Xraydiffraction 2007

Importance of crystallography:Recent Nobel prize winners: 

Importance of crystallography: Recent Nobel prize winners Roderick MacKinnon Nobel prize 2003 Ion channel structures Roger Kornberg Nobel prize 2006 Molecular basis of transcription

X-ray crystallography and drug design: 

X-ray crystallography and drug design Shiga-like toxin and Starfish complex Bacterial infection caused by eating improperly cooked meat. Can result in kidney disease. Collaborative effort from organic chemists, molecular biologists and crystallographers led to development of drug to combat disease. Kitov et al. (2000) Nature, 403(6770): 669-672.

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X-ray crystallography

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The phase diagram of proteins Nucleation Growth [protein] [precipitant]

Slide5: 

Some important parameters affecting protein crystallisation - protein concentration; - type of buffer & pH; - solution ionic strength & ionic species; - precipitants: type & concentration (PEG 400 to PEG 20000, MPD, alcohols, Jeffamines, salts - from NaCl via N2H4SO4 to Na,K-tartrate, etc.etc.); - additives (divalent ions, cofactors, chaotropes, cryo- protectants); - detergents (for membrane proteins); temperature; impurities.

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Diffraction: Huygens/Fresnel principle A small scattering object is a secondary source. The amount of scattering defines the ‘scattering cross section’. Single scatterer: F(s) = sr(x) e 2pix.s Where F(s): intensity scattered in direction ‘s’; r(x): density at position ‘x’ (single object: x=0) s: scattering cross section The emitted radiation is shifted in phase compared to the exciting radiation; the magnitude of the shift (p/2 to 3p/2) depends on the frequency of the incoming radiation and the resonance frequency of the scattering object.

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Diffraction: Young’s ‘two-slit’ interference Two scattering objects act as coherent secondary sources. They generate an interference pattern, which at every point is a vector summation of the scattered waves: F(s) = ∫V sr(x) e 2pix.s dx (Now x ≠ 0!)

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Bragg’s law - determines s, the direction of positive interference - p/2 p/2 q d Every red line is one wavelength (l) longer than the black line: Every photon that is diffracted by angle ‘q’ off a plane ‘B’ that is at distance ‘d’ from plane ‘A’ is in phase with every photon diffracted by angle ‘q’ off plane ‘A’ if: l/|d| = 2sinq Plane A Plane B s0 s

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Diffraction: The 1st order Born approximation Multiple scattering objects act as coherent secondary sources. Assuming there is no secondary scattering and all scattering is elastic, an interference pattern is generated, which at every point sufficiently far away is a vector summation of the scattered waves: F(s) = ∫Vsr(x) e 2pix.s dx (Note that this is a Fourier transform of the structure.)

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Scattering theory, Fourier transforms & convolutions: X-ray crystallography

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Laue’s condition for diffraction by a crystal q c For scattering of a one-dimensional crystal: (l/c) = l-1 2 sinq For scattering of a three-dimensional orthogonal crystal: ((h/a)2 + (k/b)2 +(l/c)2)1/2 = l-1 2 sinq Where: h, k, l are lattice indices a, b, c are unit cell constants

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The reciprocal lattice and the geometry of diffraction X-ray source X-ray detector Ewald sphere Reciprocal lattice

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Diffraction zones are visible as lunes in a projection of reciprocal space

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Spacing between diffraction spots (after projecting back on the Ewald sphere) defines unit cell 1/a 1/b

Data processing, solving phases, refinement: 

Data processing, solving phases, refinement Data processing: Indexing (finding the unit cell, orientation & space group) Integrating (determining the intensities of each spot) Merging (scaling data, averaging data & determining data quality) Calculating structure factor amplitudes from merged intensities Solving phases: Molecular replacement Isomorphous replacement Using the anomalous signal Refining phases by density modification: Solvent flattening & histogram matching Non-crystallographic symmetry Structure refinement: Free atom refinement Constrained/restrained atomic refinement

Integration of Bragg spots: 

Integration of Bragg spots Data are ‘integrated’ by measuring the difference in intensity between the Bragg spot and the local background Profile fitting improves signal-to-noise ratio of weak diffraction spots; 3D data collection (fine f-slicing) improves signal-to-noise ratio;

Merging data: 

Merging data Not all data are on the same scale due to fluctuations in beam intensity, diffracted crystal volume, crystal decay, absorption, polarisation, etc Minimise Rsym: Rsym(I) = SS|II(hkl) – I(hkl)| / SSIi(hkl) Overall Rsym depends on the multiplicity, so it is not a very good measure! (why?) hkl i hkl i

Calculating structure factor amplitudes from measured intensities: 

Calculating structure factor amplitudes from measured intensities Because of limited crystal & beam coherence, we measure intensities, which are the squares of the structure factor amplitudes. Because of counting statistics, sometimes we measure negative intensities Estimating structure factor amplitudes from intensities therefore requires some knowledge of the average relationship between resolution & measured intensity: the Wilson plot log(I(hkl)) vs d should be straight (beyond 3 Angstrom)

Phasing of protein crystals: 

Phasing of protein crystals Definition of structure factor: F(h) = ∫ r(x) e2pih∙x dx Definition of electron density: r(x) = ∫ F(h) e-2pix∙h dh

Solving the phase problem: 

Solving the phase problem Molecular replacement: determining the best orientation and position for a search model. Isomorphous replacement/Anomalous scattering: detecting and refining “heavy atom” sites to solve the phase problem. Finding the parameters which best represent the data.

Solving phases: using the anomalous signal: 

Solving phases: using the anomalous signal Anomalous scattering arises due to a phase-shift of X-rays diffracted by tightly bound electrons and mathematically is modeled by a complex contribution to the structure factor. Anomalous scattering causes very small changes in intensity between F+ and F-

Anomalous scattering: 

Anomalous scattering F+(h) =  (fj + i fj) exp(2i h . xj) F-(h) =  (fj - i fj) exp(2i h . xj)

Single-wavelength Anomalous Diffraction: 

Single-wavelength Anomalous Diffraction Solving structures using Friedel pairs collected at one wavelength from a crystal that contains an anomalous substructure.

Resolving the phase ambiguity: 

Resolving the phase ambiguity Collecting data at multiple wavelengths (a “MAD” experiment). Data sets are collected at wavelengths to maximize fj and fj. Synchrotrons are needed in order to change wavelengths of X-ray radiation. Use of intense X-rays also can lead to radiation damage of crystals.

Solving phases: molecular replacement: 

Solving phases: molecular replacement If the structure of a similar molecule is known, structure factors of this model may be calculated, allowing placement of the model in the unit cell by minimising as a function of the orientation of the model within the unit cell eg. R-factor = S ||Fobs| - k|Fcalc|| / S|Fobs| hkl hkl

Refining/extending phases: 

Refining/extending phases Additional, independent information can establish relationships between structure factors, and hence carries phase information; useful sources of such information: Solvent flatness; Histogram matching; Non-crystallographic symmetry

Building an initial model : 

Building an initial model Use prior information about sequence Use chemical knowledge (bond lengths, angles, etc) Use graphics Use ‘free atom refinement’ Automated model building – usually successful with resolution greater than 2.7 Angstroms and “good/sufficient” phase information.

Refining a model: 

Refining a model Optimise atomic parameters to fit with the experimental data: optimise the free R-factor R-free = S ||Ffobs| - k|Ffcalc|| / S|Ffobs| hkl hkl

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The mathematical background of testing and improving models Next lecture