![slide 7
A. Diffraction Data (Fobs, but no phases α)
B. Determine Phases (Fobs collected, but no phases)
Molecular Replacement (MR, requires coordinate file of related structure)
Heavy‐atom methods (SIR, MIR)
Anomalous dispersion (SAD, MAD)
C. Calculate an electron density map, based on phases above .
Build a protein model (atom type, x, y, z, B) into the map.
[electron density provides a bootstrap method for building partial models ]
D. Refine protein model.
Optimize the positions of the atoms ( x, y, z, and B) in the unit cell.
This is performed by mimimizing the difference between Fobs and Fcalc. The values
of Fcalc are derived from the atom positions of the protein model in the unit cell.
The Structure Factor equation and the Electron Density equation play important roles in all
of the steps of structure determination by X‐ray crystallography.
Steps in X‐ray structure determination](https://image.slidesharecdn.com/lecture4-230203183426-aeab95c5/85/lecture4-pdf-7-320.jpg)
lecture4.pdf
- 1. Lecture 3 Structure Factors and electron density
- 2. Review ‐ identify miller planes Green planes ‐ Blue planes ‐ Magenta planes Green arrow Magenta arrow Brown arrow
- 3. slide 3 1. Arrows on the figure correspond to dhkl 2. Arrows also are used to represent the reciprocal scattering vector s. The direction of s is identical to dhkl with length 1/dhkl. k b a ‐b ‐a 1,1,0 1,0,0 1,0,0
- 4. What defines + or ‐ miller planes? (using the 1, 1, 0 plane, +/‐ definitions define the scattering vector direction)
- 5. b a ‐b ‐a 1,‐1,0 1,1,0 ‐1,1,0 Relationship between miller planes and the reciprocal axis ‐1,‐1,0 Reciprocal Lattice Crystal lattice 1,1,0 ‐1,‐1,0 Where is the x‐ray beam for this experiment? Where is the c‐axis in this figure? Index remaining spots in the diffraction image on the right
- 6. File of structure factor amplitudes experimental data (Fobs) 6 slide PDB file atm res. Chn res# x y z occ. B Fcalc(calculated from the atoms in the pdf file) Each Fcalc is a summation of all atoms in pdb. Structure Factor Equation Crystallographic R‐Factor
- 7. slide 7 A. Diffraction Data (Fobs, but no phases α) B. Determine Phases (Fobs collected, but no phases) Molecular Replacement (MR, requires coordinate file of related structure) Heavy‐atom methods (SIR, MIR) Anomalous dispersion (SAD, MAD) C. Calculate an electron density map, based on phases above . Build a protein model (atom type, x, y, z, B) into the map. [electron density provides a bootstrap method for building partial models ] D. Refine protein model. Optimize the positions of the atoms ( x, y, z, and B) in the unit cell. This is performed by mimimizing the difference between Fobs and Fcalc. The values of Fcalc are derived from the atom positions of the protein model in the unit cell. The Structure Factor equation and the Electron Density equation play important roles in all of the steps of structure determination by X‐ray crystallography. Steps in X‐ray structure determination
- 8. Electron density equation (Density , ρ (e‐/Ǻ3) at position x,y,z in the unit cell) |F(hkl)| structure factor amplitude (scalar) measured in the diffraction experiment. α (phase) in radians. The “phase” of a hkl reflection. Lost during the diffraction experiment Thus, F(hkl) is a vector with amplitude |F(hkl)| and phase α(hkl). h k l are the miller indices!!! x y z fractional coordinates of the unit cell. Sampling from 0‐1 in each direction. electron density (ρ(xyz)) is calculated by summing all structure factors at each xyz (sampling grid) position in the unit cell. Thus, ALL reflections contribute to the electron density at each place in the cell. F hkl, α FT
- 9. h F α The electron density in the unit cells of a crystal is a periodic wave function. It may be decomposed into its component waves . Each component wave is described by a structure factor amplitude (|F|)and Phase (α). In an x‐ray diffraction experiment, |F| is measured, but α is lost. To derive the lost phases and ultimately electron density, must identify atom positions (black dots), which can be used to calculate the structure factors. (e.g. |F| and α) 108° 144° 144° 162° 234° Structure factors Highly schematic figure!!
