Molecular Biology Programme 2011 - GWDGshelx.uni-ac.gwdg.de/.../molbio/2011/pdfs/mb2011_day1.pdfMolecular Biology PhD Programme 2011 Pol II: Crystal “Snapshots” Several structures

Molecular Biology PhD Programme 2011

Molecular Biology Course 2011

Macromolecular Structure DeterminationPart I: Crystals and X-Ray Diffraction

Tim GrüneUniversity of Göttingen

Dept. of Structural Chemistry

http://[email protected]

Tim Grüne Macromolecular Structure Determination 1/87

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


Learning from Structure: Some Applications of Crystallography



Pol II: Crystal “Snapshots”

Several structures of RNA Polymerase II in differ-ent states of action lead to a concept of the modeof function.

Movie courtesy P. Cramer Lab, LMU Munich



Insulin: Quality Control

• 1982: production of recombinant human insulin (improvementof tolerance compared to bovine insulin)• recombinant and purified human insulin structurally identical• structure based point-mutations of insulin improve function-

ality (e.g. rate of release). An extensive list can be foundat http://de.wikipedia.org/wiki/Insulinpräparat (sorry, Germanpage is by far better than the English one).


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 hyper-activity 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 sophisticated

nonsense in pharmacokinetics and clinical pharmacology, EuropeanJournal of Clinical Pharmacology, 26 (1984), pp. 663–668.

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


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 func-

tion and access towards target.

The antibiotic Thiostrepton in contact with its targetDNA. Image courtesy K. Pröpper.



DNA Double Helix

• X-ray image of fibrous, crystalline DNA by R. Franklin, which led her withco-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).



“Terms and Conditions”



“Macromolecule”

A macromolecule is a protein or nucleic acid compound bigger than a couple of kDa, e.g. a protein consistingof 50 or more residues.

The term macromolecular also includes complexes, e.g. between a protein and a ligand or DNA and an antibi-otic.



“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, andshape.

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 theatoms a molecule or complex consists of.



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 somemight regard NMR and X-ray crystallography as competitive).

This course concentrates on X-ray Crystallography.



Outline of X-ray Structure Determination

Data

Deposition

Refinement

& building

collection

Data

Phasing

Crystal

growth

density map

Electron

Validation



Definition of a Crystal

The International Union of Crystallography (IUCr) defines a crystal as a solid material with an essentially dis-crete 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 thebricks can have arbitrary lengths and the sides can be inclined. But all (crystallographic) bricks must be identicalto each other.



Crystal Types



Crystal Types

All matter (including liquids and gases) is held together by electrostatic interaction, i.e. because of the attractionof positive and negative charge, also crystals. There are different sub-types of interaction. Those which areimportant for crystals can be classified as:

1. ionic2. metallic3. covalent bonds4. van-der-Waals interactions

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



Ionic Crystals

Ionic crystals are composed of negatively charged anions and positively charged cations. The net-charge of anionic crystal is always 0e, otherwise the crystal would fly apart.

NaCl is the simplest example for an ionic crystals:Na passes its outer shell electron to Cl, leaving a pos-itively charged Na+-ion and a negatively charged Cl−-ion. The total energy gain by this transition is 6.4eV .



Metals

Al13e

Al13e

Al13e

Al13e

Al13e

Al13e

Al13e

Elect

ron la

ke

(3 e

lect

rons

per Al−

atom

)Al

13e

The valence electrons dissociatefrom the atom and are sharedamongst all ionic bodies. The va-lence elctrons create an electronlake. This explains the high con-ductivity, elasticity of metals, andwhy they are shiny.



Covalent Bonds

Crystal packing of C (diamond) or Si.

(Usually) two atoms share their covalent electrons to filltheir outer electron shell. E.g. C or Si have four elec-trons in their outer shell and can therefore have up tofour bonding partners. This results in a rather compli-cated network in crystalline carbon and the mechanicalstability of diamonds.



van-der-Waals Interaction

van-der-Waals interaction is the main interaction for macromolecules, not only in crystals but also e.g. in theformation of oligomers in solution.

It is based on the random or accidental displacement of electrons which creates a temporary electric field whichpropagates through adjacent molecules.

A “snapshot” of a charge distribution threeputative, aligned molecules which inducesa temporary dipole moment by which themolecules attract each other. One momentlater the charge distribution might look differ-ent again.



Interaction between Macromolecules and their Environment

• Hydrophobic patches• negatively charged patches• positively charged patches

Schematic view of a protein Surface charge distribution ofthe nucleosome

Macromolecules are much more likely to aggregate than to crystallise.



