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GGNB Course A57: Macromolecular Crystallography

GGNB Course A57

Macromolecular Structure Determination I

Part I: Crystals and X-Ray DiffractionTim Grüne

Dept. of Structural Chemistry, University of Göttingen

September 2011

http://shelx.uni-ac.gwdg.de

[email protected]

Tim Grüne 1/87

http://shelx.uni-ac.gwdg.de


Overview Lectures & Practical

Lectures: 9am – 11am

Monday, Sept 26th

Tuesday, Sept 27th

No lecture on Wednesday

Thursday, Sept 29th

Friday, Sept 30th

Practicals 1pm–5pm

Monday, Oct 10th — Friday, Oct 14th

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Learning from Structure: Some Applications of Crystallography

Tim Grüne 3/87


Pol II: Crystal “Snapshots”

Several structures of RNA Polymerase II

in different states of action lead to a con-

cept of the mode of function.

Movie courtesy P. Cramer Lab, LMU Munich

Tim Grüne 4/87


Insulin: Quality Control

• 1982: production of recombinant human insulin (im-

provement of tolerance compared to bovine insulin)

• recombinant and purified human insulin structurally

identical

• structure based point-mutations of insulin

improve functionality (e.g. rate of re-

lease). An extensive list can be found at

http://de.wikipedia.org/wiki/Insulinpräparat (sorry,

German page is by far better than the English one).

Tim Grüne 5/87

http://de.wikipedia.org/wiki/Insulinpräparat


Small Molecules: Handedness and Purity

http://de.wikipedia.org/wiki/Methylphenidat

• Methylphenidate (aka Ritalin): drug to treat attention-

deficit hyperactivity disorder (ADHD)

• Contains two stereochemical centres, i.e. there are four

different forms

• Often only one form has the desired effect, others often

contribute to (undesired) side-effects

• see e.g. E. J. Ariëns: Stereochemistry, a basis for so-

phisticated nonsense in pharmacokinetics and clinical

pharmacology, European Journal of Clinical Pharmacol-

ogy, 26 (1984), pp. 663–668.

To my knowledge: Crystal structure only means to determine handedness and degree of

purity.

Tim Grüne 6/87

http://de.wikipedia.org/wiki/Methylphenidat


Structure Guided Drug Design

Atomic coordinates for ligand and target

enable

• fine-tuning of contact

• fine-tuning of shape: influence mode

of function and access towards target.

The antibiotic Thiostrepton in contact with its

target DNA. Image courtesy K. Pröpper.

Tim Grüne 7/87


DNA Double Helix

• X-ray image of fibrous, crystalline DNA by R. Franklin, which

led her with co-workers and Watson/Crick to the double-

helical structure of DNA

• The model is often considered the “birth of modern molecular

biology” (Voet & Voet, Biochemistry (1995), Wiley & Sons).

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“Terms and Conditions”

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“Macromolecule”

A macromolecule is a protein or nucleic acid compound bigger than a couple of kDa, e.g. a

protein consisting of 50 or more residues.

The term macromolecular also includes complexes, e.g. between a protein and a ligand or

DNA and an antibiotic.

Tim Grüne 10/87


“Structure”

Structure Determination means the description of “how something looks like”. This is a very

vague description, because it depends on the applied technique.

A microscopist may describe the compartments inside a bacterial cell, e.g in terms of colour,

composition, and shape.

For an electron microscopist, structural information of a macromolecule consists mostly of its

shape.

For a crystallographer or an NMR spectroscopist, “structure” means the determination of the

coordinates of the atoms a molecule or complex consists of.

Tim Grüne 11/87


Methods for Structure Determination

Some of the common methods for macromolecular structure determination:

Method Sample Information RemarksX-ray Crystallography Crystal atom positionsNeutron Crystallography Crystal atom positions detects H-atomsElectron Diffraction Crystal atom positions often only 2D informationNuclear Magnetic Resonance Solution atom positions size limitsElectron Microscopy Solution shape large complexes only

These methods are complementary, i.e. the information they provide add to one another

(even though some might regard NMR and X-ray crystallography as competitive).

This course concentrates on X-ray Crystallography.

