Phasing
Goal is to calculate phases using isomorphous and
anomalous differences from PCMBS and GdCl3
derivatives --MIRAS.
How many phasing triangles will we have for each
structure factor?
For example. FPH = FP+FH
for isomorphous differences
For example. FPH* = FP+ FH* for anomalous differences
-h-k-l
hkl
-h-k-l
4 Phase relationships
PCMBS FPH = FP+FH
for isomorphous differences
PCMBS FPH* = FP+ FH* for anomalous differences
-h-k-l
GdCl3
GdCl3
hkl
-h-k-l
FPH = FP+FH for isomorphous differences
FPH* = FP+ FH* for anomalous differences
-h-k-l
hkl
-h-k-l
PCMBS FPH = FP+FH
for isomorphous differences
Imaginary axis
|FPH|
|Fp |
Real axis
FH
Harker construction for SIR
phases.
FP –native measurement
FH (hkl) calculated from heavy
atom position.
FPH(hkl)–measured from
derivative. Point to these on
graph.
SIR Phasing Ambiguity
PCMBS FPH* = FP+ FH* for anomalous differences
-h-k-l
hkl
-h-k-l
Imaginary axis
Isomorphous differences
Anomalous differences
FH(-h-k-l)*
Fp (hkl)
Real axis
FH(hkl)
We will calculate SIRAS phases
using the PCMBS Hg site.
FP –native measurement
FH (hkl) and FH(-h-k-l) calculated
from heavy atom position.
FPH(hkl) and FPH(-h-k-l) –measured
from derivative. Point to these on
graph.
Harker Construction for SIRAS phasing (Single Isomorphous
Replacement with Anomalous Scattering)
GdCl3
FPH = FP+FH
for isomorphous differences
Imaginary axis
Isomorphous
Deriv 2
Anomalous
Deriv 1
Fp (hkl)
Real axis
Isomorphous
Deriv 1
Harker Construction for MIRAS phasing (Multiple Isomorphous Replacement with Anomalous Scattering)
GdCl3
FPH* = FP+ FH* for anomalous differences
-h-k-l
hkl
-h-k-l
Imaginary axis
Isomorphous
Deriv 2
Anomalous
Deriv 1
Isomorphous
Deriv 1
Fp (hkl)
Real axis
Anomalous
Deriv 2
Harker Construction for MIRAS phasing (Multiple Isomorphous Replacement with Anomalous Scattering)
Barriers to combining phase
information from 2 derivatives
1) Initial Phasing with PCMBS
1) Calculate phases using coordinates you determined.
2) Refine heavy atom coordinates
2) Find Gd site using Cross Difference Fourier map.
1) Easier than Patterson methods.
2) Want to combine PCMBS and Gd to make MIRAS phases.
3) Determine handedness (P43212 or P41212 ?)
1) Repeat calculation above, but in P41212.
2) Compare map features with P43212 map to determine
handedness.
4) Combine PCMBS and Gd sites (use correct hand of
space group) for improved phases.
5) Density modification (solvent flattening & histogram
matching)
1) Improves Phases
6) View electron density map
Center of inversion ambiguity
Remember, because the position of
Hg was determined using a Patterson
map there is an ambiguity in
handedness.
The Patterson map has an additional
center of symmetry not present in the
real crystal. Therefore, both the site
x,y,z and -x,-y,-z are equally
consistent with Patterson peaks.
Handedness can be resolved by
calculating both electron density maps
and choosing the map which contains
structural features of real proteins (Lamino acids, right handed a-helices).
If anomalous data is included, then
one map will appear significantly better
than the other.
Patterson map
Use a Cross difference Fourier to resolve
the handedness ambiguity
With newly calculated protein phases, fP, a protein
electron density map could be calculated.
The amplitudes would be |FP|, the phases would be fP.
r(x)=1/V*S|FP|e-2pi(hx+ky+lz-fP)
Answer: If we replace the coefficients with
|FPH2-FP|, the result is an electron density map
corresponding to this structural feature.
r(x)=1/V*S|FPH2-FP|e-2pi(hx-fP)
What is the second heavy atom, Alex.
When the difference FPH2-FP is taken, the protein
component is removed and we are left with only the
contribution from the second heavy atom.
This cross difference Fourier will help us in two ways:
1) It will resolve the handedness ambiguity by
producing a very high peak when phases are
calculated in the correct hand, but only noise when
phases are calculated in the incorrect hand.
2) It will allow us to find the position of the second
heavy atom and combine this data set into our
phasing. Thus improving our phases.
Phasing Procedures
1) Calculate phases for site x,y,z of PCMBS and run
cross difference Fourier to find the Gd site. Note
the height of the peak and Gd coordinates.
2) Negate x,y,z of PCMBS and invert the space
group from P43212 to P41212. Calculate a second
set of phases and run a second cross difference
Fourier to find the Gd site. Compare the height
of the peak with step 1.
3) Chose the handedness which produces the
highest peak for Gd. Use the corresponding
hand of space group and PCMBS, and Gd
coordinates to make a combined set of phases.
Lack of closure
e=(FH+FP)-(FPH)
FH-calculated from
atom position
FP-observed
FPH-observed
e is the discrepancy
between the heavy atom
model and the actual
data.
Why is it not zero?
Phasing power
|FH|/ e = phasing power.
e=(FH+FP)-(FPH)
The bigger the better.
Phasing power >1.5 excellent
Phasing power =1.0 good
Phasing power = 0.5 unusable
Rcullis
e/|FPH|-|FP|= Rcullis.
e=(FH+FP)-(FPH)
Kind of like an Rfactor for your
heavy atom model. |FPH|-|FP| is
like an observed FH, and e is
the discrepancy between the
heavy atom model and the
actual data.
Rcullis <1 is useful. <0.6 great!
Figure of Merit
0
270
0
+
+
90
180
270
+
+
90
180
0
+
+
270
180
Phase probability distribution
How far away is the center of mass from the center of the circle?
90
Density modification
A) Solvent flattening.
• Calculate an electron density map.
• If r<threshold, -> solvent
• If r>threshold -> protein
• Build a mask
• Set density value in solvent region
to a constant (low).
• Transform flattened map to structure
factors
• Combine modified phases with
original phases.
• Iterate
Density modification
B) Histogram matching.
• Calculate an electron density
map.
• Calculate the electron density
distribution. It’s a histogram.
How many grid points on map
have an electron density
falling between 0.2 and 0.3
etc?
• Compare this histogram with
ideal protein electron density
map.
• Modify electron density to
resemble an ideal distribution.
Number of times a particular electron density value is observed.
Electron density value
HOMEWORK