The Theory of X-ray Diffraction. Glaucius Oliva Instituto de Física de São Carlos Universidade de São Paulo - PDF

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The Theory of X-ray Diffraction Glaucius Oliva Instituto de Física de São Carlos Universidade de São Paulo Form and Function at the Roots of Biology Visible light in the description of nature Taxonomy

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The Theory of X-ray Diffraction Glaucius Oliva Instituto de Física de São Carlos Universidade de São Paulo Form and Function at the Roots of Biology Visible light in the description of nature Taxonomy Interaction of Electromagnetic Radiation with matter Interaction of Electromagnetic Radiation with matter Biological information that can de obtained: Structure Dynamics Energetics Analytical Optical Microscope With the optical microscope it was possible to elucidate the cellular structure of living organisms and to identify microbes as bacteria, fungi and parasites The transtion from Classical to Modern Physics Max Planck: the black-body radiation t t x i t x t x V x t x m = + ), ( ), ( ), ( ), ( ψ ψ ψ!! Classical Mechanics à Newton s Laws Quantum Mechanics à Schröedinger s Equation 2 2 dt dx m a m F = = Particle-Wave Duality Einstein: E = hν X-rays de Broglie: Electrons p= h λ Electron microscopes (scanning and transmission) showed the subcellular structure of cells and also the structure of viruses Modern Biology = Molecular Biology All the events associated with Life occur at the molecular level Biological Function Macromolecular Structure Electromagnetic radiation (X-rays, radio-frequencies, UV- Vis, infrared) Partícles (electrons, neutrons) Microscopies (visible, electrons, atomic force) Structural Molecular Biology Structural studies of biological moleculas through phisical and chemical techniques and molecular biology Structure, dinamycs, stability Structural Biology High resolution experimental structures: Protein Crystallography NMR Cryo EM Theoretical approaches: Molecular Modeling Molecular dynamics simulations Biophysical techniques CD, Fluorescence, FTIR, Raman, XAFS, SAXS, MS, EPR,... Superposição de ondas Waves o Structural Biology High resolution experimental structures: Protein Crystallography NMR Theoretical approaches: Molecular Modeling Molecular dynamics simulations Biophysical techniques CD, Fluorescence, FTIR, Raman, XAFS, SAXS, MS, EPR, Cryo EM,... Difraction: the lake experiment Diffraction Experiment detetor detector sample X-rays F.T.!!!! 2πi S r G S = ρ r e dv ρ ( r! ) ( ) ( ) Electron density Scatterred wave ρ(r) s! 0 s!!! s0 = s =1 ρ ( r! ) r! Δϕ = 2π λ ( Δ + Δ ) = r ( s s ) 1 2 2π! λ!! 0 Δ 1 Δ 2 df = ρ(! r)dv S =!! ( s ) λ s 0 ρ F! ( ) (!! S = ρ r ) e 2π ir S d 3 r!!!! ( ) ( ) r = F S e 2π i r S d 3 S! The Phase Problem!!!! Fourier Transform ϕ ρ F h = F h e i h ( r ) ( ) ( ) Diffraction Experiment the amplitudes! F h ( ) 2 are measured! Problem the phases ϕ are unknown ( h) Why X-rays? the boats example d How to measure the separation d between two boats in the water, given that we are far away from them? Using water waves, with wavelength λ λ d λ d In molecules, the separation between atoms is of the order of 1 to 2Å, therefore we have to use electromagnetic radiation with wavelengths of this magnitude, = X-rays Possible X-ray scattering/diffraction experiments to elucidate the structure of a protein 1) Sample is an isolated single molecule Very difficult to isolate and immobilize a single molecule resulting scattering is continuous and the amplitudes are so small that it may be extremely difficult to measure Possible X-ray scattering/diffraction experiments to elucidate the structure of a protein 2) Many molecules in solution as ther is no order, interference is globally destructive and we can only observe scattering at very low angles The result is a Small-Angle X-ray Scattering experiment (SAXS) It is very useful to get global inormation about your protein, as shape, size and oligomerization state Problem: we measure the amplitudes of the scattered waves but all phase information is lost - the Phase Problem Possible X-ray scattering/diffraction experiments to elucidate the structure of a protein 3) Molecules are in ordered in a crystal lattice Scattering is discrete and at high resolution : diffraction Scattered waves are measurable From the diffraction pattern we can determine with precision the atomic positions of all ordered atoms that constitute the crystal X-ray scattering by an atom Atomic Scattering Factors Scattering by a group of N atoms If the electron density ρ(r) is a group of N atoms, each one defined by its atomic type and position (x i, y i, z i ), with i=1, 2,,N Structure Factor Atomic Scattering Factor The Structure Factor Sum of all individual atom contributions Resultant F hkl imaginary Individual atom f j s B hkl real A hkl φ j = 2π ( hx j + ky j + lz j ) F hkl = N j= 1 f j e N 2πi( hx j + ky j + lz j ) = iφ f je j= 1 j Diffraction is all about Fourier Transforms and Convolution Theorem Convolution Multiplication Scattering by a crystal A crystal is a repetition of a basic unit (Unit Cell), repeated periodically along the directions of the cell parameters a,b, and c Therefore, the SCATTERING OF A CRYSTAL is the same scattering of a Unit Cell multiplied by the number of unit cell in the crystal and sampled at the reciprocal lattice 1912: Laue solves the quest about the nature of X- rays Friedrich W, Knipping P, von Laue M (1912). Interferenz- Erscheinungen bei Röntgenstrahlen . Sitzungsberichte der MathemaNsch- Physikalischen Classe der Königlich- Bayerischen Akademie der WissenschaSen zu München 1912: 303. years of the fundamental discovery of Molecular Sciences The structure of NaCl The interacnon of X- rays with atoms in a crystal Depending on the 2θ angle, the phase difference os the sca]ered waves show construcnve (les) or destrucnve (right) interference Bragg s Law Obtaining adequate protein single crystals is the major bottleneck in the process of elucidating a new structure. Adequate crystals means: highly ordered suitable size (used to be 0.1mm, nowadays can be microns or even sub-microns when using XFELs) Number of protein molecules in a typical protein crystal Steps in the analysis of protein crystals by X-ray diffraction Crystallization X-ray diffraction data collection Solution of the Phase Problem Multiple Isomorphous Replacement Anomalous Dispersion (SAD or MAD) Molecular Replacement Electron density map interpretation Structure refinement
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