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BP219 class 4 04 2011

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BioPhysics 219 Crystallography Class

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BP219 class 4 04 2011

  1. 1. B1 219 ROBERT M. STROUD<br />Understanding crystallography and structures<br />Interactive – Bring conundra –<br />Laboratory course: <br />Crystallize a protein<br />Determine structure<br />Visit ‘Advanced Light Source’ (ALS) for data <br />
  2. 2. Determining Atomic Structure<br />X-ray crystallography = optics l ~ 1.5Å (no lenses)<br />Bond lengths ~1.4Å<br />Electrons scatter X-rays; ERGO X-rays ‘see electrons’<br />Resolution –Best is l/2 Typical is 1 to 3 Å<br />Accuracy of atom center positions ±1/10 Resolution <br />2q<br />
  3. 3. Where do X-rays come from?=accelerating or decelerating electrons<br />
  4. 4. Resources:<br />http://www.msg.ucsf.edu<br />Computing<br />Calculation software-all you will ever need<br />On line course for some items:<br />http://www-structmed.cimr.cam.ac.uk/course.html<br />Dr Chris Waddling msg.ucsf.edu<br />Dr James Holton UCSF/LBNL<br />Crystallography accessible to no prior knowledge of the field or its mathematical basis. <br />The most comprehensive and concise reference <br />Rhodes' uses visual and geometric models to help readers understand the basis of x-ray crystallography.<br />http://bl831.als.lbl.gov/~jamesh/movies/<br />
  5. 5. Optical image formation, - without lenses<br />
  6. 6. Topics<br />Summmary: Resources<br />1 Crystal lattice<br /> optical analogues<br /> photons as waves/particles<br />2 Wave addition<br /> complex exponential<br />3. Argand diagram<br />Repetition ==sampling<br /> fringe function<br />5. Molecular Fourier Transform<br /> Fourier Inversion theorem<br /> sampling the transform as a product<br />6. Geometry of diffraction<br />7. The Phase problem<br /> heavy atom Multiple Isomorphous Replacement) MIR<br /> Anomalous Dispersion Multi wavelength Anomalous Diffraction MAD/SAD<br />8. Difference Maps and Errors<br />9. Structure (= phase) Refinement <br /> Thermal factors<br /> Least squares<br /> Maximum likelihood methods<br />10. Symmetry –basis, consequences<br />
  7. 7. Topics<br />11. X-ray sources: Storage rings, Free Electron Laser (FELS)<br />12. Detector systems<br />13. Errors, and BIG ERRORS! –RETRACTIONS<br />14. Sources of disorder<br />15. X-ray sources: Storage rings, Free Electron Laser (FELS)<br />
  8. 8. The UCSF beamline 8.3.1<br />UCSF mission bay<br />If automated- why are there errors?<br /> What do I trust? Examples of errors<br /> trace sequence backwards,<br />mis assignment of helices<br />etc<br />
  9. 9. NH3 sites and the role of D160 at 1.35Å Resolution<br />
  10. 10. Data/Parameter ratio is the same for all molecular sizes at the same resolution dminie. quality is the same!<br />30S<br />50S<br />
  11. 11. Macromolecular Structures<br />Growth in number and complexity of structures versus time<br />
  12. 12. The universe of protein structures: <br />Our knowledge about protein structures is increasing..<br />65,271 protein structures are deposited in PDB (2/15/2010).<br />This number is growing by > ~7000 a year <br />Growing input from Structural Genomics HT structure determination (>1000 structures a year)<br />
  13. 13. X-Ray Crystallography for Structure Determination<br />Goals: <br />1. How does it work<br />2. Understand how to judge where errors may lurk<br />3. Understand what is implied, contained in the Protein Data Bank PDB <br />http://www.pdb.org/pdb/home/home.do<br />Resolution: - suspect at resolutions >3 Å<br />R factor, and Rfree: statistical ‘holdout test’<br />Wavelength ~ atom size<br />Scattering from electrons = electron density<br />Adding atoms? how<br />Observations Intensity I(h,k,l) = F(hkl).F*(hkl)<br />Determine phases y(hkl)<br />Inverse Fourier Transform === electron density<br />Judging electron density- How to interpret?<br />Accuracy versus Reliability<br />
  14. 14. A Typical X-ray diffraction pattern<br />~100 microns<br />
  15. 15. qmax=22.5° if l=1Å<br />Resolution1.35Å<br />2qmax=45° <br />