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Outline of X-ray Structure Determination

Data

Deposition

Refinement

& building

collection

Data

Phasing

Crystal

growth

density map

Electron

Validation

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Definition of a Crystal

The International Union of Crystallography (IUCr) defines a crystal as a solid material with

an essentially discrete diffraction pattern.

For this course it is easier to think of a crystal as one motif — the unit cell containing the

molecule or molecules — which is repeated in all three directions without any gaps, like

building a house from bricks. The sides of the bricks can have arbitrary lengths and the sides

can be inclined. But all (crystallographic) bricks must be identical to each other.

Tim Grüne 14/87


Crystal Types

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Crystal Types

All matter (including liquids and gases) is held together by electrostatic interaction, i.e. be-

cause of the attraction of positive and negative charge, also crystals. There are different

sub-types of interaction. Those which are important for crystals can be classified as:

1. ionic

2. metallic

3. covalent bonds

4. van-der-Waals interactions

The categories are not “distinct": there are compounds which belong to inbetween two cate-

gories.

Tim Grüne 16/87


Ionic Crystals

Ionic crystals are composed of negatively charged anions and positively charged cations.

The net-charge of an ionic crystal is always 0e, otherwise the crystal would fly apart.

NaCl is the simplest example for an ionic crys-

tals:

Na passes its outer shell electron to Cl, leaving

a positively charged Na+-ion and a negatively

charged Cl−-ion. The total energy gain by this

transition is 6.4eV .

Tim Grüne 17/87


Metals

Al13e

Al13e

Al13e

Al13e

Al13e

Al13e

Al13e

Elect

ron la

ke

(3 e

lect

rons

per Al−

atom

)Al

13e

The valence electrons dis-

sociate from the atom and

are shared amongst all

ionic bodies. The valence

elctrons create an electron

lake. This explains the

high conductivity, elasticity

of metals, and why they are

shiny.

Tim Grüne 18/87


Covalent Bonds

Crystal packing of C (diamond) or Si.

(Usually) two atoms share their covalent elec-

trons to fill their outer electron shell. E.g. C or

Si have four electrons in their outer shell and

can therefore have up to four bonding partners.

This results in a rather complicated network in

crystalline carbon and the mechanical stability

of diamonds.

Tim Grüne 19/87


van-der-Waals Interaction

van-der-Waals interaction is the main interaction for macromolecules, not only in crystals but

also e.g. in the formation of oligomers in solution.

It is based on the random or accidental displacement of electrons which creates a temporary

electric field which propagates through adjacent molecules.

A “snapshot” of a charge distribu-

tion three putative, aligned molecules

which induces a temporary dipole mo-

ment by which the molecules attract

each other. One moment later the

charge distribution might look different

again.

Tim Grüne 20/87


Interaction between Macromolecules and their Environment

• Hydrophobic patches

• negatively charged patches

• positively charged patches

Schematic view of a proteinSurface charge distribu-

tion of the nucleosome

Macromolecules are much more likely to aggregate than to crystallise.

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Crystal Growth

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Growing Crystals

Metals Solid metals are generally crystalline, so e.g. cooling molten

metal results in crystalline metal.

Salts Drying salt dissolved in water often results in crystals because

of the strong ionic force

Proteins are difficult to crystallise. Their “natural” solid state is a

disordered aggregate, because the intermolecular forces are rel-

atively weak and the large surface of the molecule allows many

(irregular) orientations.

Tim Grüne 23/87


Crystallisation Methods

Macromolecules are usually crystallised by driving them out of solution by competition with

precipitants for solvent molecules.

Common precipitants are

salts e.g. (NH4)2SO4, NaCl, KH2PO4

organic polymers mostly polyethylen glycol

(PEG)

alcohols e.g. isopropanol

salting in salting out

salt concentration

pro

tein

so

lub

ility

good for

purification

good for

crystallisation

Example: Precipitation

with salt

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Phase Diagram Protein vs. Precipitant

Simplified phase diagram between precipitant and protein concentration.

meta−

stable

soluble(growth) (nucleation)

labile

solid(precipitation)

precipitant concentration

pro

tein

concentr

ation

protein

Crystal growth occurs in the labile and mostly

the metastable zone.

Nucleation, i.e. the formation of the initial crys-

tal seed, occurs in the labile zone.

