protein crystallography part i - gwdgshelx.uni-ac.gwdg.de/~tg/teaching/molbio/2005/day1.pdffrom...

Protein CrystallographyPart I — Technical Aspects of and Introduction to X–ray diffraction

Tim GrüneDept. of Structural Chemistry

Head of Dept.: Prof. G. SheldrickUniversity of Göttingen

November/December 2005http://shelx.uni-ac.gwdg.de

[email protected]

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

Overview

Crystals and Protein Crystals

Growing Protein Crystals

X–ray diffraction

Inside Crystals

Bragg’s Law and Resolution

From X-rays to Electron Density

Molecular Biology Course 2005 1 Protein Crystallography I


Examples of Crystals



X–ray diffraction

Inside Crystals


From X-rays to Electron Den-

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Ice Crystals Salt Crystals Artificial Monocrystal fora kJ Laser

Crystals from inorganic and especiallyionic compounds (like NaCl) tend to berather stable. They can be kept at roomtemperature, for a long time and in a dryenvironment.

Quartz Crystal



Protein Crystals



X–ray diffraction

Inside Crystals



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Protein Crystals are generally more sensitive then minerals. They are usuallygrown from solution and must be kept in a humid environment throughout theexperiment.

Protein Crystals grown in solution.

If proteins are not handled with care, they quickly degenerate/aggregate



Protein Crystals



X–ray diffraction

Inside Crystals



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Proteins are generally more sensitive then minerals. They are usually grown fromsolution and must be kept in a humid environment throughout the experiment.

Protein Crystals grown in solution.

If proteins are not handled with care, they quickly degenerate/aggregate



Growing Protein Crystals — Vapour Diffusion



X–ray diffraction

Inside Crystals



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In order to grow crystals from a protein solution, the protein must be very pure(≥ 90–95%) and very concentrated ( ≈ 15 − 25mg

ml , ranges go from 5mgml to

> 100mgml ). The probably most common method is called the vapour diffusion

method.

equilibration

reservoirsolution

before equilibration

protein/reservoir

after equilibration

The reservoir contains a buffer, a precipitant ( salts, PEG’s with M∅r = 400–

20,000 Da, organic solvents) and additives (small molecules/salts that help pack-ing, e.g. MgCl2).The drop is a (1:1)–mixture of protein and reservoir solution (mother liquor).Volatile components are exchanged until equilibrium is reached. Since at thebeginning the precipitant is diluted in the drop, the drop is being concentratedduring equilibration.



Growing Protein Crystals — Other Techniques



X–ray diffraction

Inside Crystals



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Batch method: Protein and reservoir are mixeddirectly and covered with a layer of oil⇒ no or little equilibration.

oil seal

Diffusion through Dialysis Membrane: Proteinsolution and reservoir separated by a highmolecular weight cut-off membrane⇒ also non–volatile components are ex-changed.

membrane

Diffusion through Agarose: Reservoir and Pro-tein separated through a layer of agarose⇒ concentration gradient

agarose



Growing Protein Crystals — Comparison



X–ray diffraction

Inside Crystals



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Vapour Diffusion BatchPRO:• Fast setup — robots can do up

to 100 drops in 30 min• Small amount ( 1nl / drop with

robot)

PRO:• Fast setup• water permeable oil allows con-

centration above initial proteinconcentration

CONTRA: Sudden mixing may cause protein to aggregate

Dialysis Agarose

PRO:• exchange of non–volatile com-

ponents• high degree of control about fi-

nal conditions and diffusion rate

PRO:• exchange of non–volatile com-

ponents• gradient of both protein and

reservoir: infinite number ofconditions in one setup

CONTRA: Slow setup; requires large amounts of protein


X–rays

What are X–rays?



