X-ray Crystallography-1
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Crystal Properties, space groups
Diffraction
Bragg’s law, von Laue condition
X-ray diffraction data collection
Reading:
van Holde, Physical Biochemistry, Chapter 6; the two Watson & Crick
papers
Additional optional reading:
Gale Rhodes, Crystallography Made Clear, sections of Chapters 1-4
Homework: (see next two pages), due Wednesday, Feb. 22
Remember:
Pizza & Movie, Sun, Feb. 12, 6:00 pm
Midterm 1: Monday, Feb. 27
Many slides adopted from Prof. W. Todd Lowther, Dept. of Biochemistry, Wake Forest University
Additional slides adopted from Prof. Ernie Brown, formerly in the Dept. of Chemistry, Wake Forest University
Homework 4 (Chapter 6, X-ray diffraction), due Wednesday, Feb. 22
If not stated otherwise, assume l = 0.154 nm (CuKa-radiation)
1.
van Holde 6.1
2.
van Holde 6.2a (Hint: put one atom at x, y, z,  the other atom at x+1/2, -y, -z)
3.
van Holde 6.3
4.
NaCl crystals are crushed and the resulting microcrystalline powder is placed in the Xray beam. A flat sheet of film is placed 6.0 cm from the sample and exposed. Ignoring the
possibility of forbidden reflections (which is in fact the case with NaCl, because the lattice
is centered), what would be the diameters and indices of the first two (innermost) rings
on the photograph? NaCl is cubic with unit cell dimension a = 0.56nm.
5.
You are working with a linear crystal of atoms, each spaced 6.28 nm apart. You adjust
your x-ray emitter so that it emits 0.628 nm x-rays along the axis of the array.
a. As we did in class, using the reciprocal lattice and the sphere of reflection, draw the
allowed S vectors (S = k - k0). Denote the direction of propagation of the incoming
x-rays as the positive x-axis. (k0 is direction of incoming X-rays, S is scattering
vector).
b. You place a 1 cm2 spherical detector 1 cm from the sample, centered on the x-axis,
on the opposite side of the emitter. Draw the pattern you expect to detect. Clearly
mark the expected distances.
c. If you performed the experiment on a linear crystal with atoms spaced 0.1 nm apart,
what pattern would you detect? Would you have the same pattern if your detector
were 1 m2? What does this say about the resolution of your experiment?
… continued on next page
Homework 4 (Chapter 6, X-ray diffraction), due Wednesday, Feb. 22:
… continued
If not stated otherwise, assume l = 0.154 nm (CuKa-radiation)
6.
van Holde 6.6 a-d (see Fig. 6.18)
7.
van Holde 6.9 a-c (in c), real space means on film)
8.
van Holde 6.9 d but: Sketch the fiber diffraction pattern expected for A-DNA (not Z-DNA).
X-ray crystallography – in a nutshell
• Protein is crystallized
• X-Rays are scattered by electrons in molecule
• Diffraction produces a pattern of spots on a film that must be
mathematically deconstructed (Fourier transform)
• Result is electron density (contour map) – need to know
protein sequence and match it to density
•  coordinates of protein atoms  put in protein data bank
(pdb)  download and view beautiful structures.
• Currently there are about 80,000 structures in the pdb (2012).
Check out protein data bank: (http://www.rcsb.org)
X-ray Crystallography – in a nutshell
X-ray diffraction
pattern
Protein crystal
Bragg’s
Reflections:
h k l
0
0
0
0
0
.
.
.
law
0
0
0
0
0
I
σ(I)
2 3523.1
3 -1.4
4 306.5
5 -0.1
6 10378.4
91.3
2.8
9.6
4.7
179.8
Fourier
transform
? Phase Problem ?
MIR
MAD
MR
Fit molecules (protein)
into electron density
Electron density: r(x y z) = 1/V
Electron density
SSS |F(h k l)| exp[–2pi (hx + ky + lz) + ia(h k l)]
X-ray crystallography – in a nutshell
End result!
Fourier transform of diffraction spots  electron density  fit amino acid sequence
DNA
pieces
Protein
(Dimer of
dimers)
Why determine the 3-D structure of your favorite
protein or protein-ligand complex?
• A picture is worth a thousand words.
Chicken Fibrinogen
• Insight into structure-function relationships
– Recognition and Specificity
– Might identify a pocket lined with negatively-charged residues
– Or positively charged surface – possibly for binding a
negatively charged nucleic acid
– Rossmann fold – binds nucleotides
– Zinc finger – may bind DNA.
– Aids in the design of future experiments
• Rational drug design
• Engineered proteins as therapeutics
S-Nitroso-Nitrosyl
Human hemoglobin A
Visible light vs. X-rays
Why don’t we just use a special microscope to look at proteins?