At too high protein and/or precipitant con-

centration, proteins aggregate and precipitate

without forming crystals.

Tim Grüne 25/87


Crystallisation Conditions

The phase diagram depends on many factors, e.g.

pH (buffer)ionic strength (salt concentration)

type of saltadditive compounds

temperature...

For many (most) precipitants and conditions, the labile and metastable zone are virtually

non-existant. The art of crystal growth consists of finding the right right solvent composition.

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Crystallisation Methods

The most common crystallisation methods are

1. vapour diffusion

2. liquid phase diffusion

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Vapour Diffusion

cProt = 20mg/ml

c

cPEG = 25%Prot =20mg/ml

100mM Hepes pH=7.0

Reservoir solution:

20mM CaCl 2

25% PEG 3350

1µl1µl

Protein sample:

20mM Tris pH=8.0

50mM NaCl

=10mg/mlProtc

PEGc = 12.5%

drop at setup: after equilibration:

Sealed chamber

Tim Grüne 28/87


Vapour Diffusion

cProt = 20mg/ml

c


100mM Hepes pH=7.0

Reservoir solution:

20mM CaCl 2

25% PEG 3350

1µl1µl

Protein sample:

20mM Tris pH=8.0

50mM NaCl

=10mg/mlProtc

PEGc = 12.5%


Sealed chamber

Tim Grüne 29/87


Vapour Diffusion

cProt = 20mg/ml

c


100mM Hepes pH=7.0

Reservoir solution:

20mM CaCl 2

25% PEG 3350

1µl1µl

Protein sample:

20mM Tris pH=8.0

50mM NaCl

=10mg/mlProtc

PEGc = 12.5%


Sealed chamber

Tim Grüne 30/87


Vapour Diffusion

It is usually impossible to predict the conditions that will result in crystals of the macro-

molecule.

Therefore one tests a large number of random conditions (matrix screen).

The vapour diffusion method is the most popular crystallisation method because it is easy

and fast to set up and has even been automatised to a large extent (1000 conditions in 1hr

per robot; manually about 50 conditions per 1hr).

Tim Grüne 31/87


Liquid Phase Diffusion

��

��

Dialysis

button

Protein

sample

Dialysis membrane

O−ring (seal)

solution

Reservoir

The MWCO (molecular weight cut-off) of

the dialysis membrane must be smaller

than the protein size.

By exchanging the reservoir, the condi-

tions can be very finely tuned.

Awkward to set up, requires large

amounts (≥ 5µl) of sample.

Dialysis buttons are well suited to improve/ fine-tune known crystallisation conditions.

Tim Grüne 32/87


Further Reading: Crystallisation of Macromolecules

• Drenth, Principles of Protein X-Ray Crystallography (Springer, 2007)

• Rupp, Biomolecular Crystallography: Principles, Practice, and Application to Structural

Biology (Garland Science, 2009)

• Documentation at www.jenabioscience.com

• Documentation at www.hamptonresearch.com

Tim Grüne 33/87

http://www.jenabioscience.com

http://www.hamptonresearch.com


X-Rays

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X-rays: Electromagnetic Waves

Like visible light, UV-radiation, or radiowaves, X-rays are electromagnetic waves.

800nm 400nm

Radio Micro Infrared X−raysVisible UV −raysγ

30cm10km 1mm 1nm 10pm

wavelength

123keV1.23keV3.09eV1.54eV0.00123eV4.12µV energy

According to the formula E = h cλ, a wave with a long wavelength λ has low energy E and vice versa.

The energy of X-rays lies usually between 0.5-2 Å.

Physicists measure the energy of electromagnetic waves in electronvolt, eV . 1eV = energy of one electron (or proton) accelerated

through 1V .

Tim Grüne 35/87


Why X-Rays?

Why do we use X-rays for structure determination?

• As a rule of thumb, light can only used to visualise objects greater than at least half the

wavelength of that particular light, e.g. visible light/ light microscopy (λ > 400nm) can

only be used to see objects greater than 200nm.

• The typical distance between atoms in (macro)molecules is about 1.5 Å - 2 Å. Therefore

the wavelength to investigate molecules must be below 4 Å.

• Typically X-rays between 0.5 Å and 2 Å are used for X-ray experiments with macromolec-

ular crystals.

