Pattersons The “third space” of crystallography. The “phase problem”
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Transcript of Pattersons The “third space” of crystallography. The “phase problem”
Pattersons
The “third space”
of crystallography
The “phase problem”
The “phase problem”
Phases Amplitudes
The “phase problem”
Phases Amplitudes
The “spaces” of crystallography
• Direct/real space– Distances are in Å, Angles are in degrees
• Reciprocal space– Distances in 1/Å, Angles are different
• Patterson space– Distances are in Å , Angles are in degrees– Relative distances only, origin lost– “direction” is preserved
dete
ctor
sam
ple
dete
ctor
x-ray beam
scattering
How to make a Patterson map:
1. Set all phases to zero
2. Square all structure factors
3. Calculate Fourier transform
One atom in unit cell
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Real Space
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: one atom
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: one atom
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: one atom
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: two atoms
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: three atoms
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: three atoms
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Better resolution!
Patterson: five atoms
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: five atoms
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
scattering from a structure
sample detector
forward Fourier Transforminverse Fourier Transformno phase
Patterson map!
Snapshot from single virus particle
TEM
2 keV LCLS 200 fs Mimi virus single-shot.200 nm
Reconstructed image
Resolution 20nm
Seibert, et al. (2011). Nature 470, 78-81.
lysozyme: real and reciprocal
forward Fourier Transform9 atoms
forward Fourier Transform10 atoms
Patterson map10 atoms
Patterson map9 atoms
Difference PattersonStill no phases!
forward Fourier Transform9 atoms
Harker Section of a Patterson
X-ray data are 3D!
Patterson: five atoms + 3-fold symmetry
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Real Space
Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Patterson: five atoms + 3-fold symmetry
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Patterson
Fraction across unit cell
Ele
ctro
n de
nsity
Major Phasing techniques
• Molecular Replacement
• Multiple Isomorphous Replacement
• Anomalous Diffraction
• Direct methods
2Fo-Fc maps
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True density
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
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True density
Model
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
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True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
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True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
2Fo-Fc maps
-0.10.00.10.20.30.40.50.60.70.80.91.0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
True density
Model
2Fo-Fc
Fraction across unit cell
Ele
ctro
n de
nsity
The “phase problem”
Phases Amplitudes
Fobs
The “phase problem”
Phases & Amplitudes Amplitudes
Fobs-Fcalc
The “phase problem”
Phases & Amplitudes Amplitudes
2Fobs-Fcalc
The “phase problem”
Phases Amplitudes
The “phase problem”
Phases & Amplitudes Amplitudes
2Fobs-Fcalc
The “phase problem”
Phases & Amplitudes Amplitudes
2mFobs-Fcalc (A weighting)
sam
ple
x-ray beam
anomalous scattering
dete
ctor
sam
ple
dete
ctor
x-ray beam
anomalous scattering
sam
ple
x-ray beam
anomalous scattering
dete
ctor
sam
ple
dete
ctor
x-ray beam
anomalous scattering
sam
ple
dete
ctor
x-ray beam
anomalous scattering
Harker Section of a Patterson
Summary
• Patterson = real-space representation of all information in diffraction pattern
• “small”, high-resolution structures solved with no phases!
• Difference Pattersons for finding heavy atom sites
• Native Patterson for symmetry