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Resolution is limited by wavelength. Resolution ~ l/2
– Visible light: 400-700 nm
– X-rays: 0.7-1.5 Å (0.07-0.15 nm)
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But to get images need to focus light (radiation) with lenses.
•
It is very difficult to focus X-rays (Fresnel lenses, doesn’t really work for
X-rays)  there are no lenses for X-rays  can see atoms directly.
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Getting around the problem  X-ray Crystallography
– Defined beam
– Regular structure of object (crystal)
– Result – diffraction pattern (not a focused image).
The Electromagnetic Spectrum
• Wavelength of the “radiation” needs to be
smaller than object size.
• Diffraction limit (separation of resolvable
features): ~ l/2
Crystal formation
• Start with supersaturated solution of protein
• Slowly eliminate water from the protein
• Add molecules that compete with the protein
for water (3 types: salts, organic solvents,
PEGs)
• Trial and error
• Most crystals ~50% solvent
• Crystals may be very fragile
Growing
crystals
What are crystals?
• Ordered 3D array of molecules held together by non-
covalent interactions  Unit Cell
• Sometimes see electrostatic or “salt interactions”
• Lattice network
• Defined planes of atoms
Unit cell vectors
c
b
a
What are crystals?
• Solids that are exact repeats of a
symmetric motif
• In a crystal, the level at which there is
no symmetry  asymmetric unit.
• Apply rotational or screw operators to construct lattice motif.
• Lattice motif is translated in three dimensions to form crystal lattice.
• The lattice points are connected to form the boxes  unit cell.
• The edges define a set of unit vector axes 
• Angles between axes:
a, b, c
unit cell dimensions a, b, c.
a between bc,  between ac,  between ab
What are crystals?
• Cystal  stack unit cells repeatedly without any spaces between cells
•  Unit cell has to be a parallelepiped with four edges to a face, six faces to a unit cell.
• All unit cells within a crystal are identical  morphology of (macroscopic) crystal is
defined by unit cell
• There are only seven crystal systems (describing whole (macroscopic) crystal
morphology): Triclinic, Monoclinic, Orthorhombic, Tetragonal, Trigonal, Hexagonal,
Cubic (defined by length of unit vectors and angles).
• There are only fourteen unique crystal lattices  fourteen Bravais lattices.
• P = primitive lattice point at corners of unit cell, F= face centered lattice point at all
six faces, I = lattice point in center of unit cell, C = centered, lattice point on two
opposing faces.
What are crystals?
Bravais Lattices and Space Groups
• 7 crystal systems
• 14 Bravais lattice
systems
• Space group =
Lattice identifier +
known symmetry
relationships
• Molecules within the
crystal will most likely
pack with symmetry
What are crystals?
Bravais Lattices and Space Groups
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What symmetry operations (e.g. rotation axes, (2-, 3-, 4-, 6- fold axis,
mirrors, inversion axes …, at corner, at face, … (see Table 1.4) can be
applied to the unit cell (inside crystal)? This defines the 32 point groups of
the unit cell.
•
The combination of the 32 symmetry types (point groups) with the 14
Bravais lattice, yields 230 distinct space groups.
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In biological molecules, there are really only 65 relevant space groups (no
inversion axes or mirrors allowed, because they turn L-amino acids into Damino acids.
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 The space group specifies the lattice type (Bravais lattices, outside
crystal morphology) and the symmetry of the unit cell (inside).
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Different crystals that have identical unit cell lengths and angles and are in
the same space group are isomorphous.
Examples of Symmetry
•Rotations
2-folds (dyad symbol)
3-folds (triangle)
4-folds (square)
6-folds (hexagon)
•Rotations can be combined
•Translations
-moved along fractions
of the unit cell
-see P21 example
What are crystals?
Symmetry operators
Examples
protein
(adapted from Bernhard Rupp, University of California-LLNL)
http://www-structure.llnl.gov/Xray/tutorial/Crystal_sym.htm
Two-fold axis
Bovine Pancreatic Trypsin Inhibitor P 212121
(Primitive, orthorhombic unit cell with a two-fold
screw axes along each unit cell vector)
What are crystals?
Cell Edges, Angles, and Planes
• Cell edges: a, b, c
• Cell angles: a, , 
• (100), (001), (010) planes define the unit cell
What are crystals?
Examples of 2-D Diagonal Planes, Miller planes, Miller indices
• Diagonals through the unit cell denoted by how they cross-section an axis
• e.g. 1/2 = 2, 1/3 = 3, 1/4 = 4, etc.
• e. g.: (230) plane has intercepts at 1/2x and 1/3y
• (-230) plane has intercepts at -1/2x and 1/3 y (slanted in other direction)
(230)
planes
b/3
a/2
(100)
planes
b
a
What are crystals?
