1. The document provides resources and information for determining protein crystal structures using x-ray crystallography. It discusses topics like crystal lattices, diffraction, the phase problem, and structure refinement. 2. Key resources mentioned include the Advanced Light Source for collecting diffraction data, and computing software for analyzing structures. 3. The goals of x-ray crystallography are outlined as determining how the technique works, understanding potential sources of error, and what information is contained in the Protein Data Bank structure database.

1. B1 219 ROBERT M. STROUDUnderstanding crystallography and structuresInteractive – Bring conundra –Laboratory course: Crystallize a proteinDetermine structureVisit ‘Advanced Light Source’ (ALS) for data

2. Determining Atomic StructureX-ray crystallography = optics l ~ 1.5Å (no lenses)Bond lengths ~1.4ÅElectrons scatter X-rays; ERGO X-rays ‘see electrons’Resolution –Best is l/2 Typical is 1 to 3 ÅAccuracy of atom center positions ±1/10 Resolution 2q

3. Where do X-rays come from?=accelerating or decelerating electrons

4. Resources:http://www.msg.ucsf.eduComputingCalculation software-all you will ever needOn line course for some items:http://www-structmed.cimr.cam.ac.uk/course.htmlDr Chris Waddling msg.ucsf.eduDr James Holton UCSF/LBNLCrystallography accessible to no prior knowledge of the field or its mathematical basis. The most comprehensive and concise reference Rhodes' uses visual and geometric models to help readers understand the basis of x-ray crystallography.http://bl831.als.lbl.gov/~jamesh/movies/

5. Optical image formation, - without lenses

6. TopicsSummmary: Resources1 Crystal lattice optical analogues photons as waves/particles2 Wave addition complex exponential3. Argand diagramRepetition ==sampling fringe function5. Molecular Fourier Transform Fourier Inversion theorem sampling the transform as a product6. Geometry of diffraction7. The Phase problem heavy atom Multiple Isomorphous Replacement) MIR Anomalous Dispersion Multi wavelength Anomalous Diffraction MAD/SAD8. Difference Maps and Errors9. Structure (= phase) Refinement Thermal factors Least squares Maximum likelihood methods10. Symmetry –basis, consequences

7. Topics11. X-ray sources: Storage rings, Free Electron Laser (FELS)12. Detector systems13. Errors, and BIG ERRORS! –RETRACTIONS14. Sources of disorder15. X-ray sources: Storage rings, Free Electron Laser (FELS)

8. The UCSF beamline 8.3.1UCSF mission bayIf automated- why are there errors? What do I trust? Examples of errors trace sequence backwards,mis assignment of helicesetc

9. NH3 sites and the role of D160 at 1.35Å Resolution

10. Data/Parameter ratio is the same for all molecular sizes at the same resolution dminie. quality is the same!30S50S

11. Macromolecular StructuresGrowth in number and complexity of structures versus time

12. The universe of protein structures: Our knowledge about protein structures is increasing..65,271 protein structures are deposited in PDB (2/15/2010).This number is growing by > ~7000 a year Growing input from Structural Genomics HT structure determination (>1000 structures a year)

13. X-Ray Crystallography for Structure DeterminationGoals: 1. How does it work2. Understand how to judge where errors may lurk3. Understand what is implied, contained in the Protein Data Bank PDB http://www.pdb.org/pdb/home/home.doResolution: - suspect at resolutions >3 ÅR factor, and Rfree: statistical ‘holdout test’Wavelength ~ atom sizeScattering from electrons = electron densityAdding atoms? howObservations Intensity I(h,k,l) = F(hkl).F*(hkl)Determine phases y(hkl)Inverse Fourier Transform === electron densityJudging electron density- How to interpret?Accuracy versus Reliability

14. A Typical X-ray diffraction pattern~100 microns

15. qmax=22.5° if l=1ÅResolution1.35Å2qmax=45°

16. The Process is re-iterative, and should converge-but only so far!CrystalIntensities I(h,k,l)Electron density r(x,y,z)calculated I(h,k,l)Known:Amino acid sequenceLigandsBond lengths anglesConstraints on geometryPhases f(h,k,l)Experimentalheavy atom labelsselenium for sulfurTrial & error similar structureAtom positions (x,y,z)

17. Resolution dmin = l /2 sin (qmax)differs from Rayleigh criterion dmin = l /2 sin (qmax) is the wavelength of the shortest wave used to construct the density map

18. The Rayleigh CriterionThe 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. compared withsin (qmax) = l /2 dmin

19. How do we judge the Quality of structure?2. Overall quality criteria: agreement of observations with diffraction calculated from the interpreted structure.3. Since we refine the structureTo match the Ihkloverfitting ?Define Rfree for a ‘hold-out ‘ set of observations.4. OK? R < 20%, R free< 25%5. But the experimental errors in measuring Fo are ~ 3%.inadequate models of solvent, atom motion, anharmonicisity6 Accuracy ~ 0.5*res*R

