14
9
How we get phase information from Isomorphous
replacement.
We measure ...
The amplitude of FP = |FP|
The amplitude of FPH = |FPH|
FPH(s)
FH(s)
FP(s)
real
|FP|
F
imaginary
|FPH|
We want ...
The phase of FP = αP
Tuesday, 25 March 14
Now let’s assume we can locate the positions
of the heavy atoms in the unit cell. We will skip
the details, but this is generally not too difficult, by
comparing |FP| and |FPH| . That means we can calculate
FH - both its amplitude |FH| and phase αH.
10
How we get phase information
from Isomorphous replacement.
Now we can learn something
about αP, the phase of FP.The
way we visualize this is with a
geometric construction termed
the Harker diagram - named
after its originator, David Harker.
“|FP| circle”
We star t by dr awing the
structure factor for FH, since its
amplitude and phase are known.
FH
Centered on the end point of
this vector we draw a circle with
radius |FP|
Centered on the origin, we draw
a circle with radius |FPH| ....
“|FPH| circle”
Tuesday, 25 March 14
11
How we get phase information
from Isomorphous replacement.
Since FPH = FH + FP, the
intersection of the two circles
represent the two possible ways
FP and FH can add together to
give FPH.
(I’ve drawn the
possibilities in green and blue)
Hence there are now two
possible values for the phase of
FP .
“|FP| circle”
FP
FH
FP
FPH
FPH
“|FPH| circle”
Tuesday, 25 March 14
Single isomorphous replacement
always results in this phase
ambiguity.
Only one of the
choices is correct. We can
determine the phase uniquely by
preparing a second derivative (if
you like you can add this
information to the diagram to
see how that works.)
12
What kind of compounds can we use to derivatize
protein crystals ?
From Cantor and Schimmel (1980)
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13
The method of multiple wavelength
anomalous diffraction (MAD)
The X-ray scattering from some elements is wavelength dependent. The phase
of the radiation scattered from these atoms is shifted in a predictable fashion.
This wavelength-dependent phenomenon is known as anomalous or
resonant scattering, can be exploited to determine phases.
Notes:
1. The elements normally found in proteins (N,C,O,H,S) exhibit negligible or very
small anomalous scattering effects.
2. When anomalous scattering is significant, Friedel’s law breaks down. |F(hkl)|
and |F(-h-k-l)| are no longer equivalent and the differences between the two are
exploited for phasing.
3. For any element, anomalous scattering effects are largest at wavelengths close
to an absorption edge .
Tuesday, 25 March 14
14
The method of multiple wavelength
anomalous diffraction (MAD)
Advantages.
Need one crystal,
incorporating an
anomalously scattering
atom
•Provides extremely high quality phases
•Relatively easy to incorporate anomalous scatterers (Can express
proteins in bacteria and incorporate Seleno-Methionine)
Disadvantages
•Need to travel to a synchrotron, with a tunable X-ray source, that can
provide X-rays at different wavelengths (laboratory X-ray sources can’t
do this)
•Anomalous scattering effects are often very weak, and extremely good
quality data is required to make the experiment work*.
Must usually collect X-ray
diffraction data at 2 - 3
different wavelengths from
this single crystal.
•Radiation damage can be a problem since you have to collect multiple
data sets from a single crystal.
* See the following slide ...
Tuesday, 25 March 14
15
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oscillation data.
The MAD experiment depends on accurately measuring the very small differences between the
Friedel pairs |F(hkl)| and |F(-h-k-l)|.
In the oscillation method, the Friedel equivalents of all the observations recorded on
a given oscillation image can be measured by rotating the crystal by 180°, and
recording a second equivalent oscillation image.
180º
Oscillation images taken from Shen et al. Enantiomorph determination using inverse reference-beam diffraction images. Acta Crystallogr, A, Found Crystallogr (2000) vol. 56 ( Pt 3) pp. 264-7
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If an image and its “inverse” are recorded consecutively, problems due to radiation damage will be
minimized, and collection of Friedel pairs will be highly efficient.