Tim Grüne 36/87


Carrying out an X-ray Experiment

X−raysource waves

X−ray

(sample)Crystal

Detector

beamstop(d

iffr

action)

The X-rays from an X-ray source

are “filtered” to a single wave-

length (monochromatic X-rays)

and focussed as much as (tech-

nically) possible.

Crystallography does not observe a direct image of the sample.

The crystal diffracts the X-rays which are collected as spots on the detector.

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Result of a Diffraction Experiment

• The reflections (= spots) are the data we seek to

measure: Their position and their intensity.

• The dark ring stems from scattering of solvent in

the crystal. It always lies between about 3 and 4 Å

and can be used as rough guideline for the reso-

lution of a diffraction image. However it reduces

the quality of the data and one tries to reduce the

intensity of this water ring.

The spots are the result of the interaction of the X-rays with the periodic nature of the crystal.

Tim Grüne 38/87


Light vs. X-rays

Screen

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visible light

image(focussing) lenseobject

Lenses allow us to build microscopes, telescopes, to actually see (with our own eyes’ lenses).

Tim Grüne 39/87


Light vs. X-rays

We are forced to use X-rays (wavelength λ = 0.5− 2 Å) because we want to resolve atoms

with bond distances around 1.5 Å.

��

��

��

��

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objectobject

X−raysScreen

no lense = no image, only "blur"

Lenses for X-rays do not exist.

Therefore, X-rays cannot be fo-

cussed as light can and there are not

microscopes for X-rays. Otherwise,

we could look at single molecules un-

der a microscope (and we could skip

the rest of this lecture. . . ).

Tim Grüne 40/87


Crystals and X-rays

The “blur” contains no useful information that could help us reconstruct the image of the tree.

This changes in the case of crystals:

Their periodic composition — made

up of myriads of unit cells — causes

spots (reflections) to appear on top of

the “blur”.

How this happens will be explained

later during this course.

Tim Grüne 41/87


Generating X-rays

There are two main methods to generate X-rays for crystallographic purposes:

Inhouse sources like rotating anodes. micro sources, or sealed tubes. A beam of electrons

directed at a heavy metal anode initiates the transition of inner shell electrons. Their

return to the ground state produces X-radiation.

Synchrotrons Bending of Electron Beam: An electron beam forced by a magnetic field to

drive a curve generates X-rays. This principle is exploited at Synchrotrons.

Tim Grüne 42/87


Rotating Anodes

Hitting metal (Cu, Mo, Cr,. . . ) with electrons generates two types of radiation:

1. bremsstrahlung due to the deceleration of

electrons

2. radiation due to shell transitions, usually from

L to K.

The metal is called an anode because it is posi-

tively charged to attract the electrons.

It is rotating because this facilitates cooling of the

anode which allows to generate a stronger beam.

That’s why these machines are called rotating an-

odes.Images courtesy of Jan-Olof Lill

Tim Grüne 43/87


Rotating Anodes

Inte

nsity

Wavelength [pm]

http://en.wikipedia.org/wiki/X-ray tubeRh-spectrum

Kα

Kβ

The bremsstrahlung creates a broad

spectrum at medium intensity.

The shell transitions create sharp

peaks at high intensity. The main

peak is filtered from the rest and used

for the measurement as monochro-

matic light.

Tim Grüne 44/87

http://en.wikipedia.org/wiki/X-ray_tube


Typical Inhouse Machine

Tim Grüne 45/87


Generation of X-rays: Rotating Anode

The wavelength generated from rotating anodes is exact and fixed. It can only be modified

by exchanging the type of heavy metal in use (i.e. using a different machine).

Some common metals and their wavelengths:

Metal wavelength λCopper Cu 1.5406 Å high intensityMolybdenum Mo 0.7093 Å small molecules (higher resolution)Silver Ag 0.5609 Å charge densityTungsten W 0.1795 Å medical applications (e.g. at the dentist)

Tim Grüne 46/87


Generation of X-rays: Synchrotrons

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e−

e−

e−

e−

S

SN

N

"Light"

to X−Rays)Vacuum tube

Beamlines

(from Infrared

electrons

Magnets

Electrons are circled inside a vacuum tube. At bends they generate a wide spectrum of

electro-magnetic radiation, from infrared to X-rays. The beamlines (experimental stations)

select the desired wavelength.