20. Crystal lattice is made up of many ‘Unit Cells’Unit cell dimensions are 3 distances a,b,c and angles between them a,b,ghA ‘section’ throughScattering patternof a crystal l=0Note symmetry,Absences for h=even k=evenkCauses Sampling in ‘scattering space’Repetition in ‘Real space’

21. Crystal lattice is made up of many ‘Unit Cells’Unit cell dimensions are 3 distances a,b,c and angles between them a,b,ghVabc = |a.(bxc)|Vabc = |b.(cxa)|Vabc = |c.(axb)|kCauses Sampling in ‘scattering space’Repetition in ‘Real space’

22. ScatteringAdding up the scattering of Atoms:‘interference’ of wavesWaves add out of phaseby 2p[extra path/l]

23. In general they add up to somethingamplitude In between -2f and +2f.For n atoms

24. Just 2 atoms…

25. Many atoms add by the same rules.Different in every direction.

26. Summary of the Process from beginningto structure..

27. Optical Equivalent: eg slide projector; leave out the lens..Optical diffraction = X-ray diffractionobjectImage of the objectobjectRemove the lens= observe scatteringpatternfilmobjectobject

29. ScatteringAdding up the scattering of Atoms:‘interference’ of waves

30. vectors revisited…Vectors have magnitude and directionPosition in a unit cellr = xa + yb + zcwhere a, b, c are vectors, x,y,z are scalars 0<x<1a.b = a b cos (q) projection of a onto b -called ‘dot product’axb = a b sin (q) a vector perpendicular to a, and b proportional to area in magnitude -- called cross productvolume of unit cell = (axb).c = (bxc).a = (cxa).b = -(bxa).c = -(cxb).a additivity: a + b = b + aif r = xa + yb + zc and s = ha* + kb* + lc* thenr.s = (xa + yb + zc).(ha* + kb* + lc* ) = xha.a* + xka.b* + xla.c* +yhb.a* + ykb.b* + ylb.c* + zhc.a* +zhc.a* + zkc.b* + zlc.c*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 and a.b*=0, a.c*=0 etc..r.s = (xa + yb + zc ).(ha* + kb* + lc* ) = xha.a* + xka.b* + xla.c* +yhb.a* + ykb.b* + ylb.c* + zhc.a* +zhc.a* + zkc.b* + zlc.c*= xh + yk + zl

31. Adding up the scattering of Atoms:‘interference’ of waves2pr.S

32. Adding up the scattering of Atoms:‘interference’ of wavesF(S)2pr.S

33. Revisit..

38. Revision notes on McClaurin’stheorem. It allows any function f(x) to be definedin terms of its value at some x=a valueie f(a), and derivatives of f(x) at x=a,namely f’(a), f’’(a), f’’’(a) etc

41. Extra Notes on Complex Numbers(p25-28)

45. Argand Diagram.. F(S) = |F(s)| eiqIntensity = |F(s)|2How to represent I(s)?|F(s)|2 =F(S) .F*(S) proof?Recall F*(S) is the complex conjugate of F(S) = |F(s)| e-iqso |F(s)|2 =|F(s)|[cos(q) + isin(q)].|F(s)|[cos(q) - isin(q)]=|F(s)|2 [cos2 (q) + sin2 (q)]=|F(s)|2 R.T.P.F(S)2pr.Sq

46. Argand Diagram.. F(S) = |F(s)| eiqIntensity = |F(s)|2How to represent I(s)?I(s) = |F(s)|2 =F(S) .F*(S) proof?Where F*(S) is defined to be the ‘complex conjugate’ of F(S) = |F(s)| e-iqso |F(s)|2 =|F(s)|[cos(q) + isin(q)].|F(s)|[cos(q) - isin(q)]=|F(s)|2 [cos2 (q) + sin2 (q)]=|F(s)|2 R.T.P.F(S)2pr.Sq

47. F*(S) is the complex conjugate of F(S), = |F(s)| e-iq(c+is)(c-is)=cos2q -cqisq+ cqisq+ sin2qso |F(s)|2 =F(S) .F*(S) -2pr.S-qF*(S)

49. Origin Position is arbitrary..proof..So the origin is chosen by choice of:a) conventional choice in each space group-eg Often on a major symmetry axis- BUT for strong reasons—see ‘symmetry section’.Even so there are typically 4 equivalent major symmetry axes per unit cell.. b) chosen when we fix the first heavy metal (or Selenium) atom position, -all becomes relative to that.c) chosen when we place a similar moleculefor ‘molecular replacement’ = trial and errorsolution assuming similarity in structure.