Tuesday, 25 March 14
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The method of molecular replacement
•This method relies on having an atomic model of a protein that is similar (not necessarily
identical) to the one in the crystal.
•The known structure (the search or probe molecule) must be correctly oriented and positioned
in the unit cell.
•Once that’s done we can use the search model to calculate phases, and hence electron density.
. This will hopefully be good enough that the differences between the structures are revealed, and a
model of the new structure can be completed.
Advantages.
•Very easy to do (no experimental work required)
Disadvantages
•If the molecular replacement search model is only distantly
related to the structure you’re trying to determine, the
method will provide very poor quality phases. Because of the
problem of model bias, which we’ll discuss further, maps may
be difficult to interpret or even misleading.
Dima Chirgadze, Cambridge
•It might not work. There are no universal rules, but :
With > 30% sequence identity between model and target it
should succeed.
With 15-30% sequence identity it may succeed.
With < 15% sequence identity it’s pretty hopeless.
Tuesday, 25 March 14
17
The method of molecular replacement illustrated
using cats
A regular cat
A structurally
homologous Manx cat in
the same location
The diffraction pattern
collected from the
regular cat
(Amplitudes only)
The diffraction pattern
calculated from the Manx
cat
(Amplitudes and Phases)
Amplitudes from the
Regular cat; Phases
from the Manx cat,
Resultant image after
Fourier transformation.
From Rhodes (2006)
Tuesday, 25 March 14
18
Rigid body transformations and the teapot.
To make use of molecular replacement we must correctly
position the homologous structure in the unit cell.
This
involves defining a rigid body transformation.
.
A rigid body transformation
comprises a rotation and a
translation. Six parameters are
required to specify the
transformation.
3 for the rotation and 3 for the
translation.
From Blow (2002).
For reasons both theoretical
and practical the 6 dimensional
search for the molecular
position (computationally - very
expensive) is broken down into
2 sequential 3 dimensional
searches - the first for the
rotation and the 2nd for the
translation.
So how do we search for the correct rotation and translation ?
We make use of the Patterson Function
Tuesday, 25 March 14
19
Technical aside: Rotation angle
conventions.
There are several conventions for specifying a rotation in terms of angles in 3D space. Here are
two that you will encounter in crystallography.
From Blow (2002)
Spherical polar angles: These are easy to
visualize. Two angles specify the direction of a
rotation axis, and the final angle specifies the
rotation about that axis
Tuesday, 25 March 14
Eulerian angles: Good luck visualizing these !!
Basically the rotation is described as three
successive rotations about principal axes. e.g.
rotate around z, then rotate around the new y,
then rotate around the new
z. Confused
? So is
From Blow
(2002)
everyone else.
20
The Patterson Function
The Patterson function is the autocorrelation of the electron density.
It’s simplest to think of it as a map of all the interatomic vectors in the cell.
Arthur Lindo
Patterson
Born in Nelson New
Zealand.
It was introduced by Lindo Patterson in the early 1930’s, hence the capital P.
Just for the record, here’s the equation …
1
2
P(uvw) = ∑ ∑ ∑ F(hkl) cos[2π (hu + kv + lw)]
V hk l
c.f the equation for the computation of electron density
1
ρ (xyz) = ∑ ∑ ∑ F(hkl) cos[2π (hx + ky + lz) − α (hkl)]
V hk l
The Patterson function can be calculated directly from the diffraction data.
No phases required !!
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21
The Utility of the Patterson Function
The Patterson function is integral to many of the approaches used
to overcome the phase problem.
Places where it is employed:
• The determination and characterization of non-crystallographic
symmetry.