- 10. The Isotropic Atomic Scattering Factor B=8π2<u>2 1. X‐ray scattering is proportional to the # of electrons in the atom. a. Hg is called a “heavy atom”. Lots of electrons! 2. Scattering falls off with resolution (e.g. resolution dependent). 3. Scattering falls off with increasing movement of the atom. The B‐factor (refined for each atom , found in pdb files) models the movement of atoms in crystals by the equation: atomic number # electrons Where u is the mean atomic displacement of an atom. Thus, a B‐factor of 10 corresponds to a mean atomic displacement of 0.36Å
- 11. Isotropic scattering factors in electrons found in International tables Vol. 4. (Table 2.2) The Isotropic Atomic Scattering Factor Scattering factor (f) plot for oxygen vs. resolution at B=0Å2 and B=10Å2 B=0 Å2 resolution fall off only B=10 Å2 motion and resolution fall off only
- 12. STRUCTURE FACTOR summation of scattering vectors of individual atoms in unit cell h k l 1 2 0 F is the summation of scattering vectors for each atom in the unitcell. The magnitude of f is dependent on the scattering atom. The orientation of the vector is dependent on the x, y, z position of the atom in the unit cell. F
- 13. f1, f2, and f3 are the scattering contribution of 3 different atoms in the unit cell of a crystal. All atoms contribute to the measured intensity. The phase (α) requires knowledge of the positions of atoms in the unitcell. F (hkl) corresponds to a waves scattered from all atoms in the crystal unit cell for a given set of miller planes h, k , l. STRUCTURE FACTOR EQUATION Summation over all atoms (j) in the unit cell for each h, k, l
- 14. STRUCTURE FACTOR EQUATION (algebraic calculation) It easy to break the structure factor equation down into scattering components along real (A, where A=f1cos α ) and imaginary axes (B, where B = B=f1sinα ). The summation of all A and B components (for each atom, see equations below) results in the structure factor amplitue |F|, which is equal to the SQRT (A2+B2) with resultant phase (α) = Atan2(B,A), or tan‐1 B/A. Normally the phase is converted from radians to degrees. A B
- 15. STRUCTURE FACTOR Diagrams – Graphical representation of structure factors. A=1, B=0 A=0.71, B=0.71 A=|F(hkl)|cosα B=|F(hkl)|sinα α=tan‐1 (B/A) |Fhkl| = SQRT(A2 + B2) I = A2 + B2 If |F| = 1 A B α
- 16. hkl = 200 F=12 α = 0° hkl = 200 F=12 α = 72° Note relation of e‐ density wave (green) relative to miller planes (grey) Phase diagram (left) and e‐ density and miller planes (right)
- 17. Friedels law F(hkl) = F*(hkl) Thus, F(hkl) and F(‐h‐k‐l) have equal amplitudes but opposite phases (no anomalous scattering) hkl planes 0.65 0.4 0.2 0.1 h reflection ‐h reflection a 36° 72° 144° 234° ‐36° ‐72° F(h) F(‐h) f 2*pi h x1 2*pi*(h*x1) f*cos(2*pi*(hx)) A f*sin (2*pi*(hx)) B F=SQRT(A+B) phase 6.00 6.28 1 0.10 0.63 4.85 4.85 3.53 3.53 6.00 36 6.00 6.28 ‐1 0.10 ‐0.63 4.85 4.85 ‐3.53 ‐3.53 6.00 ‐36 Assume atoms are carbons ( 6 electrons)
- 18. How do we get phases?
- 19. Experimental Data h, k , l, I and sigma I (Native Dataset used for refinement and deposited with the final coordinates in the pdb file.) Experimental Data for Phasing (generally not deposited). 1. Prior protein model for molecular replacement phasing x,y,z coordinates of model 2. Heavy‐atom experimental data (isomorphous replacement) h, k , l, I, sig.I Intensity differences between native and HA data used to find x,y,z positions of heavy‐atoms. heavy atom phases used as initial estimates of protein phases 3. Anomalous diffraction data h, k, l, I+, sig.I+ , ‐I, sig.‐I Intensity differences between I+ and I‐ (Friedel pairs) used to find x,y,z positions of anomalous scattering atoms. anomalous scatter phases used as initial estimates of protein phases.
- 20. Possible problem: No appropriate model structure or structural model too different Possible problem: heavy atoms (e.g. Hg) don’t bind OR don’t bind specifically. Heavy‐atoms change the unit cell – non‐ isomorphous, which prevents finding the positions. Generally, requires labeling of the protein (SeMet). Great if molecule can be produced in e. coli. Need anomolous scatter e.g Fe protein Positive….100% incorporation of SeMet
- 21. Collect an X‐ray diffraction data set. Calculate structure factor amplitudes for Known protein structure (Pmodel) However, don’t know where this model is in the unit cell… Molecular Replacement 1. Finds the correct rotation of Pmodel relative to Pcrystal (Pc). 2. Finds the correct translation of Pmodel in Pcrystal. 3. This is effective when Pmodel is structurally similar to Pcrystal. The breakdown occurs somewhere around ~35% or less sequence identity. model Mo d c c c c c c c c #1 #2
- 23. Fph Fp Fh FPH = FP + FH For SIR or MIR Phasing Method Dataset of Native crystal Dataset of Native Crystals derivatized With heavy‐atom Hg, or U, or Pb, or Pt (look at scattering factor table) Measure the intensity (amplitude) differences between Heavy‐atom dataset (Fph) and Native dataset (Fp). Subtraction of Fph – Fp is an estimate of the contribution of the heavy‐atom to the diffraction pattern. Find the phase of the heavy atom (Fh). Need to find the xyz position of the atom: Patterson Map. Estimate of Heavy‐atom contribution
- 24. 1. If you define the location of the heavy atom, you can calculate the vector FH. With FH you can define the phase of the protein (αP). FPH = FP + FH Remember these are each VECTORS….we only have amplitudes
- 25. |Fp| |Fph| In SIR, we have 2 circles corresponding to Fp and Fph which are native and derivative (heavy atom) data sets. The following diagram would be constructed for each hkl reflection in the dataset. We want to know the phase (α) of Fp. Because Fp and Fph are vector summations of atoms (see earlier slides) we can write the following vector equation. α |Fp| |Fph| Fh Fp + Fh = Fph This says if we know the vector for Fh, we Can solve for Fp
- 26. positional From FH, get αP Phaser……………….automate all steps + refinement
- 27. Initial Phase estimates Build initial model / refine More information in the model than in the starting map. Calc. new map.