50. Adding waves from j atoms…F(S)= G(s) =

51. and why do we care?How much difference will it make to the average intensity? average amplitude?if we add a single Hg atom?

52. The ‘Random Walk’ problem? (p33.1-33.3)What is the average sum of n steps in random directions?(What is the average amplitude<|F(s)|> from an n atom structure?)-AND why do we care?!........How much difference from adding amercury atom (f=80).

53. The average intensity for ann atom structure, each of f electronsis <I>= nf2The average amplitude is Square rootof n, times f

54. and why do we care?How much difference will 10 electrons make to the average intensity? average amplitude?average difference in amplitude?average difference in intensity?if we add a single Hg atom?

55. and why do we care?How much difference will 10 electrons make to the average intensity? 98,000 e2average amplitude? 313average difference in amplitude?average difference in intensity?if we add a single Hg atom?

56. and why do we care?How much difference will 10 electrons make to the average intensity? 98,000 e2average amplitude? 313 eaverage difference in amplitude?2.2% of each amplitude!average difference in intensity?98,100-98000=100 (1%)if we add a single Hg atom?

57. and why do we care?or Hg atom n=80eHow much difference will 80 electrons make to the average intensity? 98,000 e2average amplitude? 313 eaverage difference in amplitude?18 % of each amplitude!average difference in intensity?104,400-98000=6400 (6.5%)

59. WILSON STATISTICSWhat is the expectedintensity of scattering versus the observedfor proteins of i atoms,?on average versus resolution |s|?

61. Bottom Lines:This plot should provide The overall scale factor(to intercept at y=1)The overall B factorIn practice for proteins it has bumpsin it, they correspond to predominantor strong repeat distances in the protein.For proteins these are at 6Å (helices)3Å (sheets), and 1.4Å (bonded atoms)

62. Topic: Building up a Crystal1 DimensionScattering from an array of points, is the same as scattering from one point,SAMPLED at distances ‘inverse’ to the repeat distance in the object The fringe functionScattering from an array of objects, is the same as scattering from one object,SAMPLED at distances ‘inverse’ to the repeat distance in the object eg DNA

63. Scattering from a molecule is described byF(s)= Si fi e(2pir.s)

68. Consequences of being a crystal?Repetition = sampling of F(S)34Å1/34Å-1

69. 2/lObject repeated 1/l

72. Transform of two hoizontal lines defined y= ± y1F(s) = Int [x=0-infexp{2pi(xa+y1b).s} + exp{2pi(xa-y1b).s}dVr ]=2 cos (2py1b).s * Int [x=0-infexp{2pi(xa).s dVr]for a.s=0 the int[x=0-infexp{2pi(xa).s dVr] = total e content of the linefor a.s≠0 int[x=0-infexp{2pi(xa).s dVr] = 0hence r(r)isa line at a.s= 0 parallel to b, with F(s) = 2 cos (2py1b).s -along a vertical line perpendicular to the horizontal lines.Transform of a bilayer..b=53Å

73. 53Å Repeat distance40Å inter bilayer spacing4.6Å

74. 53Å Repeat distance40Å inter bilayer spacing4.6Å

75. The Transform of a Molecule

77. How to calculate electron density?Proof of the Inverse Fourier Transform:

80. Definitions:

82. Build up a crystal from Molecules…First 1 dimension,a direction

84. Build up a crystal from Molecules…First 1 dimension,a direction

89. a.s=hb.s=k

90. hkl=(11,5,0)Measure I(hkl)F(hkl)= √I(hkl)Determine f(s) = f(hkl)Calculate electron density map

92. The density map is made up of interfering density waves through the entire unit cell…

98. Geometry of Diffraction

100. A Typical X-ray diffraction pattern~100 microns

105. A Typical X-ray diffraction pattern~100 microns

109. l=2dhklsin(q)

113. What if ‘white radiation’? representationLaue picture.

114. Compute a Transform of a series of parallel lines

116. s0sssAdding up the scattering of Atoms:‘interference’ of wavesI(S)= F(S).F(S)*s1Ss0radius = 1/lF(S)2pr.S

119. Sum of 7 atoms scatteringResult is a wave of amplitude F(S)phase f(s) f7=8 electrons2pr7.SF(S)2pr1.Sf1=6 electrons

120. Sum of 7 atoms scatteringResult is a wave of amplitude F(S)phase F(S)f7=8 electrons2pr7.Si = √(-1)sin(q)F(S)cos(q)2pr1.Sf1=6 electronse(iq) = cos(q) + isin(q)

121. Sum of 7 atoms scatteringResult is a wave of amplitude |F(S)|phase F(S)f7=8 electrons2pr7.Si = √(-1)sin(q)F(S)cos(q)2pr1.Sf1=6 electronse(iq) = cos(q) + isin(q)F(S) = f1 e(2pir1.S) + f2 e(2pir2.S) +….