X–ray diffraction

Inside Crystals



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• (Visible) light is composed of electromagnetic waves: every colour has awavelength/energy

• X–rays are the same, but with higher energy, i.e. shorter wavelength

The spectrum of electromagnetic waves

Radio Microvisible

UV X-rays γ-raysInfra-redwavelength λ

10km1013Å

energy

125 keV1.25 keV1.25 eV4.1µeV1.6–3.1eV

30cm3 · 109Å

1mm107Å

800-400nm8− 4 · 103Å

1nm10Å

10pm0.1Å

Typical X-ray experiment:≈1Å

X–rays are generated by accelerating or decelerating electrons. This happenseither in rotating anodes or at synchrotrons.


X–rays

X–ray Diffraction — Principle of an experiment



X–ray diffraction

Inside Crystals



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X-ray beam

λ ≈ 1Å(0.1nm)

crystal ≈ (0.2mm)3

Diffraction pattern on CCDor image plate

≈ 1013 unit cells

T. Schneider

The crystals diffracts in all directions. Since the detector cannot cover the fullsphere around the crystal, the crystal is rotated during an experiment.One typical data set consists of tens to hundreds of images covering 90◦ – 360◦.


X–rays

X–ray sources — Rotating anode



X–ray diffraction

Inside Crystals



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The SMART 6000

Focussing unit

Crystal hold

Detector

(goniometer head)

Rotating Anode

Cryostream(100K)

Beam stop

G. Sheldrick

Electrons are accelerated and hit a (rotating) Cu–plate. This induces a transitionof the inner shell electrons which in turn produces characteristic CuKα radiation(1.54 Å)


X–rays

X–ray sources — Synchrotron



X–ray diffraction

Inside Crystals



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The ESRF (European Synchrotron Radiation Facility) Grenoble

G. Sheldrick

Bunches of electrons are circulating at high velocity and bent by strong magnets.This generates X–rays at range wavelengths, depending on the degree of thebent. Some set-ups allow to tune the wavelength (important for MAD– and SAD–phasing, see later in the lecture).


X–rays

X–rays: Diffraction pattern



X–ray diffraction

Inside Crystals



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Screen

object (focussing) lense image

visible light

Light is scattered from an objects in all directions. A lense (e.g., eye, a camera,a microscope, or telescope) collects some of the scattered light and focusses iton a screen (and eventually the retina). This creates an image of the object.


X–rays

X–rays: Diffraction pattern



X–ray diffraction

Inside Crystals



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Screen

object

X-rays

no image - "blur"no lenses for X-rays

For X-rays, lenses do not exist, therefore one cannot create and X-ray image.With a “normal” object, no information would be collected on the screen.

What is special about crystals so that one can retrieve information through X-raysfrom them?


Inside Crystals

The Secret of Crystals — Regularity



X–ray diffraction

Inside Crystals



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A crystal consists of a unit cell which is “piled up” like bricks (but in three direc-tions) to compose the whole crystal.

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

c

ab

γ

αβ

α: angle between b and c

β: angle between c and a

γ: angle between a and b

Irrespective of the lengths and angles a unit cell can always be “piled” withoutgaps.


Inside Crystals

How does the crystal lattice help?



X–ray diffraction

Inside Crystals



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In order to understand why crystals produce more than just a blur under X-raydiffraction, one introduces the concept of crystal planes. The corner points ofall unit cells create the crystal lattice. Three non–linear lattice points (corners ofdifferent unit cells) define a plane.

ba

(2 5)

(1 1)

One set of planes is described by the number of section it divides each side ofthe unit cell into (in the figure, side a is divided into two parts by the green set,and b into five parts). These numbers are called the Miller-Indices (hkl) (threein three dimensions).I.e., the green set of planes has the Miller Indices (250), the purple one theIndices (110).



(Bragg) Reflection



X–ray diffraction

Inside Crystals



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X-rays are reflected at the lattice planes. Since a set of planes consists of ahuge number of planes, constructive interference occurs at certain angles θ. Atall other angles the waves extinct each other. Therefore one observes the spottypattern.