Planes in 3-D and Negative Indices
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Planes extend throughout crystal with different relationships to the origin: e.g. (234)
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Negative indices tilt the plane the opposite direction:
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NOTE that (210)=(-2-10)≠(-210). “-” sign usually put as a bar above the number
Theory of X-ray diffraction
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Treat X-Rays as waves (CuKa l ~ 0.154 nm).
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Scattering: ability of an object to change the direction of a wave.
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If two objects (A and B) are hit by a wave they act as a point source of a new wave with
same wavelength and velocity (Huygens’ principle)
• Diffraction: Those two waves
interfere with each other.
 destructive and constructive
interference.
 observe where maxima and
minima are on screen.
 get position of A and B
Constructive interference:
Destructive interference:
Bragg’s law (simple model of crystal, but it works!)
Crystal is made up of crystal planes (the Miller planes we just discussed).
Assume a one-dimensional crystal:
d
Fig. 6.10
Geometric construction in class
What is the relationship between diffraction
angle 2q and unit cell dimensions?
Bragg’s
law:
Maximum at:
nl
2 sin q 
d
n… integer, l wavelength of X-ray
Reciprocal relationship between the Bragg angle q and the spacing, d, between the lattice planes.
 By measuring q, we can use Bragg’s law to determine dimension of unit cell!
von Laue condition for diffraction
Now we’ll move on to a three-dimensional crystal.
Lattice still consists of only planes, but now we have a three-dimensional grid (still just dots, no internal structure, yet)
Bragg's law in one dimension:
2sinq h

l
a
In three dimensions (pp 263-265):
von Laue condition:
2sinq
l
2q
h
k
l 
 2  2  2 
b
c 
a
2
2
2
1/ 2
h, k, l, are the Miller indices. Every discrete
diffraction spot on a film has a particular Miller
index. This are the same indices that describe the
Miller planes.
E. g. reflection (1,0,0) h=1, k=l=0; comes from
(100) Miller plane
Each cone (h=1, -1, 2, -2 …)
2q is the angle measured from the incoming Xray beam
von Laue
Condition for
Diffraction
von Laue condition:
2sinq
l
 h2 k 2 l 2 
 2  2  2 
b
c 
a
1/ 2
“One-dimensional
crystal” (horizontal
planes)
k = -2, -1, 0, 1, 2
“Three-dimensional
crystal” (horizontal
and vertical planes)
(Horizontal and
vertical diffraction
cones, dots at
intersections)
a
b
c
h=2
h=1
h=0
h = -1
h = -2
Determining the dimensions of the unit cell from the
diffraction spots.
k = -1, -2, -3
d 
k = 1, 2, 3
l
 D
2 sin 

2r


h = 1, 2, 3,…
h = -1, -2, -3,…
Precession photograph of Tetragonal crystal of
T4 lysozyme (X-ray aligned with third axis).
Note: The spacing (angle q) is not affected by the number of molecules in a unit
cell (more in a little bit).
Example
A NaCl crystal is crushed and the resulting microcrystalline powder is placed in
an 0.154 nm X-ray beam. A flat sheet of film is placed 6.0 cm from the sample.
What is the diameter of the innermost ring on the photograph. NaCl is cubic
with unit cell dimension 0.56 nm.
(A powder gives diffraction rings instead of spots, because of the random orientation of the
microcrystals in the powder.)
Diffraction image: http://www.union.edu/PUBLIC/PHYDEPT/jonesc/images/Scientific/Powder%20diff%20Al.jpg
Determining the crystal symmetry from
systematic absences
Simple, conceptual
example: P21 space group:
Has a 2-fold screw axis
along c-axis
 On 00l-axis only every
other spot is observed.
Each space group specifies its
unique set of special
conditions for observed and
unobserved reflections along
the principal and diagonal
axes. (Can be looked up in
tables).
Sometimes it is very tricky to
assign proper space group,
especially for centered cells.
Translational
symmetry elements
and their extinctions.
(Table 5.2 Jensen &
Stout)
Sometimes there is
ambiguity, i.e. two
space groups have
same pattern
Is Bragg’s law still valid for two or more atoms in a unit cell?
Conceptually:
Jensen and Stout
Two atoms in a unit cell (reflect) waves from their respective planes.
The waves combine and form a resultant wave, that looks like it has been
reflected from the original unit cell lattice plane.
Diffraction spot is in the same place, but has different intensity (intensity
of resultant wave).
We assumed the electron density is in planes. In reality it is spread throughout the unit cell.
Nevertheless, the derivation is still valid, since it can be shown that waves scattered from
electron density not lying in a plane P, can be added to give a resultant as if reflected from the
plane.
So far…
By observing the spacing and pattern of reflections on the
diffraction pattern, we can determine the lengths, and
angles between each side of the unit cell, as well as the
symmetry or space group in the unit cell.
Still, how do we find out what’s inside the unit cell?
(i.e. the interesting stuff, like proteins).