Pattersons The “third space” of crystallography. The “phase problem”

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Presentation transcript:

Pattersons The “third space” of crystallography

The “phase problem”

PhasesPhases Amplitudes

The “phase problem” PhasesPhases 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

detector sample detector 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 Fraction across unit cell Electron density

Patterson: one atom Fraction across unit cell Electron density

Patterson: one atom Fraction across unit cell Electron density

Patterson: one atom Fraction across unit cell Electron density

Patterson: two atoms Fraction across unit cell Electron density

Patterson: three atoms Fraction across unit cell Electron density

Patterson: three atoms Fraction across unit cell Electron density Better resolution!

Patterson: five atoms Fraction across unit cell Electron density

Patterson: five atoms Fraction across unit cell Electron density

scattering from a structure sample detector

forward Fourier Transforminverse Fourier Transform no 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,

lysozyme: real and reciprocal

forward Fourier Transform 9 atoms

forward Fourier Transform 10 atoms

Patterson map 10 atoms

Patterson map 9 atoms

Difference Patterson Still no phases!

forward Fourier Transform 9 atoms

Harker Section of a Patterson

X-ray data are 3D!

Patterson: five atoms + 3-fold symmetry Fraction across unit cell Electron density

Patterson: five atoms + 3-fold symmetry Fraction across unit cell Electron density

Major Phasing techniques Molecular Replacement Multiple Isomorphous Replacement Anomalous Diffraction Direct methods

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

2Fo-Fc maps Fraction across unit cell Electron density

The “phase problem” PhasesPhases Amplitudes F obs

The “phase problem” Phases & Amplitudes Amplitudes F obs -F calc

The “phase problem” Phases & Amplitudes Amplitudes 2F obs -F calc

The “phase problem” PhasesPhases Amplitudes

The “phase problem” Phases & Amplitudes Amplitudes 2F obs -F calc

The “phase problem” Phases & Amplitudes Amplitudes 2mF obs -F calc (  A weighting)

sample x-ray beam anomalous scattering detector

sample detector x-ray beam anomalous scattering

sample x-ray beam anomalous scattering detector

sample detector x-ray beam anomalous scattering

sample detector 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