Tim Grüne 47/87


Synchrotron vs. Inhouse

+ Synchrotron radiation is much stronger than inhouse sources. A full data set can

be collected in minutes as opposed to hours or days with an inhouse source.

+ Synchrotrons allow to select (tune) the wavelength. This is important for the

phasing step.

- Inhouse sources are often more stable and deliver more accurate data.

- Inhouse sources often allow more advanced settings of crystal and detector with

respect to each other, resulting in higher data quality (but not higher resolution

data).

Tim Grüne 48/87


Cryo-Crystallography

Tim Grüne 49/87


Cryo-Crystallography

The quality of data measured from X-ray crystallography has been greatly improved with the

introduction of cryo-crystallography.

The crystals are cooled to 100K (or less) during data collection.

Tim Grüne 50/87


Room Temperature Measurement: Capillary

Radiation damage by beam

E. Garman & T.R. Schneider, Macromolecular Cryocrystallography, J. Appl. Cryst. (1997). 30, 211-237

At room temperature the crystal must be kept in a humid atmosphere and is therefore mounted

in a glass capillary.

Tim Grüne 51/87


Reasons for Cryo-Crystallography

Crystal with visible consequences of

radiation damage after data collec-

tion at a synchrotron.

From E. Garman, Radiation damage in macromolec-

ular crystallography: what is it and why should we

care?, Acta Cryst. D66 (2010), p. 339

Tim Grüne 52/87


Reasons for Cryo-Crystallography

• Radiation causes radiation damage, i.e. the breaking of covalent bonds and the

generation of free radicals. This degrades the crystal. Radiation damage is not

removed but at least greatly reduced at 100 K compared to room temperature.

• The thermal motion of the atoms is reduced. Thermal motion (vibration of the

atoms) reduces the intensity of the spots at high resolution.

• Sample preparation is actually easier when frozen than at room temperature.

Tim Grüne 53/87


Sample Preparation

Macromolecular crystals always contain water. Water crystallises when it is frozen, and the

ice crystal lattice would destroy the protein crystal (they are not compatible).

Sample image with ice rings.These ice rings are actually due to superficial ice(inset image) because of a poorly adjusted or wetcryo stream.

Courtesy Stephen Curry, Imperial College London

Therefore the formation of ice crystals must be prevented by the addition of a cryo-protectant.

Tim Grüne 54/87


Sample Preparation

298K 120K, no cryo 120K, cryo

Images from E. Garman & T.R. Schneider, Macromolecular Cryocrystallography, J. Appl. Cryst. (1997). 30, 211-237

Tim Grüne 55/87


Sample Preparation

Common cryo-protectants:

glycerol PEG400 MPDsucrose 2,3-butanediol Na-malonateLiCl (2M)

Required concentration ranges between 15% and 35%, depending on the composition of the

mother liquor, and the minimum required amount should always be tested beforehand without

a crystal.

Tim Grüne 56/87


Further Reading: Freezing Crystals

Rodgers, D.W., Practical Cryocrystallography, chapter 14 in Methods in Enzymology, Vol.

276A (1997)

Garman, E.F. and Schneider, T.R., Macromolecular Cryocrystallography, J. Appl. Cryst.

(1997), 30, p. 211

Tim Grüne 57/87


Diffraction Theory

or: why do we observe these spots?

Tim Grüne 58/87


The Unit Cell

The unit cell is the smallest unit from which we can form the crystal solely by translations

(shifting).

→ →

a

γ

β

c

b

α

The unit cell is characterised by the three side lengths, a, b, c and angles α, β, γ.

α: angle between b and c

β: angle between c and a

γ: angle between a and b

Tim Grüne 59/87


Unit Cell: an X-ray Amplifier

The regular repetition of the unit cell acts as an amplifier of the X-rays and thus (indirectly)

circumvents the problem of the missing X-ray lense.

Tim Grüne 60/87


X-Ray meets Electron

X−raysource

X−ray electronwaves

ϑ

The X-rays from the source are plane waves An

electron in the crystal (sample) reacts to this in-

coming wave by emitting a spherical wave (travel-

ling in all directions) of much weaker intensity.