Crystal Growth



Growing Crystals

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

Salts Drying salt dissolved in water often results in crystals because of the strongionic force

Proteins are difficult to crystallise. Their “natural” solid state is a disorderedaggregate, because the intermolecular forces are relatively weak and the largesurface of the molecule allows many (irregular) orientations.



Crystallisation Methods

Macromolecules are usually crystallised by driving them out of solution by competition with precipitants forsolvent molecules.

Common precipitants aresalts e.g. (NH4)2SO4, NaCl, KH2PO4

organic polymers mostly polyethylen glycol (PEG)alcohols e.g. isopropanol

salting in salting out

salt concentrationpro

tein

so

lub

ility

good for

purification

good for

crystallisation

Example: Precipitation

with salt



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 themetastable zone.Nucleation, i.e. the formation of the initial crystal seed,occurs in the labile zone.At too high protein and/or precipitant concentration, pro-teins aggregate and precipitate without forming crystals.



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 artof crystal growth consists of finding the right right solvent composition.



Crystallisation Methods

The most common crystallisation methods are

1. vapour diffusion2. liquid phase diffusion



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



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



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



Vapour Diffusion

It is usually impossible to predict the conditions that will result in crystals of the macromolecule.

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 andhas even been automatised to a large extent (1000 conditions in 1hr per robot; manually about 50 conditionsper 1hr).



Liquid Phase Diffusion

��

��

Dialysis

button

Protein

sample

Dialysis membrane

O−ring (seal)

solution

Reservoir

The MWCO (molecular weight cut-off) of the dial-ysis membrane must be smaller than the proteinsize.By exchanging the reservoir, the conditions can bevery finely tuned.Awkward to set up, requires large amounts (≥ 5µl)of sample.

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



Further Reading: Crystallisation of Macromolecules

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

• Rupp, Biomolecular Crystallography: Principles, Practice, and Application to Structural Biology (GarlandScience, 2009)

• Documentation at www.jenabioscience.com

• Documentation at www.hamptonresearch.com


http://www.jenabioscience.com

http://www.hamptonresearch.com


X-Rays



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 .



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 thatparticular light, e.g. visible light/ light microscopy (λ > 400nm) can only be used to see objects greaterthan 200nm.

• The typical distance between atoms in (macro)molecules is about 1.5 Å - 2 Å. Therefore the wavelength toinvestigate molecules must be below 4 Å.

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



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 sourceare “filtered” to a single wavelength(monochromatic X-rays) and focussedas much as (technically) 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.



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 usedas rough guideline for the resolution of a diffraction image.However it reduces the quality of the data and one tries toreduce 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.



Light vs. X-rays

Screen

��

��

��

��

��

��

visible light

image(focussing) lenseobject

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



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 distancesaround 1.5 Å.

��

��

��

��

��

��

objectobject

X−raysScreen

no lense = no image, only "blur"

Lenses for X-rays do not exist. Therefore,X-rays cannot be focussed as light can andthere are not microscopes for X-rays. Other-wise, we could look at single molecules un-der a microscope (and we could skip the restof this lecture. . . ).



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: Theirperiodic composition — made up of myriadsof unit cells — causes spots (reflections) toappear on top of the “blur”.How this happens will be explained later dur-ing this course.



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 aheavy metal anode initiates the transition of inner shell electrons. Their return to the ground state producesX-radiation.

Synchrotrons Bending of Electron Beam: An electron beam forced by a magnetic field to drive a curve gener-ates X-rays. This principle is exploited at Synchrotrons.



Rotating Anodes

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

1. bremsstrahlung due to the deceleration of electrons2. radiation due to shell transitions, usually from L to K.

The metal is called an anode because it is positively chargedto attract the electrons.It is rotating because this facilitates cooling of the anodewhich allows to generate a stronger beam.That’s why these machines are called rotating anodes.

Images courtesy of Jan-Olof Lill



Rotating Anodes

Inte

nsity

Wavelength [pm]

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

Kα

Kβ

The bremsstrahlung creates a broad spec-trum at medium intensity.The shell transitions create sharp peaks athigh intensity. The main peak is filtered fromthe rest and used for the measurement asmonochromatic light.


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


Typical Inhouse Machine



Generation of X-rays: Rotating Anode

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



Generation of X-rays: Synchrotrons

��

��

��

��

��

��

��

��

��

��

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 radia-tion, from infrared to X-rays. The beamlines (experimental stations) select the desired wavelength.



Synchrotron vs. Inhouse

+ Synchrotron radiation is much stronger than inhouse sources. A full data set can be collected inminutes 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).



Cryo-Crystallography



Cryo-Crystallography

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

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



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 glasscapillary.



Reasons for Cryo-Crystallography

Crystal with visible consequences of radia-tion damage after data collection at a syn-chrotron.