  16. 16. The Process is re-iterative, and should converge-but only so far!<br />Crystal<br />Intensities I(h,k,l)<br />Electron density r(x,y,z)<br />calculated I(h,k,l)<br />Known:<br />Amino acid sequence<br />Ligands<br />Bond lengths angles<br />Constraints on geometry<br />Phases f(h,k,l)<br />Experimental<br />heavy atom labels<br />selenium for sulfur<br />Trial & error similar structure<br />Atom positions (x,y,z)<br />
  17. 17. Resolution dmin = l /2 sin (qmax)<br />differs from Rayleigh criterion <br />dmin = l /2 sin (qmax) is <br />the wavelength of the<br /> shortest wave used to <br />construct the density map<br />
  18. 18. The Rayleigh Criterion<br />The Rayleigh criterion is the generally accepted criterion for the minimum resolvable detail - the imaging process is said to be diffraction-limited when the first diffraction minimum of the image of one source point coincides with the maximum of another. <br />compared with<br />sin (qmax) = l /2 dmin<br />
  19. 19. How do we judge the <br />Quality of structure?<br />2. Overall quality criteria:<br /> agreement of observations<br /> with diffraction calculated <br /> from the interpreted structure.<br />3. Since we refine the structure<br />To match the Ihkloverfitting ?<br />Define Rfree for a ‘hold-out ‘ <br />set of observations.<br />4. OK? R < 20%, R free< 25%<br />5. But the experimental errors <br />in measuring Fo are ~ 3%.<br />inadequate models of solvent, <br />atom motion, anharmonicisity<br />6 Accuracy ~ 0.5*res*R<br />
  20. 20. Crystal lattice is made up of many ‘Unit Cells’<br />Unit cell dimensions are 3 distances a,b,c and angles between them a,b,g<br />h<br />A ‘section’ through<br />Scattering pattern<br />of a crystal l=0<br />Note symmetry,<br />Absences for <br />h=even k=even<br />k<br />Causes Sampling in<br /> ‘scattering space’<br />Repetition in <br />‘Real space’<br />
  21. 21. Crystal lattice is made up of many ‘Unit Cells’<br />Unit cell dimensions are 3 distances a,b,c and angles between them a,b,g<br />h<br />Vabc = |a.(bxc)|<br />Vabc = |b.(cxa)|<br />Vabc = |c.(axb)|<br />k<br />Causes Sampling in<br /> ‘scattering space’<br />Repetition in <br />‘Real space’<br />
  22. 22. Scattering<br />Adding up the scattering of Atoms:<br />‘interference’ of waves<br />Waves add out of phase<br />by 2p[extra path/l]<br />
  23. 23. In general they add up to something<br />amplitude In between -2f and +2f.<br />For n atoms<br />
  24. 24. Just 2 atoms…<br />
  25. 25. Many atoms add by the same rules.<br />Different in every direction.<br />
  26. 26. Summary of the <br />Process from beginning<br />to structure..<br />
  27. 27. Optical Equivalent: eg slide projector; leave out the lens..Optical diffraction = X-ray diffraction<br />object<br />Image of the object<br />object<br />Remove the lens= observe scattering<br />pattern<br />film<br />object<br />object<br />
  28. 28.
  29. 29. Scattering<br />Adding up the scattering of Atoms:<br />‘interference’ of waves<br />
  30. 30. vectors revisited…<br />Vectors have magnitude and direction<br />Position in a unit cell<br />r = xa + yb + zcwhere a, b, c are vectors, x,y,z are scalars 0<x<1<br />a.b = a b cos (q) projection of a onto b -called ‘dot product’<br />axb = a b sin (q) a vector perpendicular to a, and b proportional to area in magnitude -- called cross product<br />volume of unit cell = (axb).c = (bxc).a = (cxa).b = -(bxa).c = -(cxb).a <br />additivity: a + b = b + a<br />if r = xa + yb + zc and s = ha* + kb* + lc* then<br />r.s = (xa + yb + zc).(ha* + kb* + lc* ) <br />= xha.a* + xka.b* + xla.c* +yhb.a* + ykb.b* + ylb.c* + zhc.a* +zhc.a* + zkc.b* + zlc.c*<br />as we will see, the components of the reciprocal lattive can be represented in terms of a* + b* + c*, where a.a* = 1, b.b*=1, c.c*=1 <br />and a.b*=0, a.c*=0 etc..<br />r.s = (xa + yb + zc ).(ha* + kb* + lc* ) <br />= xha.a* + xka.b* + xla.c* +yhb.a* + ykb.b* + ylb.c* + zhc.a* +zhc.a* + zkc.b* + zlc.c*<br />= xh + yk + zl<br />
  31. 31. Adding up the scattering of Atoms:<br />‘interference’ of waves<br />2pr.S<br />
  32. 32. Adding up the scattering of Atoms:<br />‘interference’ of waves<br />F(S)<br />2pr.S<br />
  33. 33. Revisit..<br />
  34. 34.