The wave intensity is distributed as 12(1 + cos2 ϑ) around the electron, but this is not important for further understanding.

Tim Grüne 61/87


Wave Emitted by the Electron

The wave emitted by the electron is an electromagnetic wave. The electromagnetic field

travels away from the electron.

The description as wave is merely a

mathematical trick to simplify the cal-

culations. The observed intensity of

the wave is the square of the ampli-

tude. Therefore, a light-source does

not flicker.

Tim Grüne 62/87


Multiple Waves: Interference

Multiple electrons emit one wave each. The resulting wave is again a wave, but this time it is

more complicated. It is an interference pattern.

In some directions the amplitude get stronger (construc-

tive interference), but in some directions the amplitude

stays 0 at all times (destructive interference).

Note that the electrons are aligned in a regular pattern,

just like the unit cells in a crystal.

The more electrons there are the more destructive interference occurs and only certain direc-

tions remain where a signal can be detected. This is the origin of the distinct spots observed

with an X-ray crystallography experiment.

Tim Grüne 63/87


The Laue Conditions

The Laue Conditions are the main tool to predict whether or not a crystal diffracts in a certain

direction and are also the basis for the interpretation and measurement of diffraction data.

Tim Grüne 64/87


The Laue Conditions

aX−rays

inincoming

De

tecto

r

b

• Crystal and Unit Cell in some orientation

• Incoming X-rays at wavelength λ

• We want to find out if there is a reflection on thedetector at the circled position:

1. Draw input vector with length 1/λ to centreof crystal

2. Draw output vector with length 1/λ fromcentre of crystal to point on detector.

3. Scattering vector ~S = difference between outand in

4. The angle between input and output vectoris called 2θ. θ is the scattering angle (the “2”is explained shortly).

• A different point on the detector results in a dif-ferent scattering vector ~S′.

Tim Grüne 65/87


The Laue Conditions

(1/λ

)

out

a

bincoming

X−rays

De

tecto

r

(1/λ)

direct

ion o

f

obse

rvatio

n2θ

in









Tim Grüne 66/87


The Laue Conditions

ina

bincoming

X−rays

out

direct

ion o

f

S

2θ

De

tecto

r

obse

rvatio

n









Tim Grüne 67/87


The Laue Conditions

2θ′

2θ

a

bincoming

X−rays

out

direct

ion o

f

obse

rvatio

n

anoth

er direct

ion

of obse

rvatio

n

De

tecto

r

out S’

in









Tim Grüne 68/87


The Laue Conditions

The scattering vector ~S carries information about the direction of the incoming beam, the

wavelength λ and the position on the detector we are interested in. The unit cell vectors

~a,~b,~c define how the unit cell is oriented with respect to the incoming beam.

There is a reflection spot on the detector at the

position described by the scattering vector ~S only

if there are three integers h, k, l such that:

1. |~S||~a| cos(∠(~S,~a)) = h

2. |~S||~b| cos(∠(~S,~b)) = k

3. |~S||~c| cos(∠(~S,~c)) = l

Equations 1-3 are called the Laue Conditions.Tim Grüne 69/87


The Laue Conditions

The Laue conditions are if-and-only-if conditions:

• There is a spot on the detector if the numbers h, k, l are all integers.

• Each integer triplet (h, k, l) corresponds to uniquely one reflection.

An integer triplet (h, k, l) is called the Miller index of the corresponding reflection.

Tim Grüne 70/87


The origin of “2” in 2θ

inS

θ

a

bout

θ in θout

2θ

in

out

By rotating the picture on the left by θ, the incoming and the outgoing wave vectors become

much more symmetrical and the picture looks like a light-ray reflected by a mirror plane. Like

in optics the θin = θout = θ. This also justifies the term “reflection” for the diffraction spots.

Tim Grüne 71/87


Lattice Planes

There is a connection between the aforementioned “mirror plane” and the Miller indices. Con-

sider the crystal lattice with the unit cell highlighted in green:

• Pick one corner of the unit cell.

• Pick a corner from a second unit cell (in

3D, pick two other ones)

• Shift the line (plane) so that it hits all unit

cell corners as long as it passes through

the original unit cell.