From E. Garman, Radiation damage in macromolec-

ular crystallography: what is it and why should we

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



Reasons for Cryo-Crystallography

• Radiation causes radiation damage, i.e. the breaking of covalent bonds and the generation offree radicals. This degrades the crystal. Radiation damage is not removed but at least greatlyreduced 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.



Sample Preparation

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



Sample Preparation

298K 120K, no cryo 120K, cryo

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



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, andthe minimum required amount should always be tested beforehand without a crystal.



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



Diffraction Theory

or: why do we observe these spots?



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 cβ: angle between c and aγ: angle between a and b



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 theproblem of the missing X-ray lense.



X-Ray meets Electron

X−raysource

X−ray electronwaves

ϑ

The X-rays from the source are plane waves An electron inthe crystal (sample) reacts to this incoming wave by emittinga spherical wave (travelling in all directions) of much weakerintensity.

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



Wave Emitted by the Electron

The wave emitted by the electron is an electromagnetic wave. The electromagnetic field travels away from theelectron.

The description as wave is merely a mathe-matical trick to simplify the calculations. Theobserved intensity of the wave is the squareof the amplitude. Therefore, a light-sourcedoes not flicker.



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 (constructive inter-ference), 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 theunit cells in a crystal.

The more electrons there are the more destructive interference occurs and only certain directions remain wherea signal can be detected. This is the origin of the distinct spots observed with an X-ray crystallography experi-ment.



The Laue Conditions

The Laue Conditions are the main tool to predict whether or not a crystal diffracts in a certain direction and arealso the basis for the interpretation and measurement of diffraction data.



The Laue Conditions

aX−rays

inincoming

Dete

cto

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′.



The Laue Conditions

(1/λ

)

out

a

bincoming

X−rays

Dete

cto

r

(1/λ)

direct

ion o

f

obse

rvatio

n2θ

in











The Laue Conditions

ina

bincoming

X−rays

out

direct

ion o

f

S

2θ

Dete

cto

r

obse

rvatio

n











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

Dete

cto

r

out S’

in











The Laue Conditions

The scattering vector ~S carries information about the direction of the incoming beam, the wavelength λ and theposition on the detector we are interested in. The unit cell vectors ~a,~b,~c define how the unit cell is oriented withrespect to the incoming beam.

There is a reflection spot on the detector at the position de-scribed by the scattering vector ~S only if there are three in-tegers 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.



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.



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 sym-metrical and the picture looks like a light-ray reflected by a mirror plane. Like in optics the θin = θout = θ. Thisalso justifies the term “reflection” for the diffraction spots.



Lattice Planes

There is a connection between the aforementioned “mirror plane” and the Miller indices. Consider the crystallattice 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 originalunit cell.



Lattice Planes


• 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




Lattice Planes


• 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




Lattice Planes


• 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




Lattice Planes


a

b

The planes divide the side ~a 1x, the ~b side 2x, andthe ~c side 0x.The planes we thus constructed are the mirror planesfor the reflection with the Miller index (1,2,0).From the incoming beam direction and the unit cellwe could now predict the orientation of the crystalin the beam so that the reflection (1,2,0) can becollected.



Lattice Planes

For every such plane (which runs through three unit cell corners) there is a scattering vector ~S and integer Millerindices (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.



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 purple latticeplanes 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 theresolution 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.



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.



Spot Position and Intensity

dd

A A’

B’B

Atoms A and its corresponding atom A’ in the next unit cell areboth on the plane (120) and contribute with their small wavesto the spot (120).The shifted atoms B and B’ contribute to the same spot (the shiftdoes not change the Laue conditions!).

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



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 wavelengthλ:

d =λ

2 sin θ≥λ

2

This assumes a perfectly ordered crystal. Unfortunately, the molecules inside the crystal do not know aboutcrystallography and the concept of the unit cell (or they do and only want to tease you).




A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic crystal, however, onlyas a limited order, and spots with a small lattice distance d are not formed beyond a certain limit, the resolutionlimit of the crystal.




A small lattice distance d corresponds to a long-distance order of the unit cells. A realistic crystal, however, onlyas a limited order, and spots with a small lattice distance d are not formed beyond a certain limit, the resolutionlimit of the crystal.



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



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 freez-

ing)• Multiple lattices (twin)



Sample Images

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

α = β = γ = 90◦

• Small cell⇒ few (large) spots (but beyondthe edge of the detector)



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)


Documents

Molecular Biology Programme 2011 - GWDGshelx.uni-ac.gwdg.de/.../molbio/2011/pdfs/mb2011_day1.pdfMolecular Biology PhD Programme 2011 Pol II: Crystal “Snapshots” Several structures