# lecture 2: crystallization & symmetry crystal growth theory symmetry symmetry operator symmetry...

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- Lecture 2: Crystallization & Symmetry crystal growth theory symmetry symmetry operator symmetry operators space groups
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- protein crystals cellulase subtilisin The color you see is birefringence, the wavelength- dependent rotation of polarized light. ~1mm
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- 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
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- Protein crystal growth 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. protein concentration precipitant concentration Arrows indicate different diffusion experients. A,B,D,F,G. Vapor diffusion. E. Bulk C. Microdialysis L=liquid S=solid m=metastable state (supersaturated)
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- vapor diffusion setup Volatiles (i.e. water) evaporate from one surface and condence on the other. Drop has higher water concentration than reservoir, so drop slowly evaporates. a Linbro plate
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- Other ways to supersaturate slowly Sitting drops Microdialysis Gel filtration
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- precipitants A precipitant (r) causes proteins (p) to stick to each other by competing for solvent. p p r r r p p r r r r = EtOH, (NH 4 ) 2 SO 4, methylpentanediol, polyethylene glycol, etc
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- http://www.ccp14.ac.uk/ccp/web-mirrors/llnlrupp/crystal_lab/hampton_screen.htm 50 of the most successful crystallization conditions:
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- Crystallization theory Nucleation takes higher concentration than crystal growth. R R R R R R R R R R R R R slow fast RRRRRRRRRRRRRRRR R R RRRRRRRRRRRRRRR R RRRRRRRRRRRRRRR R RRRRRRRRRRRRRRR R RRRRRRRRRRRRRRR R RRRRRRRRRRRRRRR R fast not so slow After nucleation, the large size of a face makes the weak bond more likely.
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- 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
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- Periodic Bond Chain theory Growth is unfavorable directions increases as the crystal grows. Weak bonds in Z favor growth in XY, forming plate xtal. Growing cross- section in XY favors growth in Z. Ratio of cross sections is inverse to ratio of bond strength.
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- 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 SiO 2 (amethyst) are concentrated in the part of the crystal that formed last (the tip).
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- Better crystals in micro gravity? More at: http://science.nasa.gov/ssl/msad/pcg/ Higher concentration of protein = higher density Differences in densty cause convection currents, which might cause crystal defects. Microgravity eliminates convection currents.
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- mounting crystals Thin-walled glass capillaries (
- Crystals must be flash frozen Water must be frozen to < 70C 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 N 2 gas at 70C. hexagonal ice...to prevent glass->ice transition
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- Centering the crystal in the beam machine center is the intersection of the beam and the two goniostat rotation axes. Must be set by manufacturer! xrays 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. whoops its off center. Fix it!
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- Aligning crystal lattice with the beam. Rotate the crystal until the zero-layer disappears and the 1-layer is centered on the beam. This is where the a* axis is pointed h=1 h=0 h=-1 h=1 misaligned aligned beam is here Concentric circles around beam means axis is aligned with beam.
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- Precession photograph Note symmetrical pattern. Crystal symmetry leads to diffraction pattern symmetry. Spacing of spots is used to get unit cell dimensions.
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- symmetry An object or function is symmetrical if a spatial transformation of it looks identical to the original. X X This is the original This is rotated by 180
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- 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
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- 3x3 Matrix multiplication by the way REMINDER
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- Types of symmetry operations Point of inversion mirror plane glide plane rotation (2,3,4 or 6-fold) screw axis lattice symmetry
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- 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. Fractional coordinates are always measured parallel to each axis. The axes are not necessarilly 90 apart! (0.33,0.25,0.55) a c b
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- point of inversion centric symmetry
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- mirror plane centric symmetry
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- glide plane x centric symmetry
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- rotation non-centric symmetry
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- screw- rotation 1/3 of a unit cell non-centric symmetry 120
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- 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. N CC H R C O
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- Rotational symmetry A 2-fold (180) rotation around the Z-axis
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- rotation
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- Rotation matrices... the mathematical description of a rotation. In polar coordinates, a rotation is the addition of angles. x r (x,y) (x,y) y axis of rotation atom starts here......goes here..rotates by
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- REMINDER: sum of angles rules cos ( = cos cos sin sin sin ( = sin cos sin cos
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- Adding angles in Cartesian space x y r (x,y) (x,y) x' = |r| cos ( = |r|(cos cos sin sin = (|r| cos cos |r| sin sin = x cos y sin y' = |r| sin ( = |r|(sin cos sin cos = (|r| sin cos |r| cos sin = y cos x sin x = |r|cos y = |r|sin in matrix notation... converting internal motion to Cartesian motion rotation matrix
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- 2D rotation using matrix notation 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 ( row times column
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- Transposing the matrix reverses the rotation To rotate the opposite direction, flip the matrix about the diagonal. inverse rotation matrix = transposed rotation matrix. the transpose...because cos cos + sin sin = 1
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- A 3D rotation matrix Is the product of 2D rotation matrices. Rotation around z-axis Rotation around y-axis 3D rotation
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- Example: Rotate v=(1.,2.,3.) around Z by 60, then rotate around Y by -60
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- Examples: x y z z x y y z x 90 rotation around X Y Z For a R-handed rotation, the minus sine is the one on the Right. Helpful hint:
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- Euler angles, Order of rotations: 123 axis of rotation: zxz 123 z-z-y yz 4 5 Polar angles, 3D angle conventions: Net rotation =
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- Properties of rotation matrices More at http://mathworld.wolfram.com/RotationMatrix.html Square, 2x2 or 3x3 The product of any two rotation matrices is a rotation matrix The inverse equals the transpose, R -1 = R T 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.
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- 2-fold rotation R R 180 rotation. Called a 2-fold because doing it twice brings you back to where you started. 2-fold symbol Equivalent positions in fractional coordinates: x,y,z-x,-y,z P2
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- 3-fold rotation R 3-fold symbol R R In fractional coordinates: P3 Equivalent positions : x,y,z-y,x-y,z-x+y,-x,z
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- 4-fold rotation R 4-fold symbol R In fractional coor

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