- 28. Crystallographer observes incorrect fits between improved map and model. Refinement program cannot “fix” all problems . Manual intervention through molecular graphics Correct model, refine model, recalculate electron density map. R‐factor should be lower, improved map quality, protein geometry improved.
- 30. Anomalous scattering factors are determined experimentally by monitoring Fluorescence as A function of energy (wavelength). Performed at a synchrotron beamlines. Requires a Tunable x‐ray source.
- 31. Selenium, atomic # = 34, Sulfur mimick Edge keV Å K 12.6578 0.9795 L-I 1.6539 7.4965 L-II 1.4762 8.3989 L-III 1.4358 8.6352
- 32. 12595.00 ‐4.793897 0.5074802 12600.00 ‐4.879107 0.5071128 12605.00 ‐4.972221 0.5067459 12610.00 ‐5.074862 0.5063793 12615.00 ‐5.189123 0.5060131 12620.00 ‐5.317906 0.5056473 12625.00 ‐5.465356 0.5052819 12630.00 ‐5.637684 0.5049169 12635.00 ‐5.844824 0.5045523 12640.00 ‐6.104173 0.5041881 12645.00 ‐6.450631 0.5038243 12650.00 ‐6.972532 0.5034610 12655.00 ‐8.055866 0.5030980 12660.00 ‐8.319967 3.846461 peak 12665.00 ‐7.052211 3.843189 12670.00 ‐6.486044 3.839924 12675.00 ‐6.116062 3.836665 12680.00 ‐5.840367 3.833412 Energy Kev f’ f’’ 12900.00 ‐3.196559 3.696152 remote Energy Kev f’ f’’
- 33. f”, varies strongly near the absorption edge, becoming most positive at energies > E. f” is the component of scattered radiation 90° out of phase with the normally scattered component fo fo f” f” fo f” fo f” fo f” fo E Se 2 2 " v v g f B when >B Else, 0 Peak High energy remote Low energy remote Scattering factor f” is maximal at the peak, which corresponds to maximal Fph+ and Fph‐ intensities observed in the diffraction data of Friedel pairs. SAD data is collected at energies/λ just above the peak.
- 34. Friedels law F(hkl) = F*(hkl) Thus, F(hkl) and F(‐h‐k‐l) have equal amplitudes but opposite phases (no anomalous scattering) hkl planes 0.65 0.4 0.2 0.1 h reflection ‐h reflection a 36° 72° 144° 234° ‐36° ‐72° F(h) F(‐h) f 2*pi h x1 2*pi*(h*x1) f*cos(2*pi*(hx)) A f*sin (2*pi*(hx)) B F=SQRT(A+B) phase 6.00 6.28 1 0.10 0.63 4.85 4.85 3.53 3.53 6.00 36 6.00 6.28 ‐1 0.10 ‐0.63 4.85 4.85 ‐3.53 ‐3.53 6.00 ‐36 Assume atoms are carbons ( 6 electrons)
- 35. Friedel pairs At the appropriate X‐ray energy, the white atom Scatters anomalously and Friedel’s law breaks down. This is because the phase of the anomalous scattering atom is advanced by 90 degrees relative to the other atoms (Δf”). This results in different intensities at I h,k,l and I –h,‐k, ‐l, which are measured in the x‐ray diffraction experiment. As a result, FPH+ and FPH‐ are different. These differences can be used to locate the x, y, z position of the selenium atom, providing phase estimates for the entire protein structure. h k l plane ‐h ‐k ‐l plane Breakdown in Friedel’s law FPH (hkl)≠FPH (‐h‐k‐l) when an anomalous scattering atom is present Δf” Δf” 0.9
- 36. MAD data collection statistics