•The determination of the positions of heavy or anomalously
scattering atoms within the crystal (i.e. the determination of the
heavy atom sub-structure)
• The determination of the position and orientation of molecular
fragments within the crystal (i.e molecular replacement)
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22
The Patterson Function
1. A simple crystal.
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23
The Patterson Function
1. It’s Patterson function (slightly schematic)
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24
The Patterson Function
•Immediately we see that the Patterson function is much more complicated than the structure
itself. This precludes using it to directly solve the structure in the case of proteins.
•If a structure has N atoms there will be N2 peaks in the Patterson, N of them at the origin.
•However the Patterson does contain important information about the orientation and position
of molecules.
There is a direct correspondence between the symmetry and orientation of
molecules in the cell, and the symmetry and orientation of the
intramolecular vectors (the “self ”vectors) in the Patterson function.
Tuesday, 25 March 14
25
The Patterson function carries information
about molecular orientation
.
Note
particularly that the
symmetry and
orientation of the
intramolecular vectors
(the “self ”vectors) will not be
influenced by the position of a
molecule - i.e. they are
translation independent.
This means we can tackle the
problem of determining
molecular orientation, before
we know anything about
position.
From Toth E.A. Methods in Molecular Biology Vol 364
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26
Use of the Patterson function in molecular
replacement
•Just as the intramolecular (“self ”) vectors carry information about the
orientation of molecules so the intermolecular (“cross”) vectors carry information
. about the position of molecules. We’ll skip the demonstration ...
•As you might suspect, we use the Patterson function to help us sequentially determine the rotation
and translation which will correctly position the search model in the unit cell. So we break the 6dimensional search down to two 3-dimensional searches.
•Basically we look for agreement (correlation) between the Patterson functions calculated from the
search model, and from the experimental data. We will not consider the detailed calculation of the
Rotation function, and the Translation function, further.
•Programs for performing molecular replacement (e.g. Phaser) are now highly automated, and quite
robust.
If the molecular replacement calculations are successful we can calculate phases, but there is a need
to be careful …
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27
The problem of model bias.
Dominance of the phases in the Fourier synthesis
The following picture gives a particularly dramatic illustration of the importance of phase. On the
top are photographs of Jerome Karle (left) and Herb Hauptman (right), who won the Nobel Prize
for their work on solving the phase problem for small molecule crystals. We can treat the
photographs as density maps and calculate their Fourier transforms, to get amplitudes and phases.
Karle
Hauptman
Karleman
Hauptle
(Amplitudes of Karle & Phases of Hauptman)
(Amplitudes of Hauptman & Phases of Karle)
Images courtesy of Randy Read
If we combine the phases from the picture of Hauptman with the amplitudes from the picture of Karle, we get the picture on the
bottom left. The bottom right picture combines the phases of Karle with the amplitudes of Hauptman. Clearly the phases are
dominating what we see. This is worrying when we consider using phases from a model to help image an unknown structure
Tuesday, 25 March 14
28
Dominance of the phases in the Fourier synthesis
An alternate illustration
Cat
Duck
FT(Duck)
FT(Cat)
Duck Amplitudes
+ Cat Phases
FT(Duck
Amplitudes +
Cat Phases)
From Rhodes (2006)
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29
Dominance of the phases in the Fourier
synthesis
Finally let’s consider a crystallographic application. Watch what happens to this 1.5 Å map as the
structure factor amplitudes are slowly randomized (while the phases are left untouched)
.
(James Holton, Berkeley)
The take home message … phases dominate the appearance of the Fourier synthesis, therefore maps
calculated by molecular replacement need to be very carefully inspected because they are
inherently biased toward the search model. It is often best to remove troublesome (poorly
conserved) regions of the model, to help clarify the electron density.
Tuesday, 25 March 14
30
Slightly better ways to calculate electron
density maps
In practice if you calculate electron density maps using the experimentally observed structure factor
amplitudes |Fobs(hkl)|, and phases from the positioned model αcalc(hkl), the model bias is severe. So
instead, two slight variants are in common use. We state these results without proof.
