EM Diffraction

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EM Diffraction

• As applied to proteins

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Diffraction from a grating
Molecular Expressions
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Braggs Law

• N 2dsin(?)/?
• Derivation http//www.eserc.stonybrook.edu/Projec
  tJava/Bragg/
• Applies to electron diffraction from crystals
• Therefore, the diffraction of electrons from a
  crystal is quantized

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Scattering factors

• What within the planes of the crystal is
  scattering electrons? The electrons of the atoms
  of the molecules.
• If we can map the electron density distribution
  of the scattering material, we can determine
  crystal structure

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Fourier Transform
We can model any function by adding together
waves that are integral multiples in
frequency F(x) C0 ?Cncos(2px/(?/n) an).
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Scattering of waves by an object gives rise to a
Fourier transform

• http//micro.magnet.fsu.edu/primer/java/interferen
  ce/doubleslit/
• The Fourier transform of a single spacing is a
  single cosine wave (This is how diffraction
  gratings work)
• Note that small spacings in real space give rise
  to large spacings in reciprocal space
• This is the origin of the Rayleigh limit the
  cone of scattering that can be collected by a
  microscope is finite.

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Inverse Fourier Transform

• A Fourier transform of a Fourier transform
  generates the negative of the original function
• We can therefore multiply this by -1 to give an
  inverse Fourier transform Optically, this is
  accomplished by a lens
• However, all we can measure in the rear focal
  plane of a microscope is the amplitudes of
  scattered beams, not their phases
• This is the origin of the phase problem in
  diffraction

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Electron diffraction crystal structure

• Electron diffraction is advantageous because the
  diffracted electrons average over many repeats of
  a structure
• We can measure diffracted electron intensities
• How can we get their phases, so that we can use
  an inverse Fourier transform to retrieve the
  structure?

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Electron Diffraction from BR
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Electron Crystallography

• One way Do electron diffraction to measure
  amplitude, and low-dose transmission electron
  microscopy to get a direct image
• Use the direct image to calculate phases
• This is extremely useful for proteins that for
  microcrystals, or two dimensional crystals

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Flow Chart (Henderson et al, 1990)
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Examples

• 2-D crystals
• Purple membrane protein (bacteriorhodopsin) from
  Halobacter halobium.
• Light harvesting complex
• http//blanco.biomol.uci.edu/Membrane_Proteins_xta
  l.html
• Microcrystals
• Prion protein
• Catalase

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Catalase crystals
Brink Lab, Baylor
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Purple membrane protein (1990)
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Techniques (Fujiyoshi)

• Membranes prepared from Halobacter halobium
• Crystals suspended in 3 trehalose
• Applied to carbon-coated melybdenum grids
• Plunged into liquid ethane to give vitreous ice
• Transferred to cryostage of EM (FEG)

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Projection density map of BR
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How to access the third dimension?Tilt Series
Note that because this is a 2-Dimensional
crystal, you are sampling the continuous Fourier
transform in the third dimension.
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Section of 3 Å Electron Density map of BR
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LHC-II
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Aquaporin
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Problems

• Stage needs to be chilled with liquid helium
  (radiation damage)
• Tilt series limited to 60 - limits Z resolution
• Field emission gun necessary