X-ray Crystallography-1

• Crystal Properties, space groups • Diffraction • Bragg’s law, von Laue condition• X-ray diffraction data collection

Many slides adopted from Prof. W. Todd Lowther, Dept. of Biochemistry, Wake Forest UniversityAdditional slides adopted from Prof. Ernie Brown, formerly in the Dept. of Chemistry, Wake Forest University

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 pmMidterm 1: Monday, Feb. 27

Homework 4 (Chapter 6, X-ray diffraction), due Wednesday, Feb. 22

If not stated otherwise, assume = 0.154 nm (CuK-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 X-ray 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 = 0.154 nm (CuK-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 nutshellReflections:

h k l I σ(I) 0 0 2 3523.1 91.30 0 3 -1.4 2.80 0 4 306.5 9.60 0 5 -0.1 4.70 0 6 10378.4 179.8 . . .

? Phase Problem ?MIRMADMR

Electron density:(x y z) = 1/V |F(h k l)| exp[–2i (hx + ky + lz) + i(h k l)]

Bragg’s

law

Fourier transform

Protein crystalX-ray diffraction

pattern

Electron densityFit molecules (protein) into electron density

End result!Fourier transform of diffraction spots electron density fit amino acid sequence

Protein

DNA pieces

(Dimer of dimers)

X-ray crystallography – in a nutshell

Why determine the 3-D structure of your favorite protein or protein-ligand complex?

• A picture is worth a thousand words.

• 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

Chicken Fibrinogen

S-Nitroso-Nitrosyl Human hemoglobin A

Visible light vs. X-raysWhy don’t we just use a special microscope to look at proteins?

• Resolution is limited by wavelength. Resolution ~ /2

– Visible light: 400-700 nm

– X-rays: 0.7-1.5 Å (0.07-0.15 nm)

• 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.

• 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): ~

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?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

a

bc

• 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 unit cell dimensions a, b, c.

• Angles between axes:

a, b, c

between bc, between ac, between ab

What are 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 What are crystals??

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 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.

• In biological molecules, there are really only 65 relevant space groups (no

inversion axes or mirrors allowed, because they turn L-amino acids into D-

amino acids.

The space group specifies the lattice type (Bravais lattices, outside

crystal morphology) and the symmetry of the unit cell (inside).

• Different crystals that have identical unit cell lengths and angles and are in

the same space group are isomorphous.

What are crystals?Bravais Lattices and Space Groups

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

Two-fold axisprotein

Bovine Pancreatic Trypsin Inhibitor P 212121

(Primitive, orthorhombic unit cell with a two-fold

screw axes along each unit cell vector)

(adapted from Bernhard Rupp, University of California-LLNL)

http://www-structure.llnl.gov/Xray/tutorial/Crystal_sym.htm

• Cell edges: a, b, c• Cell angles: , , • (100), (001), (010) planes define the unit

cell

What are crystals?

Cell Edges, Angles, and Planes

• 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)

What are crystals?Examples of 2-D Diagonal Planes, Miller planes, Miller indices

a

b

(100) planes

a/2

b/3

(230) planes

• Planes extend throughout crystal with different relationships to the origin: e.g. (234)

• Negative indices tilt the plane the opposite direction:

• NOTE that (210)=(-2-10)≠(-210). “-” sign usually put as a bar above the number

What are crystals?Planes in 3-D and Negative Indices

Theory of X-ray diffraction• Treat X-Rays as waves (CuK ~ 0.154 nm).

• Scattering: ability of an object to change the direction of a wave.

• 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:

Reciprocal relationship between the Bragg angle and the spacing, d, between the lattice planes.

By measuring , we can use Bragg’s law to determine dimension of unit cell!

d

Fig. 6.10

Geometric construction in class

What is the relationship between diffraction angle 2 and unit cell dimensions?

Maximum at:

n2 sin

d

Bragg’s law:

n… integer, wavelength of X-ray

von Laue condition for diffractionNow 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)

In three dimensions (pp 263-265):

1/ 22 2 2

2 2 2

von Laue condition:

2sin h k l

a b c

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

is the angle measured from the incoming X-ray beam

Bragg's law in one dimension:

2sin h

a

Each cone (h=1, -1, 2, -2 …)

2

von Laue Condition for

Diffraction

“One-dimensional crystal” (horizontal planes)

“Three-dimensional crystal” (horizontal and vertical planes)

(Horizontal and vertical diffraction cones, dots at intersections) a

bc

h = 2h = 1h = 0h = -1 h = -2

k = -2, -1, 0, 1, 2

1/ 22 2 2

2 2 2

von Laue condition:

2sin h k l

a b c

Determining the dimensions of the unit cell from the

diffraction spots.

Precession photograph of Tetragonal crystal of

T4 lysozyme (X-ray aligned with third axis).

Note: The spacing (angle ) is not affected by the number of molecules in a unit cell (more in a little bit).

h = 1, 2, 3,…

h = -1, -2, -3,…

k = 1, 2, 3 k = -1, -2, -3

dD

2 sin2r

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?

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).

Conceptually:

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).