.
The “2Fo-Fc” synthesis
A more useful electron density map is obtained using 2 |Fobs(hkl)| - |Fcalc(hkl)| as the amplitudes,
where the |Fcalc(hkl)| are calculated from the model.
The “Fo-Fc” or Difference synthesis
A second very useful map is obtained using |Fobs(hkl)| - |Fcalc(hkl)| as the amplitudes.This difference
map has the following property …
•There are strong negative peaks where atoms should not be (i.e. are wrongly placed)
•There are strong positive peaks where atoms should be (i.e. are missing)
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31
An example of 2Fo-Fc and Fo-Fc maps
Neuraminidase from Salmonella typhimurium refined at 0.92 Å
.
Blue = 2|Fo|-|Fc| map
Green = |Fo|-|Fc| map,
positive density
There is clear evidence
for an un-modeled
alternate conformation
for this tyrosine side
chain.
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32
Density modification
Experimentally determined phases contain errors, and the resultant electron density maps are often
noisy. A variety of “density modification” procedures exist. All of these adjust electron density
maps to make them conform to some reasonable physical assumptions.
In this example, density modification
procedures have eliminated a break in
the main chain density, and improved
the density for several side chains.
From Blow (2002)
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33
Solvent Flattening
Density in the bulk solvent region is
expected to be smooth and featureless.
Solvent flattening procedures impose
this reasonable physical constraint.
From Sanderson & Skelly (2007)
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34
Histogram matching
For any image, it is possible to calculate a histogram
which shows the relative frequency of each density
value.
Density histograms calculated from reliable protein
electron density maps always look similar.
In the histogram matching procedure, the
density values of an experimental electron density
map are transformed, so that the map has the
histogram expected for a protein.
From Blow (2002)
Tuesday, 25 March 14
35
Non-crystallographic
symmetry averaging
If there is more than one molecule in the
asymmetric unit of the crystal, and these molecules
have essentially the same structure (likely), then we
can average the electron density for the two
copies.
This requires that both the NCS operations, and
the regions of the map related by the NCS, are
known.
From Sanderson & Skelly (2007)
Tuesday, 25 March 14
36
Mask definition
Underpinning practical application of density
modification is the binary partitioning of the map
into protein and solvent regions (mask
definition).
In such a scheme the solvent and protein regions
are mutually exclusive. The union of the two regions
covers the entire cell.
Without defining this mask we do not know which
regions should be made flat; which regions should
be adjusted so that the density conforms to the
expected histograms; and which regions should be
averaged to make the density equivalent.
But how do we determine a mask from a poorly
phased electron density map, into which we can’t
build a model?
Tuesday, 25 March 14
37
Mask definition
R
The sliding sphere used in smoothing an electron
density map. Typically a sphere of radius R = 8-10 Å
is centered on each grid point in turn, and the
average at all surrounding grid points within the
sphere is taken. The average is weighted such that
those grid points furthest from the center make
the least contribution - indicated by the shading.
(After Sherwood and Cooper, 2011)
Tuesday, 25 March 14
B.C. Wang introduced the idea
of smoothing an experimental
electron density map prior to
performing a thresholding step
to determine the protein and
solvent regions (protein will be
high density and solvent low
density)
This works pretty well in
practice.
38
Density modification as an iterative process
You may be wondering how flattening the solvent region in an electron
density map can possibly improve the appearance of the protein region.
Good question !!
In practice, density modification is performed in an iterative fashion, cycling
between a Fourier space representation of the data (Amplitudes and Phases)
and a Real Space representation (electron density) until convergence is
achieved. We will not discuss the details.
However note that each structure factor F(hkl) contains contributions from
the scattering of all atoms in the unit cell (protein and solvent). So it
shouldn’t be hard to intuit that modification of the density in the solvent
region can influence the phases α(hkl) of the Structure Factors, when the
process is made iterative.
Tuesday, 25 March 14
39