One set of planesinside the crystal

dX-rays θ

The angle θ, the plane separa-tion d and the wavelength of theincident X-rays are connectedby Bragg’s law or the Bragg con-ditiona:

λ = 2d sin θ

aRemark: Bragg’s law can easily be derived from the above pictur by using the fact that the pathdifference between rays reflected from two adjacent planes must be an integer multiple of thewavelength. This leads to the actual law nλ = 2d sin θ, but higher order reflections (n > 1) aregenerally too weak to be detected.



Indexing of Reflections



X–ray diffraction

Inside Crystals



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Given a setup of crystal orientation, detector, and beam, each set of planesleads to one reflection. Therefore the reflection that belongs to one plane islabelled with the same Miller Indices (hkl). Assigning these three numbers toeach recorded reflection is called indexing.

beam 2θmax

crystal

recordedimage

(hkl)

The distance d which can be calculated via θ from Bragg’s law is called the reso-lution of that reflection. The resolution of the reflection with maximal θ determinesthe resolution of the data set.



The Meaning of Resolution



X–ray diffraction

Inside Crystals



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The resolution of data collected by X-ray diffraction is a measure for how muchdetail can be seen. It is related with the plane distance d via Bragg’s law by

d =λ

2 sin θmax

θmax is the maximum angle to which data (i.e. a reasonable number of reflec-tions) could be collected.The final goal of data collection is to calculate an electron density map for thecrystallised molecule. The resolution corresponds quite well to the minimum dis-tance between two atoms that can still be resolved in the electron density map.

high resolution density map

dd’

low resolution density map



Examples for the Resolution of Electron Density Maps



X–ray diffraction

Inside Crystals



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Low Resolution Medium Resolution

High Resolution

G. Sheldrick

The images show three times the same region of a protein map at different res-olutions.N.B.: Crystallographers usually speak of “high resolution” when the number issmall (e.g. 1.2Å) and of “low resolution” when the number is large (e.g. 3Å).



How to calculate an Electron Density from Diffraction Data



X–ray diffraction

Inside Crystals



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An X-ray experiment measures intensities of many reflections. The intensity of areflection (hkl) is proportional to the square modulus of the structure factors:

I(hkl) ∝ |F (hkl)|2

Structure factors are complex quantities introduced in order to facilitate the cal-culation of the electron density, i.e.

F (hkl) = |F (hkl)| eiφ(hkl)

|F (hkl)| is called the amplitude and φ(hkl) the phase of the reflection.The electron density within the unit cell of the crystal is related to the structurefactors via a Fourier transformation:

ρ(x, y, z) =1

Vunit cell

h,k,l=∞∑h,k,l=−∞

|F (hkl)| eiφ(hkl)e−2πi(hx+ky+lz)

The Fourier transform can also be inverted:

F (h, k, l) =∫

Vunit cell

d3x ρ(x, y, z)e2πi(hx+ky+lz)



Limits of Crystallographic Data Collection



X–ray diffraction

Inside Crystals



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If we knew all structure factors, we could calculate a perfect electron density andbuild a perfect model.Obviously it is not possible to collect data from an infinite number of reflectionsat infinite accuracy. This leads to truncation errors of the Fourier transformation,which basically means noise in the electron density map. Even with the bestof equipment there are physical limits why we cannot collect data to arbitraryresolution:

1. Bragg’s Law limits the resolution to λ/2, because |sin θ| ≤ 1

2. Crystal imperfections, mostly mosaicity. Mosaicity describes that the unitcells do not pack without gaps. This leads to a broadening of the signalwhich at high resolution vanishes in the background.

3. Detectors also contribute to a broadening of the signal.

4. Every crystal has a limited size. This means that there is only a limitednumber of sets of planes that can cause reflections. While this is usually nota limiting factor, it becomes an issue for very small crystals ( < 50µm sidelengths)


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