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

-0.10.00.10.20.30.40.50.60.70.80.91.0

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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

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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

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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