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
Structure factors Reciprocal lattice: http://skuld.bmsc.washington.edu/scatter/
http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
http://www.doitpoms.ac.uk/tlplib/reciprocal_lattice/index.php
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1
Crystals
Data
Coordinates
1UBQ
P212121
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.
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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.
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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)
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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
slide #
7
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)
slide #
3. Intensities ‐ translational symmetries/centering (systematic absences)
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)
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9
Orthogonal vs. fractional coordinates (hexagonal lattice) Unit cell lengths
(0,1)
yo
a = 100Å
b = 100Å
(1,0)
xo
Convert fractions to orthogonal
0, 1 = 0Å, 100Å
1, 0 = 86.6Å, ‐50Å
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10
Proper and improper rotational symmetry
Improper Rotation
Proper Rotation
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.
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 Cn = rot. 360°/n
4. Cyclic symmetry C2, often referred to as a 2fold axis
5. C1 is the special case called the identity.
Note: C2 corresponds to a mirror plane
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11
Why is this important?
Improper Rotation
Proper 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. C5, C7 etc.).
( why is this true?)
1. X‐ray diffraction patterns are centrosymmetric. (Laue groups)
2. Patterson maps are centrosymmetric.
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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
21
180 deg. rotation
180° rotation
Translate ½ unit cell axis
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16
Screw axes must conform to the repetitive nature of the crystal
Thus, observe only 21, 31, 41, 61 screws.
Stacking of unit cells along
Z would allow the 2fold
screw axis to continue
Without interruption
throughout the crystal.
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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. b axis 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°
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19
7
14
Luckily, only 65 of 230 possible space groups can
be used to describe protein crystals
because proteins
are chiral! , handed.
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20
International Tables for Crystallography P212121 Space Group Representation
P212121, 3 mutually perpendicular 2fold screw axes. 2fold screw axis perpendicular to paper is shown in red box.
General Equivalent Positions for space group P212121: 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
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21
International Tables for Crystallography P212121 Space Group Representation
P212121, 3 mutually perpendicular 2fold screw axes Z=4
General Equivalent 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
positions: x=0.1, y=0.2, z=0.3
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22
You can populate the unit cell from these positions.. How do you finish this?
HEADER
TITLE
CHROMOSOMAL PROTEIN
02-JAN-87
1UBQ
STRUCTURE OF UBIQUITIN REFINED AT 1.8 ANGSTROMS RESOLUTION
CRYST1
SCALE1
SCALE2
SCALE3
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
50.840
42.770
28.950 90.00 90.00 90.00 P 21 21 21
0.019670 0.000000 0.000000
0.00000
0.000000 0.023381 0.000000
0.00000
0.000000 0.000000 0.034542
0.00000
atom res ch res#
x
y
z
Q
B
1 N
MET A
1
27.340 24.430
2.614 1.00 9.67
2 CA MET A
1
26.266 25.413
2.842 1.00 10.38
3 C
MET A
1
26.913 26.639
3.531 1.00 9.62
4 O
MET A
1
27.886 26.463
4.263 1.00 9.62
5 CB MET A
1
25.112 24.880
3.649 1.00 13.77
Unit cell, spacegroup, and fractionalization matrix (scale123) in each 6 CG MET A
1
25.353 24.860
5.134 1.00 16.29
pdb file.
7 SD MET A
1
23.930 23.959
5.904 1.00 17.17
8 CE MET A
1
24.447 23.984
7.620 1.00 16.11
9 N
GLN A
2
26.335 27.770
3.258 1.00 9.27
10 CA GLN A
2
26.850 29.021
3.898 1.00 9.07
11 C
GLN A
2
26.100 29.253
5.202 1.00 8.72
12 O
GLN A
2
24.865 29.024
5.330 1.00 8.22
13 CB GLN A
2
26.733 30.148
2.905 1.00 14.46
14 CG GLN A
2
26.882 31.546
3.409 1.00 17.01
15 CD GLN A
2
26.786 32.562
2.270 1.00 20.10
16 OE1 GLN A
2
27.783 33.160
1.870 1.00 21.89
17 NE2 GLN A
2
25.562 32.733
1.806 1.00 19.49
18 N
ILE A
3
26.849 29.656
6.217 1.00 5.87
19 CA ILE A
3
26.235 30.058
7.497 1.00 5.07
20 C
ILE A
3
26.882 31.428
7.862 1.00 4.01
21 O
ILE A
3
27.906 31.711
7.264 1.00 4.61
22 CB ILE A
3
26.344 29.050
8.645 1.00 6.55
23 CG1 ILE A
3
27.810 28.748
8.999 1.00 4.72
24 CG2 ILE A
3
25.491 27.771
8.287 1.00 5.58
slide #
25 CD1 ILE A
3
27.967 28.087 10.417 1.00 10.83
4
N
C
C
O
C
C
S
C
N
C
C
O
C
C
C
O
N
N
C
C
O
C
C
C
23
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
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25
Crystallographic symmetry and the Biologically relevant unit
Crystal packing or biologically relevant Structure?
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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
27
slide #
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
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28
CRYSTAL SOLVENT CONTENT Matthews’ number Vm or Ǻ3/dalton protein for the unit cell
Matthews, J.Mol.Biol 33, 491‐497 (1968). Percent‐solvent.xls
Vm = V / MW of protein * Z * X
where V is unit cell volume (Ǻ3), 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
1
2
3
Vm
3.94
1.97
1.31
%solvent
68.81%
37.62%
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‐cos2α ‐cos2β‐cos2γ + 2cos α cos β cos γ)
Webserver:
http://csb.wfu.edu/tools/vmcalc/vm.html
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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