Transcript Solving Crystal Structures From Two-wavelength X

Phasing
Today’s goal is to calculate phases (ap) for proteinase K using
PCMBS and EuCl3 (MIRAS method) .
What experimental data do we need?
1) from native crystal we measured |Fp| for native crystal
2) from PCMBS derivative we measured |FP•Hg(h k l)|
3) also from PCMBS derivative we measured |FP•Hg(-h-k-l)|
4) fHg for Hg atom of PCMBS
5) from EuCl3 derivative we measured |FP•Eu(h k l)|
6) also from EuCl3 derivative we measured |FP•Eu(-h-k-l)|
7) FEu for Eu atom of EuCl3
How many phasing triangles will we have for each structure factor if
we use all the data sets ?
FP ( h k l) = FPH ( h k l) - FH ( h k l) is one type of phase triangle.
FP ( h k l) = FPH(-h-k-l) * - FH (-h-k-l)* is another type of phase triangle.
Four Phase Relationships
PCMBS
FP ( h k l) = FP•Hg ( h k l) - fHg ( h k l) isomorphous differences
FP ( h k l) = FP•Hg(-h-k-l)* - fHg (-h-k-l)*
from Friedel mates (anomalous)
EuCl3
FP ( h k l) = FP•Eu( h k l) - fEu ( h k l)
isomorphous differences
FP ( h k l) = FP•Eu (-h-k-l)* - fEu (-h-k-l)* from Friedel mates (anomalous)
FP ( h k l) = FPH1 ( h k l) - fH1 (h k l)
Imaginary axis
|Fp ( h k l) | = 1.8
2
1
Real axis
-2
-1
1
2
-1
-2
Harker construction
1) |FP ( h k l) | –native measurement
FP ( h k l) = FPH1 ( h k l) - fH1 ( h k l)
|Fp ( h k l) | = 1.8
|fH ( h k l) | = 1.4
aH (hkl)
= 45°
Imaginary axis
|Fp |
Real axis
FH
Harker construction
1) |FP ( h k l) | –native measurement
2) FH (h k l) calculated from heavy
atom position.
FP ( h k l) = FPH1 ( h k l) - fH1 ( h k l)
Imaginary axis
|FP ( hkl) | = 1.8
|fH1 ( hkl) | = 1.4
aH1 (hkl) = 45°
|FPH1 (hkl) | = 2.8
|Fp |
|FPH|
Real axis
fH
|FPH|
Let’s look at the quality
of the phasing statistics
up to this point.
Harker construction
1) |FP ( h k l) | –native measurement
2) fH1 (h k l) calculated from heavy
atom position.
3) |FPH1(hkl)|–measured from
derivative.
Which of the following
graphs best represents
the phase probability
distribution, P(a)?
SIR
Imaginary axis
|Fp |
a)
|FPH|
0
90
180
270
360
0
90
180
270
360
0
90
180
270
360
Real axis
b)
fH
|FPH|
c)
The phase probability
distribution, P(a) is sometimes
shown as being wrapped
around the phasing circle.
SIR
Imaginary axis
|Fp |
0
|FPH|
90
180
270
Real axis
fH
90
|FPH|
0
180
270
360
Which of the following is
the best choice of Fp?
SIR
Imaginary axis
90
|Fp |
a)
90
|FPH|
270
0
180
Real axis
90
b)
fH
0
180
0
180
|FPH|
270
270
90
c)
0
180
270
Radius of circle is approximately |Fp|
SIR
Imaginary axis
|Fp |
90
|FPH|
a
0
180
fH
|FPH|
270
|Fp|eia•P(a)da
best F =
Real axis
Sum of
probability
weighted
vectors Fp
Usually shorter than Fp
SIR
|Fp|eia•P(a)da
best F =
90
a
0
180
270
Best phase
Sum of
probability
weighted
vectors Fp
SIR
Which of the following is
the best approximation
to the Figure Of Merit
(FOM) for this reflection?
90
0
180
a)
b)
c)
d)
1.00
2.00
0.50
-0.10
270
FOM=|Fbest|/|FP|
Radius of circle is approximately |Fp|
Which phase probability distribution would
yield the most desirable Figure of Merit?
b)
a)
0
90
270
0
180
270
+
+
90
180
c)
0
+
+
270
180
90
SIR
Which of the following is
the best approximation
to the phasing power for
this reflection?