  35. 35.
  36. 36.
  37. 37.
  38. 38. Revision notes on McClaurin’s<br />theorem. <br />It allows any function f(x) to be defined<br />in terms of its value at some x=a value<br />ie f(a), and derivatives of f(x) at x=a,<br />namely f’(a), f’’(a), f’’’(a) etc<br />
  39. 39.
  40. 40.
  41. 41. Extra Notes on Complex Numbers<br />(p25-28)<br />
  42. 42.
  43. 43.
  44. 44.
  45. 45. Argand Diagram.. F(S) = |F(s)| eiq<br />Intensity = |F(s)|2<br />How to represent I(s)?<br />|F(s)|2 =F(S) .F*(S) <br />proof?<br />Recall F*(S) is the complex conjugate of F(S) = |F(s)| e-iq<br />so |F(s)|2 =|F(s)|[cos(q) + isin(q)].|F(s)|[cos(q) - isin(q)]<br />=|F(s)|2 [cos2 (q) + sin2 (q)]<br />=|F(s)|2 R.T.P.<br />F(S)<br />2pr.S<br />q<br />
  46. 46. Argand Diagram.. F(S) = |F(s)| eiq<br />Intensity = |F(s)|2<br />How to represent I(s)?<br />I(s) = |F(s)|2 =F(S) .F*(S) <br />proof?<br />Where F*(S) is defined to be the ‘complex conjugate’ of F(S) = |F(s)| e-iq<br />so |F(s)|2 =|F(s)|[cos(q) + isin(q)].|F(s)|[cos(q) - isin(q)]<br />=|F(s)|2 [cos2 (q) + sin2 (q)]<br />=|F(s)|2 R.T.P.<br />F(S)<br />2pr.S<br />q<br />
  47. 47. F*(S) is the complex conjugate of F(S), = |F(s)| e-iq<br />(c+is)(c-is)=cos2q -cqisq+ cqisq+ sin2q<br />so |F(s)|2 =F(S) .F*(S) <br />-2pr.S<br />-q<br />F*(S)<br />
  48. 48.
  49. 49. Origin Position is arbitrary..<br />proof..<br />So the origin is chosen by choice of:<br />a) conventional choice in each space group<br />-eg Often on a major symmetry axis- BUT for <br />strong reasons—see ‘symmetry section’.<br />Even so there are typically 4 equivalent <br />major symmetry axes per unit cell.. <br />b) chosen when we fix the first heavy <br />metal (or Selenium) atom position, <br />-all becomes relative to that.<br />c) chosen when we place a similar molecule<br />for ‘molecular replacement’ = trial and error<br />solution assuming similarity in structure.<br />
  50. 50. Adding waves from j atoms…<br />F(S)= G(s) = <br />
  51. 51. and why do we care?<br />How much difference will <br />it make to the <br />average intensity? <br />average amplitude?<br />if we add a single Hg atom?<br />
  52. 52. The ‘Random Walk’ problem? (p33.1-33.3)<br />What is the average sum of <br />n steps in random directions?<br />(What is the average amplitude<br /><|F(s)|> from an n atom structure?)<br />-AND why do we care?!........<br />How much difference from adding a<br />mercury atom (f=80).<br />
  53. 53. The average intensity for an<br />n atom structure, each of f electrons<br />is <I>= nf2<br />The average amplitude is Square root<br />of n, times f <br />
  54. 54. and why do we care?<br />How much difference will <br />10 electrons make to the <br />average intensity? <br />average amplitude?<br />average difference in amplitude?<br />average difference in intensity?<br />if we add a single Hg atom?<br />
  55. 55. and why do we care?<br />How much difference will <br />10 electrons make to the <br />average intensity? 98,000 e2<br />average amplitude? 313<br />average difference in amplitude?<br />average difference in intensity?<br />if we add a single Hg atom?<br />
  56. 56. and why do we care?<br />How much difference will <br />10 electrons make to the <br />average intensity? 98,000 e2<br />average amplitude? 313 e<br />average difference in amplitude?<br />2.2% of each amplitude!<br />average difference in intensity?<br />98,100-98000=100 (1%)<br />if we add a single Hg atom?<br />
  57. 57. and why do we care?<br />or Hg atom n=80e<br />How much difference will <br />80 electrons make to the <br />average intensity? 98,000 e2<br />average amplitude? 313 e<br />average difference in amplitude?<br />18 % of each amplitude!<br />average difference in intensity?<br />104,400-98000=6400 (6.5%)<br />
  58. 58.
