Protein Strucutre Determination

1. Protein Strucutre Determination lectire 6 -- Ewald sphere, data collection, scaling

2. Scattering factor of an atom An atom is a spherically symmetrical cloud of electron density which is densest in the center. By integrating over the electron cloud, we get the Fourier transform of the atom. If we define r to be a vector relative to the center of the atom, then f(S) can be thought of as a single wave coming from the center of the atom.

3. Scattering from atoms and other centro-symmetric objects... ... has phase equal to either 0° or 180° The imaginary part cancels out because sin(–2pS•r) = –sin(2pS•(–r)) + net sine part is zero

4. Temperature factor, B The sharper the electron density distribution, the broader the scattering factor. The temperature factor, B, modifies the scattering factor by spreading out the electron density. Correction factor for atomic scattering factor: A B=0. B=10. B=20. B=30. 2sinq

5. Fourier transform as a sum over all atoms One wave for each atom. The amplitude of the scattering factor f(g) depends on how many electrons that atom has. Each f(g) is positive and real (i.e. not complex).Using Miller indeces:

7. What reflections can we see without moving the beam? crystal X-ray source beam X-ray detector

8. Behavior of S versus 2q with the beam fixed crystal 2q = 0. beam s s0 2q = 30° 2q = 90° 2q = 170° For a given direction of the incoming x-rays, the Ewald sphere is the set of possible scattering vectors S given s0. The radius of the sphere is 1/l (reciprocal units!!)

9. y x z The Ewald Sphere is defined as the sphere of radius 1/l that is the locus of possible scattering vectors (S) when the beam (s0) is fixed. s0 Crystal position defines the coordinate axes. This sphere is in scattering space (reciprocal space). If a scattered wave has S on the Ewald sphere, it is visible on the film/detector.

10. y y x x z z Moving the beam moves the Ewald Sphere For a given direction of the incoming x-rays, the set of possible scattering vectors S is the surface of a sphere of radius 1/l, passing through the crystallographic origin. s0 Keeping the crystal fixed, we rotate the X-ray source. The Ewald Sphere moves in parallel with the X-ray source. The new set of S vectors describe the phase vs. direction of scatter for that position of the source. s0

11. s0 Moving the beam. Crystal fixed. s0 s0 By moving the X-ray source relative to the crystal, we can sample every possible S

12. limit = 2/l(q=180°) The visible part of reciprocal space The set of all vectors S (red), given all possible directions of the beam (black arrows), is called reciprocal space. Remember: in this view, the crystal is fixed (center of image, where the X-rays are pointed). In real life, we find it easier to move the crystal, not the source. It doesn’t matter which one you move, the crystal or the source. The results are the same.

13. Determining the space group (1) Find the high symmetry axes using precession photos. (2) Are the axes... all 90° apart?: orthorhombic, tetragonal, or cubic 90, 90 and 120°?: trigonal or hexagonal 90, 90 and ≠90°?: monoclinic none-of-the-above?: triclinic, P1

14. Precession camera geometry Seeing one plane of the reciprocal lattice at a time By using a screen, all but one lattice plane is masked out. If the angle of the beam with the a axis is w, then the correct angle setting for the screen is . a s0  Ewald sphere This method NOT used for data collection. Too slow.

15. Precession photography A precession photograph contains one complete reciprocal plane on one film. a  for example:b*c* plane at h=0. path of beam relative to crystal

16. Precession photograph note systematic absences, relative space of spots, angle between axes. 0-level. What crystal form is this?

17. p p p q p p p q p p p q q p p q p q p p q q Determining the space group (3)If trigonal of hexagonal, take h01 precession photo. 6-fold symmetry: P6n 6m symmetry: P6n2n2 3-fold symmetry: P3, P31, P32 3m symmetry: P3n21 or P3n12 If orthorhombic or tetragonal. Are cell dimensions.. a ≠ b≠c? : orthorhombic a=b≠c? : tetragonal a=b=c? : cubic p p p q p

18. Systematic absences (4)To get screw axis “n”, look at systematic absences. examples: orthorhombic + odd h reflections missing in h00 line --> 21 trigonal + only l=3n reflections present in 00l line--> 31 or 32 hexagonal + only l=2n reflections present in 00l line-->63 tetragonal + only l=4n reflections present in 00l line --> 41 or 43 Enantiomeric space groups (P41 and P43) can’t be distinguished until the phases have been solved.

19. Data collection = Measuring the intensity (amplitude squared) of each reflection. Output of data collection, thousands of reflections, each with 5 parameters: h k l F sigma

20. Diffractometers yesterday and today Counter moves in 2. Crystal moves in 3 angles ,, and  Single photon counter (photo multiplier tube)

21. Collecting data on photographic film still widely used! oscillation image Raw images are scanned into digital images. Each image has three angle associated with it (, and ). A series of films, each with a different  angle, are collected and digitized.

22. Today: Image plates Image plates are ultra-sensitive, reusable films. Data collection is done the same way as for photographic film.

23. Area detectors Position sensitive X-ray detectors give a 3D image of each spot, where fild or image plates give 2D images.

