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## EM Diffraction

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**EM Diffraction**As applied to proteins**Diffraction from a grating**Molecular Expressions**Bragg’s Law**• N = 2d∙sin(Ө)/λ • Derivation: http://www.eserc.stonybrook.edu/ProjectJava/Bragg/ • Applies to electron diffraction from crystals • Therefore, the diffraction of electrons from a crystal is quantized**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**Fourier Transform**We can model any function by adding together waves that are integral multiples in frequency F(x) = C0 + ∑Cncos(2πx/(λ/n) +αn).**Scattering of waves by an object gives rise to a Fourier**transform • http://micro.magnet.fsu.edu/primer/java/interference/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.**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**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?**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**Examples**• 2-D crystals • Purple membrane protein (bacteriorhodopsin) from Halobacter halobium. • Light harvesting complex • http://blanco.biomol.uci.edu/Membrane_Proteins_xtal.html • Microcrystals • Prion protein • Catalase**Catalase crystals**Brink Lab, Baylor**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)**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.**Problems**• Stage needs to be chilled with liquid helium (radiation damage) • Tilt series limited to 60° - limits Z resolution • Field emission gun necessary