Lecture 2: Crystallization & Symmetry

1. Lecture 2: Crystallization & Symmetry crystal growth theory symmetry symmetry operator symmetry operators space groups

2. protein crystals ~1mm cellulase subtilisin The color you see is “birefringence”, the wavelength-dependent rotation of polarized light.

3. Crystallization robot • High-throughput crystallography labs use pipeting robots to explore thousnds of “conditions”. Each condition is a formulation of the crystal drop and the reservoir solution. • Conditions can have different: • protein concentration • pH • precipitant, precipitant concentration • detergents • organic co-solvents • metal ions • ligands • concentration gradient

4. Protein crystal growth Arrows indicate different diffusion experients. A,B,D,F,G. Vapor diffusion. E. Bulk C. Microdialysis L=liquidS=solidm=metastable state (supersaturated) protein concentration precipitant concentration blue line = saturation of protein red line = supersaturation limit Crystal growth occurs between these two limits. Above the supersaturation limit, proteins form only disordered precipitate.

5. vapor diffusion setup a Linbro plate Volatiles (i.e. water) evaporate from one surface and condence on the other. Drop has higher water concentration than reservoir, so drop slowly evaporates.

6. Other ways to supersaturate slowly Sitting drops Microdialysis Gel filtration

7. p p p r r r precipitants A precipitant (r) causes proteins (p) to stick to each other by competing for solvent. p r r r r = EtOH, (NH4)2SO4, methylpentanediol, polyethylene glycol, etc

8. 50 of the most successful crystallization conditions: http://www.ccp14.ac.uk/ccp/web-mirrors/llnlrupp/crystal_lab/hampton_screen.htm

9. R R R R R R R R R R R R R Crystallization theory Nucleation takes higher concentration than crystal growth. slow slow fast After nucleation, the large size of a face makes the weak bond more likely. R RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR not so slow fast

10. Periodic Bond Chain theory Bonds A,B are stronger than P,Q. Dimensions of crystal at equilibrium are proportional. More on Periodic Bond Chain theory: http://www.che.utoledo.edu/nadarajah/webpages/PBC.htm

11. Periodic Bond Chain theory Growth is unfavorable directions increases as the crystal grows. Ratio of cross sections is inverse to ratio of bond strength. Growing cross-section in XY favors growth in Z. Weak bonds in Z favor growth in XY, forming “plate” xtal.

12. diffusion depletion Crystal growth depletes the surrounding solution of protein, while concentrating impurities. Local depletion... ...prevents nucleations close to a growing crystal ...slows and eventually stops crystal growth ...concentrates impurities on the surface of the crystal ...causes convection currents. Cobalt impurities in SiO2 (amethyst) are concentrated in the part of the crystal that formed last (the tip).

13. Better crystals in micro gravity? Higher concentration of protein = higher density Differences in densty cause convection currents, which might cause crystal defects. Microgravity eliminates convection currents. More at: http://science.nasa.gov/ssl/msad/pcg/

14. mounting crystals Protein crystals are extremely fragile!!! They may break upon sudden contact with a solid object. Tiny pipets are used to pull crystals from drops. Thin-walled glass capillaries (<1mm in diameter) are filled with “mother liquor”(the fluid in which the crystal was grown) and a crystal is carefully dropped in. The mother liquor is removed using filter paper cut to fine strips. The crystal sticks to the glass, immobilized. The xtal remains in vapor diffusion contact with the mother liquor. If not it will dryout and crack.

15. Crystal mounting If not freezing If freezing (preferred) Xtal is mounted on a thin film of water in a wire loop. The loop is fixed to a metal or glass rod. Xtal is mounted in a thin-walled glass capillary tube Mounted xtal is attached to a goniometer head for precise adjustment. Low-melting hard wax is used to ‘glue’ the rod or capillary here. Small wrenches fit here, here, here and here. Must freeze immediately or film will dry out.! wax eucentric goniometer head(made by Nonius) Crystal must be kept at proper humidity and temperature!! Very fragile!

16. Why freeze? Essentially eliminates X-ray damage to crystal. Crystals do not decay during data collection. Why not? wire-loop crystal catcher Cryo equipment is expensive. Ice crystals may form if freezing is not done properly, ruining data.

17. Crystals must be flash frozen ...to prevent glass->ice transition Water must be frozen to < –70°C very fast to prevent the formation of hexagonal ice. Water glass forms. How? Crystals, mounted on loops, are flash frozen by dipping in liquid propane or freon at –70°, or by instant exposure to N2 gas at –70°C. hexagonal ice

18. Centering the crystal in the beam whoops it’s off center. Fix it! xrays “machine center” is the intersection of the beam and the two goniostat rotation axes. Must be set by manufacturer! To place crystal at machine center, rotate  and  and watch the crystal. If it moves from side to side, it is off center. If it is off-center, we adjust the screws on the goniometer head.

19. Aligning crystal lattice with the beam. Rotate the crystal until the zero-layer disappears and the 1-layer is centered on the beam. aligned misaligned h=1 h=0 h=-1 h=1 This is where the a* axis is pointed beam is here Concentric circles around beammeans axis is aligned with beam.

20. Precession photograph Spacing of spotsis used to get unit celldimensions. Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry.

