Overview of the Phase Problem
Protein
Crystal
Data
Phases
Structure
John Rose
ACA Summer School 2006
Reorganized by Andy Howard, Biology 555, Spring 2008
Part 2 of 2
Remember
We can measure reflection intensities
We can calculate structure factors from the intensities
We can calculate the structure factors from atomic positions
We need phase information to generate the image
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Biology 555:
Crystallographic Phasing II
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Finding the Heavy Atoms
or Anomalous Scatterers
The Patterson function
- a F2 Fourier transform
with f = 0
- vector map
(u,v,w instead of x,y,z)
- maps all inter-atomic vectors
- get N2 vectors!!
(where N= number of
atoms)
1
Puvw  | Fhkl |2 cos2 (hu  kv  lv)
V hkl
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Biology 555:
Crystallographic Phasing II
From Glusker, Lewis and Rossi
p. 2 of 38
The Difference Patterson Map
SIR : |DF|2 = |Fnat - Fder|2
SAS : |DF|2 = |Fhkl - F-h-k-l|2
Patterson map is centrosymmetric
- see peaks at u,v,w & -u, -v, -w
Peak height proportional to ZiZj
Peak u,v,w’s give heavy atom x,y,z’s
- Harker analysis
Origin (0,0,0) maps vector of atom to itself
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Harker analysis
• Certain relationships apply in Patterson
maps that enable us to determine some of
the coordinates of our heavy atoms
David
• They depend on looking at differences
Harker
between atomic positions
• These relationships were worked out by
Lindo Patterson and David Harker
• Patterson space is centrosymmetric but
otherwise similar to original symmetry; but
Patterson symmetry has no translations
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Example: space group P21
• P21 has peaks at
R1=(x,y,z) and R2=(-x,y+1/2,-z)
• Therefore we’ll get Patterson (difference) peaks at
R1-R1, R1-R2, R2-R1, R2-R1:
• (0,0,0), (2x,-1/2,2z), (-2x,1/2,-2z),(0,0,0)
• So if we look at the section of the map at Y=1/2,
we can find peaks at (-2x,1/2,-2z) and thereby
discern what the x and z coordinates of a real atom
are
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How do we actually use this?
• Compute difference Patterson map,
i.e. map with coefficients derived from
FhklPH - FhklP or Fhkl - F-h-k-l
• Examine Harker sections
• Peaks in Harker sections tell us where the
heavy atoms or anomalous scatterers are
• Automated programs like BNP, SOLVE,
SHELX can do the heavy lifting for us
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A Note About Handedness
• We identify each reflection by an index, hkl.
• The hkl also tells us the relative location of that reflection
in a reciprocal space coordinate system.
• The indexed reflection has correct handedness if a data
processing program assigns it correctly.
• The identity of the handedness of the molecule of the
crystal is related to the assignment of the handedness of the
data, which may be right or wrong!
• Note: not all data processing programs assign handedness
correctly!
• Be careful with your data processing.
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The Phase Triangle Relationship
DOLM = DOLN
M
QLM  LON  
Q
LON     H
L
O

From Glusker, Lewis and Rossi
N

FPH = FP + FH
Need value of FH
FP, FPH, FH and -FH are vectors (have direction)
FP <= obtained from native data
FPH <= obtained from derivative or anomalous data
FH <= obtained from Patterson
analysis
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The Phase Triangle Relationship
M
Q
L
O
From Glusker, Lewis and Rossi
N
• In simplest terms, isomorphous replacement finds the
orientation of the phase triangle from the orientation of one
of its sides. It turns out, however, that there are two
possible ways to orient the triangle if we fix the orientation
of one of its sides.
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Single Isomorphous Replacement
Note:
FP = protein
FH = heavy atom
FP1 = heavy atom derivative
The center of the FP1circle is
placed at the end of the
vector -FH1.
X1 ftrueor ffalse
X2 ftrueor ffalse
From Glusker, Lewis and Rossi
• The situation of two possible SIR phases is called the
“phase ambiguity” problem, since we obtain both a true
and a false phase for each reflection. Both phase
solutions are equally probable, i.e. the phase probability
distribution is bimodal. Biology 555:
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Crystallographic Phasing II
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Resolving the Phase Ambugity
Note:
FP = protein
FH = heavy atom
FP1 = heavy atom derivative
The center of the FP1circle
is placed at the end of the
vector -FH1.
X1 ftrueor ffalse
X2 ftrueor ffalse
From Glusker, Lewis and Rossi
Add more information:
(1) Add another derivative (Multiple Isomorphous Replacement)
(2) Use a density modification technique (solvent flattening)
(3) Add anomalous data (SIR with anomalous scattering)
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Multiple Isomorphous Note:
Replacement
FP = protein
FH1 = heavy atom #1
FH2 = heavy atom #2
FP1 = heavy atom derivative
FP2 = heavy atom derivative
The center of the FP1 and FP1
circles are placed at the end of the
vector -FH1 and -FH2, respectively.