Imaginary axis
|Fp |
Fbest
|FPH|
Real axis
fH
|Fp |
a)
b)
c)
d)
2.50
1.00
0.50
-0.50
|FPH|
Phasing Power =
|fH|
Lack of closure
|fH ( h k l) | = 1.4
Lack of closure = |FPH|-|FP+FH| = 0.5
(at the aP of Fbest)
SIR
Which of the
following is the most
desirable phasing
power?
Imaginary axis
|Fp |
Fbest
|FPH|
Real axis
fH
|Fp |
a)
b)
c)
d)
2.50
1.00
0.50
-0.50
|FPH|
Phasing Power =
|fH|
Lack of closure
What Phasing Power is sufficient to solve the structure?
>1
|FP ( hkl) | = 1.8
|FP•Hg (hkl) | = 2.8
SIR
Imaginary axis
Which of the following is
the RCullis for this
reflection?
a)
b)
c)
d)
|Fp |
Fbest
|FPH|
Real axis
fH
|Fp |
-0.5
0.5
1.30
2.00
|FPH|
RCullis =
Lack of closure
isomorphous difference
From previous page, LoC=0.5
Isomorphous difference= |FPH| - |FP|
1.0 = 2.8-1.8
SIRAS
Isomorphous differences
Anomalous differences
Imaginary axis
FP ( h k l) = FPH(-h-k-l)* - fH (-h-k-l)*
Real axis
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.
To Resolve the phase ambiguity
SIRAS
Isomorphous differences
Anomalous differences
Imaginary axis
Real axis
Which P(a) corresponds to SIR?
Which P(a) corresponds to SIRAS?
0
90
180
270
360
SIRAS
Isomorphous differences
Anomalous differences
Imaginary axis
Which graph represents FOM of SIRAS phase?
Which represents FOM of SIR phase?
Which is better? Why?
Real axis
90
90
0
180
270
0
180
270
Calculate phases and refine
parameters (MLPhaRe)
A Tale of Two Ambiguities
• We can solve both ambiguities in one
experiment.
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
Note: Inversion of the space group symmetry
(P43212 →P41212) accompanies inversion of the
coordinates (x,y,z→ -x,-y,-z)
Choice of origin ambiguity
• I want to include the Eu data (derivative 2)
in phase calculation.
• I can determine the Eu site x,y,z coordinates
using a difference Patterson map.
• But, how can I guarantee the set of
coordinates I obtain are referred to the same
origin as Hg (derivative 1)?
• Do I have to try all 48 possibilities?
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 Hg and run
cross difference Fourier to find the Eu site. Note
the height of the peak and Eu coordinates.
2) Negate x,y,z of Hg 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 Eu site. Compare the height of
the peak with step 1.
3) Chose the handedness which produces the
highest peak for Eu. Use the corresponding
hand of space group and Hg, and Eu
coordinates to make a combined set of phases.
fH1(hkl)
FPH1(hkl)
MIRAS
fH1(-h-k-l)*
FPH1(-h-k-l)*
Imaginary axis
Isomorphous
Deriv 2
fH2(hkl)
FPH2(hkl)
Fp (hkl)
Anomalous
Deriv 1
Real axis
Isomorphous
Deriv 1
EuCl3
FPH = FP+fH
for isomorphous differences
fH1(hkl)
FPH1(hkl)
EuCl3
FP ( h k l) = FPH1 (-h-k-l)* - fH1 (-h-k-l)*
fH1(-h-k-l)*
FPH1(-h-k-l)*
Imaginary axis
Isomorphous
Deriv 2
fH2(hkl)
FPH2(hkl)
fH2 (-h-k-l)*
FPH2(-h-k-l)*
Anomalous
Deriv 1
Isomorphous
Deriv 1
Fp (hkl)
Real axis
Anomalous
Deriv 2
Harker Construction for MIRAS phasing (Multiple Isomorphous Replacement with Anomalous Scattering)
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
MIRAS phased map
MIR phased map +
Solvent Flattening +
Histogram Matching
MIRAS phased map
MIR phased map +
Solvent Flattening +
Histogram Matching
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
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 Eu site using Cross Difference Fourier map.
1) Easier than Patterson methods.
2) Want to combine PCMBS and Eu 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 Eu 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