Tim Grüne 72/87


Lattice Planes





3D, pick two other ones)




Tim Grüne 73/87


Lattice Planes





3D: two other ones)




Tim Grüne 74/87


Lattice Planes





3D: two other ones)




Tim Grüne 75/87


Lattice Planes



a

b

The planes divide the side ~a 1x, the ~b side

2x, and the ~c side 0x.

The planes we thus constructed are the mir-

ror planes for the reflection with the Miller

index (1,2,0).

From the incoming beam direction and the

unit cell we could now predict the orienta-

tion of the crystal in the beam so that the

reflection (1,2,0) can be collected.

Tim Grüne 76/87


Lattice Planes

For every such plane (which runs through three unit cell corners) there is a scattering vector~S and integer Miller indices (hkl) which fulfil the Laue conditions.

Any other plane never fulfils the Laue conditions.

The construction also helps to understand the resolution limit of a realistic crystal.

Tim Grüne 77/87


Bragg’s Law

Another important consequence from the Laue conditions is Bragg’s Law:

θ

θ

d

In order that the reflection that belongs to the pur-

ple lattice planes can be measured, the planes

(and hence the crystal) must be oriented to the

beam such that

λ = 2d sin θa

d : distance between two adjacent planes. It is

called the resolution of the reflection.

λ : wavelength of the X-raysaThe exact law is nλ = 2d sin θ, but n > 1 corresponds to multiplerefraction in the crystal and can usually be neglected.

Tim Grüne 78/87


Spot Position and Intensity

Bragg’s law and the Laue conditions depend on the unit cell parameters ~a,~b,~c, but not the

unit cell content, i.e. the molecule inside.

The diffraction pattern tells us about the unit cell parameters ~a,~b,~c.

The spot intensities tell us about what is inside the unit cell.

Tim Grüne 79/87


Spot Position and Intensity

dd

A A’

B’B

Atoms A and its corresponding atom A’ in the next

unit cell are both on the plane (120) and contribute

with their small waves to the spot (120).

The shifted atoms B and B’ contribute to the same

spot (the shift does not change the Laue conditions!).

Depending on the small shift, the contribution interferes constructively or destructively and

therefore changes the spot intensity: Its intensity changes depending on the number and

positions of the atoms inside the unit cell, i.e. depending on the molecule in the unit cell.

Tim Grüne 80/87


Resolution Limit: Theory and Practice

Bragg’s law λ = 2d sin θ sets a lower limit for the plane distance d that can be measured

with a fixed wavelength λ:

d =λ

2 sin θ≥λ

2

This assumes a perfectly ordered crystal. Unfortunately, the molecules inside the crystal do

not know about crystallography and the concept of the unit cell (or they do and only want to

tease you).

Tim Grüne 81/87



A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic

crystal, however, only as a limited order, and spots with a small lattice distance d are not

formed beyond a certain limit, the resolution limit of the crystal.

Tim Grüne 82/87



A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic

crystal, however, only as a limited order, and spots with a small lattice distance d are not

formed beyond a certain limit, the resolution limit of the crystal.

Tim Grüne 83/87


Sample Images

• Resolution: 1.5 Å at edge

• Cell: a = 92.6Å, b = 92.6Å, c = 128.9Å, α =

β = 90◦, γ = 120◦

• sharp and small spots

• Some overloads (saturated counter)

• white bar: beam stop

• white lines: detector tiling

Tim Grüne 84/87


Sample Images

• Resolution: 2.5 Å at edge

• Cell: a = 111.7Å, b = 80.5Å, c = 70.3Å, α =

γ = 90◦, β = 94.2◦

• Smeared spots (very common)

• Ice rings (from cryo stream or poor

freezing)

• Multiple lattices (twin)

Tim Grüne 85/87


Sample Images

• Cell: a = 10.56Å, b = 11.64Å, c = 16.14Å,

α = β = γ = 90◦

• Small cell ⇒ few (large) spots (but

beyond the edge of the detector)

Tim Grüne 86/87


Further Reading: Diffraction Theory

• Drenth, Principles of Protein X-Ray Crystallography (Springer, 2007)

• T. L. Blundell & L. N. Johnson, Protein Crystallography (Academic Press London, 1976)

Tim Grüne 87/87

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