Helpful resources for all X ray lectures Crystallization http://www.hamptonresearch.com under tech support: crystal growth 101 literature Spacegroup tables http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Crystallography 101 http://www.ruppweb.org/xray/101index.html Xray anomalous scattering http://skuld.bmsc.washington.edu/scatter/ Structure factors http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html Reciprocal lattice: http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/index.php slide # 1

Crystals Data Coordinates 1UBQ P2 1 2 1 2 1

Righted handed coordinate systems Used to describe atomic coordinates unit cells, and diffraction geometry c β α γ b a c Note: convention for the angles between the vectors. slide # 3

Important definitions used to describe crystals. Remember..You can t pick your family or the parameters of your crystal. 1. Asymmetric unit Smallest unit operated on by spacegroup symmetry; also called crystallographic Symmetry. Contents reported in a pdb file 2. Spacegroup Mathematical descriptors of the positions of all atoms in the unit cell 3. Unitcell Volume element repeated by translational symmetry to describe a crystal. Experimentally, we determine unit cell first,,lattice / spacegroup, then asymmetric unit. slide # 4

The Crystal Lattice 1. Lattice point: a point in a crystal repeated many times (the first point chosen is arbitrary). 2. The translational component of the repeated points are given by lattice vectors. Lattice vectors connect two lattice points. 3. Any lattice point may be reached from any other by the vector addition of an integral number of lattice vectors. 5. Fractional lattice indices indicate atomic positions within the unit cell. 6. The minimum repeating unit (lattice vectors) is the primitive cell, which contains 1 lattice point (1/8 of a point at each intersection). 7. Unit cells are made by defining a set of three non colinear lattice vectors. Most unit cells are primitive (P, contain 1 lattice point), but other are possible (Centered cells, C,I, F, R) slide # 5

Lattice points and lattice vectors Choose atom a ( )as a lattice point. Which other atoms are lattice points? Draw a few lattice vectors. How many lattice vectors are there? What are the shortest lattice vectors? Draw a bounding box with the shortest lattice vectors. This is the primitive cell. slide # 6

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Difference between Primitive and C Centered Lattice Primitive Centered The lattice elements are determined from the x ray diffraction pattern. 1. Repeating unit vectors unit cell parameters (a, b, c, α, β, γ) 2. Point group symmetry (constrained by unit vectors) 3. Intensities translational symmetries/centering (systematic absences) slide # 8

Orthogonal vs. fractional coordinates PDB coordinate files are on orthogonal axes in units of Å However, crystallographic Analysis often uses a fractional Coordinates on the true crystal basis (unit cell )vectors. Fractionalization matrix ( scale123 matrix in pdb file) slide # 9

Orthogonal vs. fractional coordinates (hexagonal lattice) yo (0,1) Unit cell lengths a= 100Å b = 100Å (1,0) xo Convert fractions to orthogonal 0, 1 = 0Å, 100Å 1, 0 = 86.6Å, 50Å slide # 10

Proper and improper rotational symmetry Proper Rotation Improper Rotation 1. Rotation axis perpendicular to the page 2. Rotating each object by 180 gives the other object (in this case a hand) 3. 360 /180 = 2= n or C n = rot. 360 /n 4. Cyclic symmetry C2, often referred to as a 2fold axis 5. C1 is the special case called the identity. 1. Center of symmetry or inversion symmetry C1 2. Moves an atom from x, y, z to x, y, z. (Rotoinversion) 3. 2fold rotation, then mirror across plane in the paper. Note: C2 corresponds to a mirror plane slide # 11

Why is this important? Proper Rotation Improper Rotation 1. Proteins are chiral/handed. 2. Thus, they only undergo proper rotations. 3. This limits the spacegroups accessible to proteins to 65 instead of 230. 4. While crystal p.g. symmetries are limited to C1,C2,C3,C4,C6 rotational symmetry, proteins can use of any rotational symmetry element (e.g. C 5, C 7 etc.). ( why is this true?) 1. X ray diffraction patterns are centrosymmetric. (Laue groups) 2. Patterson maps are centrosymmetric. slide # 12

Point Group Symmetry Plane representation of various point groups (e.g. no translational symmetry) Relationships between identical points Write the general coordinate positions for these point groups. Principle axis normal to plane of paper. Open circled below the plane of the board. 13