122. Sum of 7 atoms scatteringResult is a wave of amplitude |F(S)|phase F(S)f7=8 electrons2pr7.Si = √(-1)sin(q)F(S)cos(q)2pr1.Sf1=6 electronse(iq) = cos(q) + isin(q)F(S) = Sjfj e(2pirj.S)

123. Sum of 7 atoms scatteringf7=8 electrons2pr7.SF(S)2pr1.Sf1=6 electrons

124. f7=8 electrons2pr7.SF(S)2pr1.Sf1=6 electrons

125. e(iq) = cos(q) + isin(q)F(S) = Sjfj e(2pirj.S)

126. The Process is re-iterative, and should converge-but only so far!CrystalIntensities I(h,k,l)Electron density r(x,y,z)Known:Amino acid sequenceLigandsBond lengths anglesConstraints on geometryPhases f(h,k,l)Experimentalheavy atom labelsselenium for sulfurTrial & error similar structureAtom positions (x,y,z)

127. Scattering pattern is the Fourier transform of the structure F(S) = Sjfj e(2pirj.S)Structure is the ‘inverse’Fourier transform of the Scattering pattern r(r) = SF(S) e(-2pir.S)

135. Rosalind Franklin and DNA

141. This is all there is? YES!!Scattering pattern is the Fourier transform of the structure FTF(S) = Sjfj e(2pirj.S)FT-1Structure is the ‘inverse’Fourier transform of the Scattering pattern 1/ar(r) = SF(S) e(-2pir.S)aFTb1/bFT-1

142. But we observe |F(S)|2 and there are ‘Phases’Scattering pattern is the Fourier transform of the structure FTF(S) = Sjfj e(2pirj.S)FT-1Structure is the ‘inverse’Fourier transform of the Scattering pattern 1/ar(r) = SF(S) e(-2pir.S)aFTWhere F(S) has phaseAnd amplitude b1/bFT-1

143. This is all there is?Scattering pattern is the Fourier transform of the structure FTF(S) = Sjfj e(2pirj.S)SFT-1Structure is the ‘inverse’Fourier transform of the Scattering pattern 1/ar(r) = SF(S) e(-2pir.S)aFTbF(h,k,l) = Sjfj e(2pi(hx+ky+lz))S1/bh=15, k=3,r(x,y,z) = SF(h,k,l) e(-2pir.S)FT-1

144. Relative Information in Intensities versus phasesr(r) duckr(r) cat|F(S)|F(S)= Sjfj e(2pirj.S)F(S) duckF(S) catf(s)|F(S)|duckr(r) = SF(S) e(-2pir.S)f(s) catLooks like a …..

145. Relative Information in Intensities versus phasesr(r) duckr(r) cat|F(S)|F(S)= Sjfj e(2pirj.S)F(S) duckF(S) catf(s)|F(S)|duckr(r) = SF(S) e(-2pir.S)f(s) catLooks like a CATPHASES DOMINATE:-Incorrect phases = incorrect structure-incorrect model = incorrect structure-incorrect assumption = incorrect structure

146. The Process is re-iterative, and should converge-but only so far!CrystalIntensities I(h,k,l)Electron density r(x,y,z)Known:Amino acid sequenceLigandsBond lengths anglesConstraints on geometryPhases f(h,k,l)Experimentalheavy atom labelsselenium for sulfurTrial & error similar structureAtom positions (x,y,z)

147. Consequences of being a crystal?Repetition = sampling of F(S)34Å

148. Consequences of being a crystal?Repetition = sampling of F(S)34Å1/34Å-1

149. Consequences of being a crystal?Sampling DNA = repeating of F(S)1/3.4Å-13.4Å34Å1/34Å-1

150. 2/lObject repeated 1/l

152. Build a crystalObjectScattering

153. Early Definitions:

157. a.s=hb.s=k

158. hkl=(11,5,0)Measure I(hkl)F(hkl)= √I(hkl)Determine f(s) = f(hkl)Calculate electron density map

160. The density map is made up of interfering density waves through the entire unit cell…

161. Measure I(hkl)

162. The Process is re-iterative, and should converge-but only so far!CrystalIntensities I(h,k,l)Electron density r(x,y,z)Known:Amino acid sequenceLigandsBond lengths anglesConstraints on geometryPhases f(h,k,l)Experimentalheavy atom labelsselenium for sulfurTrial & error similar structureAtom positions (x,y,z)

168. Anomalous diffraction