  59. 59. WILSON STATISTICS<br />What is the expected<br />intensity of scattering <br />versus the observed<br />for proteins of i atoms,?<br />on average versus resolution |s|?<br />
  60. 60.
  61. 61. Bottom Lines:<br />This plot should provide <br />The overall scale factor<br />(to intercept at y=1)<br />The overall B factor<br />In practice for proteins it has bumps<br />in it, they correspond to predominant<br />or strong repeat distances in the protein.<br />For proteins these are at 6Å (helices)<br />3Å (sheets), and 1.4Å (bonded atoms)<br />
  62. 62. Topic: Building up a Crystal<br />1 Dimension<br />Scattering from an array of points, is the same as scattering from one point,<br />SAMPLED at distances ‘inverse’ to the repeat distance in the object <br />The fringe function<br />Scattering from an array of objects, is the same as scattering from one object,<br />SAMPLED at distances ‘inverse’ to the repeat distance in the object <br />eg DNA<br />
  63. 63. Scattering from a molecule is described by<br />F(s)= Si fi e(2pir.s)<br />
  64. 64.
  65. 65.
  66. 66.
  67. 67.
  68. 68. Consequences of being a crystal?<br />Repetition = sampling of F(S)<br />34Å<br />1/34Å-1<br />
  69. 69. 2/l<br />Object repeated <br />1/l<br />
  70. 70.
  71. 71.
  72. 72. Transform of two hoizontal lines defined y= ± y1<br />F(s) = Int [x=0-infexp{2pi(xa+y1b).s} + exp{2pi(xa-y1b).s}dVr ]<br />=2 cos (2py1b).s * Int [x=0-infexp{2pi(xa).s dVr]<br />for a.s=0 the int[x=0-infexp{2pi(xa).s dVr] = total e content <br />of the line<br />for a.s≠0 int[x=0-infexp{2pi(xa).s dVr] = 0<br />hence r(r)isa line at a.s= 0 parallel to b, with <br />F(s) = 2 cos (2py1b).s -along a vertical line <br />perpendicular to the horizontal lines.<br />Transform of a bilayer..<br />b=53Å<br />
  73. 73. 53Å Repeat distance<br />40Å inter bilayer spacing<br />4.6Å <br />
  74. 74. 53Å Repeat distance<br />40Å inter bilayer spacing<br />4.6Å <br />
  75. 75. The Transform of a Molecule<br />
  76. 76.
  77. 77. How to calculate electron density?<br />Proof of the Inverse Fourier Transform:<br />
  78. 78.
  79. 79.
  80. 80. Definitions:<br />
  81. 81.
  82. 82. Build up a crystal from <br />Molecules…<br />First 1 dimension,<br />a direction<br />
  83. 83.
  84. 84. Build up a crystal from <br />Molecules…<br />First 1 dimension,<br />a direction<br />
  85. 85.
  86. 86.
  87. 87.
  88. 88.
  89. 89. a.s=h<br />b.s=k<br />
  90. 90. hkl=<br />(11,5,0)<br />Measure I(hkl)<br />F(hkl)= √I(hkl)<br />Determine f(s) = f(hkl)<br />Calculate electron density map<br />
  91. 91.
  92. 92. The density map is made up of interfering density waves through the entire <br />unit cell…<br />
  93. 93.