Point Group Symmetry Plane representation of various point groups (e.g. no translational symmetry) Relationships between identical points 2m=mm2 Or mmm Write the general coordinate positions for these point groups. Principle axis normal to plane of paper. Open circled below the plane of the board. 14

What constraints are placed on a cell with 222 Point Group Symmetry? 2m=mm2 Or mmm 1) 3 perpendicular two folds 2) Angles between cell axes must be 90 3) Unequal cell axes 15

Point Group Symmetry & Translational Symmetry Defines a new Symmetry Element (The Screw Axis) (2fold axis) (2fold screw axis) 2 2 1 180 deg. rotation 180 rotation Translate ½ unit cell axis slide # 16

Screw axes must conform to the repetitive nature of the crystal Thus, observe only 2 1, 3 1, 4 1, 6 1 screws. Stacking of unit cells along Z would allow the 2fold screw axis to continue Without interruption throughout the crystal. slide # 17

Crystal System. P.G. sym., Bravais la ce, Trans. Sym. Space Group Conditions imposed on cell geometry Unique axis b; α=γ=90 1. 7 crystal systems (lattice), which defines unit cell vector lengths/angles Laue group / translational symmetry / bravais lattice Spacegroup There are 230 possible spacegroups. However, because proteins are chiral, only 65 are possible for proteins. Why is this true again? slide # 18

65 non enantiogenic spacegroups Example Convention using P222 P = primitive lattice 222 corresponds to symmetry along the a, b, and c principle axes. (In other space groups symmetry along specific Axes eg. baxis in monoclinic and c axis in tetragonal etc.) Because it is an orthorhombic spacegroup, we know that a b c and α=β=γ = 90 slide # 19

7 14 Luckily, only 65 of 230 possible space groups can be used to describe protein crystals because proteins are chiral!, handed. slide # 20

International Tables for Crystallography P2 1 2 1 2 1 Space Group Representation P2 1 2 1 2 1, 3 mutually perpendicular 2fold screw axes. 2fold screw axis perpendicular to paper is shown in red box. General Equivalent Positions for space group P2 1 2 1 2 1 : Z=4 1) x, y, z 2) x+ ½, y, z+1/2 3) x, y+1/2, z+1/2 4) x+1/2, y+1/2, z 1) How many molecules in the unit cell? 2) Discriminate translational symmetry. 3) Describe the correspondence between symmetry in the spacegroup and the equivalent positions slide # 21

International Tables for Crystallography P2 1 2 1 2 1 Space Group Representation Z=4 P2 1 2 1 2 1, 3 mutually perpendicular 2fold screw axes General Equivalent positions: x=0.1, y=0.2, z=0.3 1) x, y, z 2) x+ ½, y, z+1/2 3) x, y+1/2, z+1/2 4) x+1/2, y+1/2, z slide # 22 You can populate the unit cell from these positions.. How do you finish this?

HEADER CHROMOSOMAL PROTEIN 02-JAN-87 1UBQ TITLE STRUCTURE OF UBIQUITIN REFINED AT 1.8 ANGSTROMS RESOLUTION CRYST1 50.840 42.770 28.950 90.00 90.00 90.00 P 21 21 21 4 SCALE1 0.019670 0.000000 0.000000 0.00000 SCALE2 0.000000 0.023381 0.000000 0.00000 SCALE3 0.000000 0.000000 0.034542 0.00000 atom res ch res# x y z Q B ATOM 1 N MET A 1 27.340 24.430 2.614 1.00 9.67 N ATOM 2 CA MET A 1 26.266 25.413 2.842 1.00 10.38 C ATOM 3 C MET A 1 26.913 26.639 3.531 1.00 9.62 C ATOM 4 O MET A 1 27.886 26.463 4.263 1.00 9.62 O ATOM 5 CB MET A 1 25.112 24.880 3.649 1.00 13.77 C ATOM Unit 6 cell, CG spacegroup, MET A 1 and fractionalization 25.353 24.860 matrix 5.134 (scale123) 1.00 in 16.29 each C ATOM pdb file. 7 SD MET A 1 23.930 23.959 5.904 1.00 17.17 S ATOM 8 CE MET A 1 24.447 23.984 7.620 1.00 16.11 C ATOM 9 N GLN A 2 26.335 27.770 3.258 1.00 9.27 N ATOM 10 CA GLN A 2 26.850 29.021 3.898 1.00 9.07 C ATOM 11 C GLN A 2 26.100 29.253 5.202 1.00 8.72 C ATOM 12 O GLN A 2 24.865 29.024 5.330 1.00 8.22 O ATOM 13 CB GLN A 2 26.733 30.148 2.905 1.00 14.46 C ATOM 14 CG GLN A 2 26.882 31.546 3.409 1.00 17.01 C ATOM 15 CD GLN A 2 26.786 32.562 2.270 1.00 20.10 C ATOM 16 OE1 GLN A 2 27.783 33.160 1.870 1.00 21.89 O ATOM 17 NE2 GLN A 2 25.562 32.733 1.806 1.00 19.49 N ATOM 18 N ILE A 3 26.849 29.656 6.217 1.00 5.87 N ATOM 19 CA ILE A 3 26.235 30.058 7.497 1.00 5.07 C ATOM 20 C ILE A 3 26.882 31.428 7.862 1.00 4.01 C ATOM 21 O ILE A 3 27.906 31.711 7.264 1.00 4.61 O ATOM 22 CB ILE A 3 26.344 29.050 8.645 1.00 6.55 C ATOM 23 CG1 ILE A 3 27.810 28.748 8.999 1.00 4.72 C ATOM 24 CG2 ILE A 3 25.491 27.771 8.287 1.00 5.58 C slide # 23 ATOM 25 CD1 ILE A 3 27.967 28.087 10.417 1.00 10.83 C