21. symmetry An object or function is symmetrical if a spatial transformation of it looks identical to the original. This is the original This is rotated by 180° X X

22. Symmetry operators A spatial transformation can be expressed as an operator that changes the coordinates of every point in the object the same way. Symmetry operators do not distort the object. In other words, the distance between any two points is the same before and after being moved by the symmetry operation. Here is the operator for a 180° rotation around Z. equivalent positions

23. REMINDER by the way 3x3 Matrix multiplication

24. Types of symmetry operations • Point of inversion • mirror plane • glide plane • rotation (2,3,4 or 6-fold) • screw axis • lattice symmetry

25. Fractional coordinates The crystallographic coordinate system is defined by the unit cell. The location of a point is defined by fraction of traveled (from 0 to 1) along each unit cell axis. (0.33,0.25,0.55) c b a Fractional coordinates are always measured parallel to each axis. The axes are not necessarilly 90° apart!

26. point of inversion Object Object centric symmetry

27. mirror plane Object Object centric symmetry

28. glide plane Object Object x centric symmetry

29. rotation Object Object non-centric symmetry

30. screw-rotation Object 120° Object 1/3 of a unit cell non-centric symmetry

31. Why proteins cannot have centric symmetry Mirror images and points of inversion cannot be re-created by pure rotations. Centric operations would change the chirality of chiral centers such as the alpha-carbon of amino acids or the ribosal carbons of RNA or DNA. R H Ca C N O

32. Rotational symmetry A 2-fold (180°) rotation around the Z-axis

33. rotation

34. Rotation matrices ... the mathematical description of a rotation. atom starts here... ..rotates by b.. y ...goes here (x’,y’) axis of rotation (x,y)  r  x In polar coordinates, a rotation is the addition of angles.

35. REMINDER: sum of angles rules cos (= cos cos sin sin  sin (= sin cos sin cos 

36. converting internal motion to Cartesian motion Adding angles in Cartesian space y x = |r|cos  y = |r|sin  (x’,y’) (x,y)  r  x y' = |r| sin ( = |r|(sin cos sin cos  = (|r| sin cos |r| cos sin  = ycos x sin  x' = |r| cos ( = |r|(cos cos sin sin  = (|r| cos cos |r| sin sin  = xcos y sin  in matrix notation... rotation matrix

37. 2D rotation using matrix notation “row times column” x'= xcos y sin  = = (|r| cos cos |r| sin sin   = |r| cos (  y'= ycos x sin  = (|r| sin cos |r| cos sin  = |r| sin (

38. Transposing the matrix reverses the rotation To rotate the opposite direction, flip the matrix about the diagonal. the “transpose” inverse rotation matrix = transposed rotation matrix. ...because cosb cosb + sinb sinb = 1

39. A 3D rotation matrix Is the product of 2D rotation matrices. Rotation around y-axis Rotation around z-axis 3D rotation

40. Example: Rotate v=(1.,2.,3.) around Z by 60°, then rotate around Y by -60°

41. Examples: z 90° rotation around X y x x Y z y y Z x z Helpful hint: For a R-handed rotation, the minus sine is the one on the “Right.”

42. 3D angle conventions: Euler angles, a b g axis of rotation: z’’ x’ z Order of rotations: 3 2 1 Polar angles, fyk z’’’’ y’’’ z’’ -y’ -z 4 5 3 2 1 Net rotation = k

43. Square, 2x2 or 3x3 • The product of any two rotation matrices is a rotation matrix • The inverse equals the transpose, R-1 = RT • orthogonality • The dot-product of any row or column with itself is one. • The dot-product of any row or column with a different row or column is zero. • |x| equals |Rx|, for any rotation R. Properties of rotation matrices More at http://mathworld.wolfram.com/RotationMatrix.html

44. 2-fold rotation R 2-fold symbol P2 R Equivalent positions in fractional coordinates: x,y,z -x,-y,z 180° rotation. Called a 2-fold because doing it twice brings you back to where you started.

45. 3-fold rotation R R 3-fold symbol P3 R In fractional coordinates: Equivalent positions : x,y,z -y,x-y,z -x+y,-x,z

46. 4-fold rotation R R R P4 R 4-fold symbol In fractional coordinates (same as orthogonal coords): Equivalent positions: x,y,z -x, -y,z-y, x,z y,-x,z

47. 6-fold rotation R R R R R P6 R 6-fold symbol In fractional coordinates: Equivalent positions: x,y,z -y,x-y,z -x+y,-x,z-x,-y,z y,-x+y,z x-y,x,z

48. In class exercise: rotating a point (a) Choose a point r=(0.1,0.2,0.3) [orthogonal coordinates] Rotate the point by 30° in x. Then rotate it by -90° in y. What are the new coordinates? (b) Choose a point r=(0.1,0.2,0.3) [fractional coordinates] Multiply by the symmetry operator: What are the new fractional coordinates?

49. No 5-fold symmetry in crystals?? A crystal lattice must be space-filling and periodic. This “Penrose tile pattern” is spacefilling but not periodic. Look for translational symmetry in this image. Is there any?

50. Quasicrystals: 5-fold point group symmetry, but no space group symmetry