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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
tg@shelx.uni-ac.gwdg.de
Tim Grüne 1/87
GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
Learning from Structure: Some Applications of Crystallography
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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).
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
“Terms and Conditions”
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GGNB Course A57: Macromolecular Crystallography
“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.
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GGNB Course A57: Macromolecular Crystallography
“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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
Outline of X-ray Structure Determination
Data
Deposition
Refinement
& building
collection
Data
Phasing
Crystal
growth
density map
Electron
Validation
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
Crystal Types
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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 .
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
Crystal Growth
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
Crystallisation Methods
The most common crystallisation methods are
1. vapour diffusion
2. liquid phase diffusion
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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).
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GGNB Course A57: Macromolecular Crystallography
Liquid Phase Diffusion
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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.
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
X-Rays
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GGNB Course A57: Macromolecular Crystallography
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 .
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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).
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GGNB Course A57: Macromolecular Crystallography
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. . . ).
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
Typical Inhouse Machine
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GGNB Course A57: Macromolecular Crystallography
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)
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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).
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GGNB Course A57: Macromolecular Crystallography
Cryo-Crystallography
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
Sample Preparation
298K 120K, no cryo 120K, cryo
Images from E. Garman & T.R. Schneider, Macromolecular Cryocrystallography, J. Appl. Cryst. (1997). 30, 211-237
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
Diffraction Theory
or: why do we observe these spots?
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GGNB Course A57: Macromolecular Crystallography
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
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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.
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GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
The Laue Conditions
(1/λ
)
out
a
bincoming
X−rays
De
tecto
r
(1/λ)
direct
ion o
f
obse
rvatio
n2θ
in
• 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 66/87
GGNB Course A57: Macromolecular Crystallography
The Laue Conditions
ina
bincoming
X−rays
out
direct
ion o
f
S
2θ
De
tecto
r
obse
rvatio
n
• 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 67/87
GGNB Course A57: Macromolecular Crystallography
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
• 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 68/87
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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 73/87
GGNB Course A57: Macromolecular Crystallography
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: 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 74/87
GGNB Course A57: Macromolecular Crystallography
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: 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 75/87
GGNB Course A57: Macromolecular Crystallography
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:
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
Resolution Limit: Theory and Practice
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
GGNB Course A57: Macromolecular Crystallography
Resolution Limit: Theory and Practice
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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
GGNB Course A57: Macromolecular Crystallography
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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