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

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Patterson: one atom

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Real Space

Patterson

Fraction across unit cell

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Patterson: one atom

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Real Space

Patterson

Fraction across unit cell

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Patterson: one atom

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Real Space

Patterson

Fraction across unit cell

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Patterson: two atoms

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Real Space

Patterson

Fraction across unit cell

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Patterson: three atoms

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Real Space

Patterson

Fraction across unit cell

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Patterson: three atoms

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Real Space

Patterson

Fraction across unit cell

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Better resolution!

Patterson: five atoms

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Real Space

Patterson

Fraction across unit cell

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Patterson: five atoms

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Real Space

Patterson

Fraction across unit cell

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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

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Patterson: five atoms + 3-fold symmetry

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Patterson

Fraction across unit cell

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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

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2Fo-Fc maps

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True density

Model

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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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

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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

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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

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2Fo-Fc maps

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True density

Model

2Fo-Fc

Fraction across unit cell

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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

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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

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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

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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