  94. 94.
  95. 95.
  96. 96.
  97. 97.
  98. 98. Geometry of Diffraction<br />
  99. 99.
  100. 100. A Typical X-ray diffraction pattern<br />~100 microns<br />
  101. 101.
  102. 102.
  103. 103.
  104. 104.
  105. 105. A Typical X-ray diffraction pattern<br />~100 microns<br />
  106. 106.
  107. 107.
  108. 108.
  109. 109. l=2dhklsin(q)<br />
  110. 110.
  111. 111.
  112. 112.
  113. 113. What if ‘white radiation’? representation<br />Laue picture.<br />
  114. 114. Compute a Transform of a series of parallel lines<br />
  115. 115.
  116. 116. s0sss<br />Adding up the scattering of Atoms:<br />‘interference’ of waves<br />I(S)= F(S).F(S)*<br />s1<br />S<br />s0<br />radius = 1/l<br />F(S)<br />2pr.S<br />
  117. 117.
  118. 118.
  119. 119. Sum of 7 atoms scattering<br />Result is a wave of <br />amplitude F(S)<br />phase f(s) <br />f7=8 electrons<br />2pr7.S<br />F(S)<br />2pr1.S<br />f1=6 electrons<br />
  120. 120. Sum of 7 atoms scattering<br />Result is a wave of <br />amplitude F(S)<br />phase F(S)<br />f7=8 electrons<br />2pr7.S<br />i = √(-1)<br />sin(q)<br />F(S)<br />cos(q)<br />2pr1.S<br />f1=6 electrons<br />e(iq) = cos(q) + isin(q)<br />
  121. 121. Sum of 7 atoms scattering<br />Result is a wave of <br />amplitude |F(S)|<br />phase F(S)<br />f7=8 electrons<br />2pr7.S<br />i = √(-1)<br />sin(q)<br />F(S)<br />cos(q)<br />2pr1.S<br />f1=6 electrons<br />e(iq) = cos(q) + isin(q)<br />F(S) = f1 e(2pir1.S) + f2 e(2pir2.S) +…. <br />
  122. 122. Sum of 7 atoms scattering<br />Result is a wave of <br />amplitude |F(S)|<br />phase F(S)<br />f7=8 electrons<br />2pr7.S<br />i = √(-1)<br />sin(q)<br />F(S)<br />cos(q)<br />2pr1.S<br />f1=6 electrons<br />e(iq) = cos(q) + isin(q)<br />F(S) = Sjfj e(2pirj.S)<br />
  123. 123. Sum of 7 atoms scattering<br />f7=8 electrons<br />2pr7.S<br />F(S)<br />2pr1.S<br />f1=6 electrons<br />
  124. 124. f7=8 electrons<br />2pr7.S<br />F(S)<br />2pr1.S<br />f1=6 electrons<br />
  125. 125. e(iq) = cos(q) + isin(q)<br />F(S) = Sjfj e(2pirj.S)<br />
  126. 126. The Process is re-iterative, and should converge-but only so far!<br />Crystal<br />Intensities I(h,k,l)<br />Electron density r(x,y,z)<br />Known:<br />Amino acid sequence<br />Ligands<br />Bond lengths angles<br />Constraints on geometry<br />Phases f(h,k,l)<br />Experimental<br />heavy atom labels<br />selenium for sulfur<br />Trial & error similar structure<br />Atom positions (x,y,z)<br />
  127. 127. Scattering pattern is the <br />Fourier transform of the structure <br />F(S) = Sjfj e(2pirj.S)<br />Structure is the ‘inverse’<br />Fourier transform of the <br />Scattering pattern <br />r(r) = SF(S) e(-2pir.S)<br />
  128. 128.
  129. 129.
  130. 130.
  131. 131.
  132. 132.
  133. 133.
  134. 134.
  135. 135. Rosalind Franklin and DNA<br />
  136. 136.
  137. 137.
  138. 138.
  139. 139.
  140. 140.