Generating symmetry related molecules in pymol slide # 24

Generating symmetry related molecules in pymol Show unit cell with the command show cell (s tab) b c a slide # 25

Crystallographic symmetry and the Biologically relevant unit Crystal packing or biologically relevant Structure? slide # 26

NON CRYSTALLOGAPHIC SYMMETRY ASU contains more than 1 ( e.g. 4 in this case) protein chain related by symmetry. There are NO restrictions on this symmetry ( rotation or translational). Chains A, B, C, and D are related by n.c.s. symmetry ~colinear with fourfold crystal axis. Rather than 360/n=4 90 rotation, n.c.s. tetramer may adopt pseudo fourfold symm.~84. However, the crystallographic symmetry (general equivalent positions) is still strictly obeyed. C slide # 27

PDBePISA server: (Protein Interfaces, Surfaces and Assemblies) This server calculates details of crystal contacts from a pdb file (the asymmetric unit). Crystal contacts: hydrogen bonds, salt bridges, VDW, hydrophobic interactions between proteins related by crystallographic symmetry. http://www.ebi.ac.uk/msd srv/prot_int/pistart.html slide # 28

CRYSTAL SOLVENT CONTENT Matthews number Vm or Ǻ 3 /dalton protein for the unit cell Matthews, J.Mol.Biol 33, 491 497 (1968). Vm = V / MW of protein * Z * X Percent solvent.xls where V is unit cell volume (Ǻ 3 ), X is number of molecules in ASU, and Z is number of equivalent positions. Provides a way to estimate the contents of the ASU prior to structure solution. Vm values typically in the range of 2 3 Å/Dalton. % solvent in crystal= (1 1.23/Vm)*100 1.850398 0.961165 Z Vm %solvent 1 3.94 68.81% 2 1.97 37.62% 3 1.31 6.44% Calculate the Vm and solvent content of the following cell a=65.57, b=30.92, c=34.81 beta=106.02 SG=P21,res = 1.7A MW 17,200 V=abc SQRT(1 cos 2 α cos 2 β cos 2 γ + 2cos α cos β cos γ) Webserver: http://csb.wfu.edu/tools/vmcalc/vm.html slide # 29

The goal is to understand the following concepts. 1. Asymmetric unit. 2. Unit cell, unit cell translations. 3. Crystal lattice, lattice points and vectors, types of lattices observed in crystals. 4. Coordinate systems. Conversion between fraction and orthogonal coordinates. 5. General equivalent positions. 6. Symmetry (point symmetry, translational symmetry including screw axes) observed in crystals and how to describe it with equivalent positions. 7. Spacegroup designations. 8. How to read space group tables from international tables. 9. Understand pdb files and how to evaluate crystal symmetry using pymol. 10. Understand the difference between crystallographic and non crystallographic symmetry. 11. Understand how to calculate solvent content in protein crystals. slide # 30