From Glusker, Lewis and Rossi
•
X1 ftrue
X2 ffalse
X3 ffals
We still get two solutions, one true and one
false for each reflection from the second
Exact overlap at X1
derivative. The true solutions should be
dependent on data accuracy
consistent between the two derivatives
dependent on HA accuracy
while the false solution should show a
called lack of closure
random variation.
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Crystallographic Phasing II
Solvent Flattening
Similar to noise filtering
Resolve the SIR or SAS phase ambiguity
Electron density can’t be negative
Use an iterative process to enhance true phase!
From Glusker, Lewis and Rossi
B.C. Wang, 1985
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How does solvent flattening
resolve the phase ambiguity?
• Solvent flattening can locate and enhance the protein
image—viz., whatever is not solvent must be protein!
• From the protein image, the phases of the structure
factors of the protein can be calculated
• These calculated phases are then used to select the true
phases from sets of true and false phases
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Using the structure to solve the
phase ambiguity
• Thus, in essence, the phase
ambiguity is resolved by the protein
image itself!
• This solvent-flattening process was
made practical by the introduction of
the ISIR/ISAS program suite (Wang,
1985) and other phasing programs
such DM and PHASES are based on
this approach.
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Handedness from solvent flattening
• The ISAS process is performed twice, once with
heavy atom sites @ refined locations, once in
their inverted locations
Data
FOM1 Handed-
FOM2
Rfactor
Corr.
Coeff.
ness
RHE
0.54
Correct
0.82
0.26
0.958
0.54
Wrong
0.80
0.30
0.940
0.54
Correct
0.80
0.27
0.955
0.54
Wrong
0.76
0.36
0.919
NP+I+S4 0.56
Correct
0.82
0.24
0.964
0.56
Wrong
0.78
0.35
0.926
NP + I3
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Notes on the handedness table
•
1:
Figure of merit before solvent flattening
•
2:
Figure of merit after one filter and four cycles of solvent
flattening
•
3:
Four Iodine were used for phasing
• 4: Four Iodine and 56 Sulfur atoms were used for phasing
• Heavy Atom Handedness and Protein Structure Determination
using Single-wavelength Anomalous Scattering Data, ACA
Annual Meeting, Montreal, July 25, 1995.
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Does the correct hand make a difference?
• Yes!
• The wrong
hand will give
the mirror
image!
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Anomalous Dispersion Methods
• All elements display an anomalous dispersion
(AD) effect in X-ray diffraction
• For light elements (H, C, N, O), anomalous
dispersion effects are negligible; they’re small
even for S and P at typical X-ray energies
• For heavier elements, especially when the X-ray
wavelength approaches an atomic absorption edge
of the element, these AD effects can be very large.
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Scattering power when
anomalous scattering exists
The scattering power of an atom exhibiting AD effects is:
fAD = fn + Df' + iDf”
where:
fnis the normal scattering power of the atom in absence of
AD effects
Df' arises from the AD effect and is a real factor
(+/- signed) added to fn
Df" is an imaginary term which also arises from the AD
effect
Df" is always positive and 90° ahead of (fn + Df') in phase
angle
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Df’ and Df”
• The values of Df' and Df" are highly dependent
on the wavelength of the X-radiation.
• In the absence of AD effects, Ihkl = I-h-k-l
(Friedel’s Law).
• With AD effects, Ihkl ≠ I-h-k-l (Friedel’s Law breaks
down).
• Accurate measurement of Friedel pair differences
can be used to extract starting phases if the AD
effect is large enough.
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Breakdown of Friedel’s Law
f’
f’
(Fhkl Left) Fn represents the total scattering by "normal" atoms without AD effects,
f’ represents the sum of the normal and real AD scattering values (fn + Df'), Df"
is the imaginary AD component and appears 90° (at a right angle) ahead of the f’
vector and the total scattering is the vector F+++.
(F-h-k-l Right) F-n is the inverse of Fn (at -hkl) and f’ is the inverse of f’, the Df"
vector is once again 90° ahead of f’. The resultant vector, F--- in this case, is
obviously shorter than the F+++ vector.
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Crystallographic Phasing II
Collecting Anomalous Scattering Data
• Anomalous scatterers, such as
selenium, are generally incorporated
into the protein during expression of
the protein or are soaked into the
crystals in a manner similar to
preparing a heavy atom derivative.
• Bromine, iodine, xenon and
traditional heavy atom compounds
are also good anomalous scatterers.
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How strong is the signal?
• The anomalous signal, the difference between
|F+++| and |F---| is generally about one order of
magnitude smaller than that between |FPH(hkl)|,
and |FP(hkl)|.
• Thus, the signal-to-noise (S/n) level in the data
plays a critical role in the success of anomalous
scattering experiments, i.e. the higher the S/n in
the data the greater the probability of
producing an interpretable electron density
map.
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Why does it work at all?
The lack of isomorphism problem is much
milder for anomalous data than for
isomorphous replacement:
• One sample, not two or more
• Unit cell is by definition (?) identical
• Molecule is in the same place within that
unit cell
• That partly compensates for the low S/N
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Why is selenium a good choice?