  141. 141. This is all there is? YES!!<br />Scattering pattern is the <br />Fourier transform of the structure <br />FT<br />F(S) = Sjfj e(2pirj.S)<br />FT-1<br />Structure is the ‘inverse’<br />Fourier transform of the <br />Scattering pattern <br />1/a<br />r(r) = SF(S) e(-2pir.S)<br />a<br />FT<br />b<br />1/b<br />FT-1<br />
  142. 142. But we observe |F(S)|2 and there are ‘Phases’<br />Scattering pattern is the <br />Fourier transform of the structure <br />FT<br />F(S) = Sjfj e(2pirj.S)<br />FT-1<br />Structure is the ‘inverse’<br />Fourier transform of the <br />Scattering pattern <br />1/a<br />r(r) = SF(S) e(-2pir.S)<br />a<br />FT<br />Where F(S) has phase<br />And amplitude <br />b<br />1/b<br />FT-1<br />
  143. 143. This is all there is?<br />Scattering pattern is the <br />Fourier transform of the structure <br />FT<br />F(S) = Sjfj e(2pirj.S)<br />S<br />FT-1<br />Structure is the ‘inverse’<br />Fourier transform of the <br />Scattering pattern <br />1/a<br />r(r) = SF(S) e(-2pir.S)<br />a<br />FT<br />b<br />F(h,k,l) = Sjfj e(2pi(hx+ky+lz))<br />S<br />1/b<br />h=15, k=3,<br />r(x,y,z) = SF(h,k,l) e(-2pir.S)<br />FT-1<br />
  144. 144. Relative Information in Intensities versus phases<br />r(r) duck<br />r(r) cat<br />|F(S)|<br />F(S)= Sjfj e(2pirj.S)<br />F(S) duck<br />F(S) cat<br />f(s)<br />|F(S)|duck<br />r(r) = SF(S) e(-2pir.S)<br />f(s) cat<br />Looks like a ….. <br />
  145. 145. Relative Information in Intensities versus phases<br />r(r) duck<br />r(r) cat<br />|F(S)|<br />F(S)= Sjfj e(2pirj.S)<br />F(S) duck<br />F(S) cat<br />f(s)<br />|F(S)|duck<br />r(r) = SF(S) e(-2pir.S)<br />f(s) cat<br />Looks like a CAT<br />PHASES DOMINATE:<br />-Incorrect phases = incorrect structure<br />-incorrect model = incorrect structure<br />-incorrect assumption = incorrect structure <br />
  146. 146. The Process is re-iterative, and should converge-but only so far!<br />Crystal<br />Intensities I(h,k,l)<br />Electron density r(x,y,z)<br />Known:<br />Amino acid sequence<br />Ligands<br />Bond lengths angles<br />Constraints on geometry<br />Phases f(h,k,l)<br />Experimental<br />heavy atom labels<br />selenium for sulfur<br />Trial & error similar structure<br />Atom positions (x,y,z)<br />
  147. 147. Consequences of being a crystal?<br />Repetition = sampling of F(S)<br />34Å<br />
  148. 148. Consequences of being a crystal?<br />Repetition = sampling of F(S)<br />34Å<br />1/34Å-1<br />
  149. 149. Consequences of being a crystal?<br />Sampling DNA = repeating of F(S)<br />1/3.4Å-1<br />3.4Å<br />34Å<br />1/34Å-1<br />
  150. 150. 2/l<br />Object repeated <br />1/l<br />
  151. 151.
  152. 152. Build a crystal<br />Object<br />Scattering<br />
  153. 153. Early Definitions:<br />
  154. 154.
  155. 155.
  156. 156.
  157. 157. a.s=h<br />b.s=k<br />
  158. 158. hkl=<br />(11,5,0)<br />Measure I(hkl)<br />F(hkl)= √I(hkl)<br />Determine f(s) = f(hkl)<br />Calculate electron density map<br />
  159. 159.
  160. 160. The density map is made up of interfering density waves through the entire <br />unit cell…<br />
  161. 161. Measure I(hkl)<br />
  162. 162. The Process is re-iterative, and should converge-but only so far!<br />Crystal<br />Intensities I(h,k,l)<br />Electron density r(x,y,z)<br />Known:<br />Amino acid sequence<br />Ligands<br />Bond lengths angles<br />Constraints on geometry<br />Phases f(h,k,l)<br />Experimental<br />heavy atom labels<br />selenium for sulfur<br />Trial & error similar structure<br />Atom positions (x,y,z)<br />
  163. 163.
  164. 164.
  165. 165.
  166. 166.
  167. 167.
  168. 168. Anomalous diffraction<br />

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