24. s0 Moving the crystal is like moving the Ewald sphere s0 s0 By moving the X-ray source relative to the crystal, we can sample every possible S

25. data collection frame 1 10 a* 5 b* h=0 4 visible part of transform k=0 Ewald sphere -4

26. data collection frame 4 10 a* 5 As the crystal is rotates, the reciprocal lattice rotates. b* h=0 4 k=0 Ewald sphere -4

27. Try this... How far can you rotate the crystal before two spots fall on top of each other on the film? • Draw an Ewald sphere and a lattice. • Put your pencil on a hkl position that is on the sphere. • Rotate sphere and pencil together until pencil hits the next reciprocal lattice plane. • How far did you rotate? 10 a* 5 b* h=0 4 k=0 Ewald sphere -4 For this exercise to work in reality, the lattice and sphere must be drawn to scale. Ewald sphere radius = 1/

28. X-ray diffractometer with area detector Axis of two-theta arm The detector (or film) sits on a “Two-theta arm” that can swing out, away from the beam to collect high-resolution data.

29. Schematic diffractometer beamstop crystal X-ray source beam 2 detector setting low-resolution limit goniostat X-ray detector high-resolution limit

30. The film is a window on the Ewald sphere All lattice points that fall on the Ewald sphere are “on” (meaning photons are being scattered that direction), but most of those reflections are “off camera” (they end up on the laboratory wall). The detector or film collects a window of the Ewald sphere.

31. s0 As the crystal moves, the window “moves” ...relative to the reciprocal lattice. path of detector through reciprocal lattice

32. s0 One sweep through reciprocal space collects a donut-shaped volume of data This is the volume of reciprocal space that has been “seen” by the detector. s0  axis = machine axis, center of “donut” Red circle shows the low-resolution limit for this detector position. A low-angle setting of the detector would be necessary to collect the low resolution data.

33. Multiplesweeps are usually necessary Gray area = volume of reciprocol space that has been seen by the Ewald sphere (and thus, the detector). Intersecting “donuts” of data add up to the whole Unique Set or more. Most reflections have multiple copies. “Completeness” = what fraction of the unique set has been collected, at least once. (Number of reflections depends on resolution cutoff.)

34. Data reduction hkl F s 200 99.0 0.2210 65.1 0.3201 78.5 0.2220 6.3 0.1221 19.9 0.2222 88.1 0.2 ... • indexing • background estimation • integration of spots • merging of partials • scaling • merging of syms raw images reflection data “Structure factors”

35. Data reduction • indexing = finding the location of each reciprocol lattice point HKL • background estimation = like subtracting the baseline, in 2D • integration of spots = intensity is proportional to F2 • merging of partials = One reflection may be split between two films. • Scaling = If there is significant decay, then data is scaled in blocks of time. • Averaging of syms = Symmetry-related reflections are averaged

36. Indexing the data A reciprocal lattice is initialized using the known cell dimensions. Spots are predicted to be at the places where the lattice intersects the Ewald sphere. A systematic search (rotation of the lattice) is done until the predictions match the observations. Small refinements in the beam position might be required. When the solution is found, every spot has an index (hkl).

37. Calibrating the film, or detector. For photographic film, or any type of X-ray counter, a calibration curve has been pre-calculated. The pixels are counted, multiplied by “I” from the calibration curve, to get I(hkl) for each spot. absorbance I

38. Merging partials If films were switched while a spot was on the Ewald sphere, both copies (“partials” are summed together to get I(hkl). =153.0° =153.5° First half of spot hits the Ewald sphere. Other half of spot passes through.

39. How can ‘partials’ exist? Reflections are points in reciprocal space, right? Wrong. Reflections have size and shape in all three S directions (a*, b*, c*), because the crystal lattice is not perfect and infinite. Reflections have additional shape in the laboratory dimensions (x,y on the film or detector), because the beam is not infinitely small and the crystal is not infinitely small. beam The spot shape is partly the shape of the intersection of the beam and crystal.

40. Spot profiles in 3D-1 Profile of the average spot, summed over all spots with similar x,y,2, Intensity of each spot (blue) is the summed only within the spot profile limits (grey). This prevents counting spurious data like this.

41. Scaling within a dataset • Reflections may have errors in amplitude within a dataset because: • Xray intensity varied. • Film/detector sensitivity varied. • Crystal orientation/ cross section varied with w. • Crystal decayed over time. • Exposure time varied. • Background radiation varied. • Scaling assumes: • (1) Symmetry-related reflections have the same amplitude • (2) Reflections that were collected together are scaled together (i.e. applied the same scale factor). • Quality of the data set = • Should be < 2% a sym op

42. Example Structure Factor file data_r1pkqsf #------------------------------------------------------------ _audit.revision_id 1_0 _audit.creation_date 2003-07-15 _audit.update_record 'Initial release' # loop_ _refln.wavelength_id _refln.crystal_id _refln.index_h _refln.index_k _refln.index_l _refln.F_meas_au _refln.F_meas_sigma_au _refln.status 1 1 -39 0 26 70.300 34.700 0 1 1 -39 0 27 158.300 25.740 0 1 1 -39 1 1 156.000 15.800 0 1 1 -39 1 25 54.100 23.690 0 1 1 -39 1 26 201.400 11.450 0 1 1 -39 2 25 151.900 11.970 0 1 1 -39 3 22 202.800 22.730 0 1 1 -38 0 26 75.900 37.400 0 Structure Factors are deposited in the PDB (www.rcsb.org) along with the atomic coordinates.