• Methionine is a relatively rare amino acid: 2.4%
(vs. average of 5%)
• So there aren’t a huge number of mets in a typical
protein, but there generally are a few
• It’s possible to make E.coli auxotrophic for
methionine and then feed it selenomethionine in
its place
• This incorporates SeMet stoichiometrically and
covalently, which is definitely good!
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Anomalous data collection
• The anomalous signal can be optimized by data
collection at or near the absorption edge of the
anomalous scatterer. This requires a tunable X-ray
source such as a synchrotron.
• The S/n of the data can also be increased by
collecting redundant data.
• The two common anomalous scattering experiments
are Multiwavelength Anomalous Dispersion
(MAD) and single wavelength anomalous
scattering/diffraction (SAS or SAD)
• The SAS technique is becoming more popular since
it does not require a tunable X-ray source.
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Increasing Number of SAS Structures
MAD
SAD
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Increasing S/n with Redundancy
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Multiwavelength Anomalous Dispersion
.
Note:
FP = protein
FH1 = heavy atom
F+PH = F+++
F-PH = F--F+H” = Df”+++
F-H” = Df”---
From Glusker, Lewis and Rossi
•
The center of the F+PH and F-PH
circles are placed at the end of
the vector -F+H” and -F-H”
respectively
In the MAD experiment a strong anomalous scatterer is introduced into the crystal
and data are recorded at several wavelengths (peak, inflection and remote) near
the X-ray absorption edge of the anomalous scatterer. The phase ambiguity
resolved a manner similar to the use of multiple derivatives in the MIR technique
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Single Wavelength Anomalous Scattering
• The SAS method, which combines the use of SAS data
and solvent flattening to resolve phase ambiguity was
first introduced in the ISAS program (Wang, 1985).
The technique is very similar to resolving the phase
ambiguity in SIR data.
• The SAS method does not require a tunable source
and successful structure determination can be carried
out using a home X-ray source on crystals containing
anomalous scatterers with sufficiently large Df” such
as iron, copper, iodine, xenon and many heavy atom
salts.
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Sulfur S-SAS:
experimental realities
• The ultimate goal of the SAS method is the use of SSAS to phase protein data since most proteins contain
sulfur. However sulfur has a very weak anomalous
scattering signal with Df” = 0.56 e- for Cu X-rays. The
S-SAS method requires careful data collection and
crystals that diffract to 2Å resolution.
• A high symmetry space group (more internal
symmetry equivalents) increases the chance of success.
• The use of soft X-rays such as Cr K (= 2.2909Å)
X-rays doubles the sulfur signal (Df” = 1.14 e-).
• There over 20 S-SAS structures in the Protein Data
Bank.
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What is the Limit of the SAS
Method?
• Electron density maps of Rhe by Sulfur-ISAS
• Calculated using simulated data in 1983
• Df” = 0.56e- using Cu K X-rays
19 Feb 2008
Biology Enzymol.
555:
Wang (1985), Methods
115: 90-112 p. 33 of 38
Crystallographic Phasing II
Molecular Replacement
• Molecular replacement has proven effective for
solving macromolecular crystal structures based
upon the knowledge of homologous structures.
• The method is straightforward and reduces the time
and effort required for structure determination
because there is no need to prepare heavy atom
derivatives and collect their data.
• Model building is also simplified, since little or no
chain tracing is required.
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Molecular Replacement:
Practical Considerations
• The 3-dimensional structure of the search model must be
very close (< 1.7Å r.m.s.d.) to that of the unknown
structure for the technique to work.
• Sequence homology between the model and unknown
protein is helpful but not strictly required. Success has
been observed using search models having as low as 17%
sequence similarity.
• Several computer programs such as AmoRe, XPLOR/CNS PHASER are available for MR calculations.
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How Molecular Replacment works
• Use a model of the
protein to estimate
phases
• Must be a structural
homologue
(RMSD < 1.7Å)
• Two-step process:
rotation and translation
• Find orientation of model
(red black)
• Find location of oriented
model (black blue)
px.cryst.bbk.ac.uk/03/sample/molrep.htm
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Using a protein model to estimate phases:
the rotation function
• We need to determine the model’s
orientation in X1’s unit cell
• We use a Patterson search approach in
(,,), which are Euler angles
associated with the rotational space
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Euler angles for
rotation function
The coordinate system
is rotated by:
• an angle  around
the original z axis;
• then by an angle 
around the new y
axis;
• and then by an angle
 around the final z
axis.
zyz convention
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Using a protein model to estimate phases:
translation function
• We need to determine the oriented model’s
location in X1’s unit cell
• We do this with an R-factor search, where
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Translation functions
• Oriented model is stepped through the X1 unit
cell using small increments in x, y, and z (e.g.
x  x+ step)
• The point where R is lowest represents the
correct location
• There exists an alternative method that uses
maximum likelihood to find the translation
peak; this notion is embodied in the software
package PHASER by Randy Read
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