Obtaining the diffraction patternObtaining the diffraction pattern

Instruments

Collecting diffraction data

Fourier analysis and the phase problem

Measurement of the diffraction pattern

Information obtainable: Angles (or direction) of scattering (2) : used to measure the size and shape of the unit cell. Intensities of diffracted beams : analysis gives position of atoms within unit cell (expressed as fractions of unit-cell edges)The diffraction pattern measured is unique to the atomic identities and arrangement in a particular crystal.

Maximum number of reflections is 32V/33n, where V is unit cell volume, is wavelength of radiation used and n is a constant (1 for primitive, 2 for body-centred or end-centred and 4 for face-centred)

Instruments & data collection strategiesGenerators and detectors X-rays are produced by focussing a beam of electrons on to a metal target (anode) in a vacuum. Electrons are produced by a heated filament (cathode) and accelerated by electric field towards anode. Electrons cause transitions in the metal atoms of the anode, which release photons of ‘characteristic’ radiation as well as a general spectrum (‘white’ radiation). Commonly use Cu K (1.542 Å)

K

K

X-ray sources (In-house)

• Typically used with image plate detectors

• Efficient cooling is achieved by Rotating Anode generators - metal plate revolves during the experiment

• Data collection is typically 30minutes per degree exposure UWC - Rotating anode

generator

Rotating anode generators used to generate intense X-rays for lab use.More intense radiation can be obtained at synchrotrons.

“Home” source useful for

• Characterizing new crystals (determining unit cell and symmetry)

• Determine freeze properties and cryo-conditions• Assessing potential heavy atom derivatives• Assessing potential substrate/inhibitor/drug

derivatives• Routine “low priority” data collection

Advantages

• Cheap• Permanent access• Very low radiation damage• Can save huge quantities of time at synchrotron

(increase efficiency)

Synchrotron sources• These are large-scale facilities which use a particle accelerator to produce a

continuous spectrum of high-energy X-rays

If electrons are accelerated to near the speed of light in a ring shaped particle accelerator called an electron storage ring and then quickly decelerated by forcing them into a curved trajectory, they emit their excess radiation in the form of a powerful stream of X-rays tangential to the ring; this is called synchrotron radiation

The particle accelerators necessary to perform this function are large, extremely expensive and are located in a small number of large-scale facilities internationally such as at APS (USA), Grenoble (France), Diamond (England), Trieste (Italy) and Hamburg (Germany), Spring 8 (Japan)

The European Synchrotron Radiation Facility (ESRF)

at Grenoble

Advantages of synchrotron radiationcompared to the beam of X-rays which can be generated from the laboratory, the beam from a synchrotron has some thousand fold greater energy

more highly collimated which means that the diffraction spots measured are both smaller and more intense, thus enhancing the resolution and precision of the diffraction pattern

data-sets consisting of thousands of Bragg reflections can be collected in minutes rather than the hours required with laboratory sources, , typically used with CCD detectors

The synchrotron beam is extremely narrow and intense allowing collection of data from much smaller crystals than can be used with a laboratory-scale instrument. Moreover, the beam is TUNABLE and can generate a wider range of wavelengths than laboratory-scale sources which may be used in multiple wavelength anomalous diffraction experiments for phase angle calculations.

X-ray detectors respond to the diffracted beam intensities. Original : photographic filmMore common : diffractometer (allows crystal orientation and detector to be controlled)Image plate stores diffraction data temporarily

•Atoms promoted to an excited state when they absorb an X-ray photon•Diffraction data read off by fluorescent technique : intensity of fluorescence total X-ray energy that arrived at each point•Plate ‘erased’ by flooding it with light

Exposure

Readout

Erase

Mounting crystal for diffractometer

Standard: On glass fibre (a), or two-stage (b) or multiple fibre (c).

Air-sensitive: In a capillary tube (with mother liquor, if necessary)

X-ray source Diffractometer• Freeze a crystal on a loop & mount in an X-ray beam.

The crystal environment

Choosing cryoprotectant

Usually the mother liquor with an anti-freeze agent added (e.g. glycerol, ethylene glycol, MPD, light PEGs) in high enough concentration to prevent the formation of ordered ice and to promote vitrification of the sample.

A test with the putative cryo-solution, flash-cooled without the crystal, will give the correct concentration needed

A drop, rather than a film, should be used, as this better simulates the presence of a crystal. A clear film in the loop is a necessary but not sufficient criterion. A diffraction image should always be taken to check for diffuse rings from ice crystallites.

Soaking methodsTransfer the sample from its growing drop straight into the final concentration of cryo-solution and leave it for between 30 s to 4 minutes.

soak time empirical; a longer soak allows more time for equilibration but prolongs the time for which the cryo-solution can degrade the crystal, whereas a short soak maximises the osmotic shock to the crystal but minimises the degradation time.

Times between 0.5 s (crystal dragged through cryo-solution) to several days have been used successfully, and there is a huge variation in soaking times used in different laboratories practising crystallography. Sequential soaks in increasing concentrations can also be used, and these lessen the osmotic shock to the crystal.

X-rays for crystallography aren’t very penetrating – almost all absorbed in a few mm of protein or water. But whenever an X-ray is scattered incoherently, it destroys that molecule. A crystal contains many molecules (ca. 1015) so many photons can be scattered before the overall structure is destroyed.

Ideally, all diffraction data will be obtained before radiation damage is significant, but it may be necessary to combine measurements from many crystals to obtain a complete data set.

Radiation damage

At fixed , a Bragg reflection occurs only if the incident beam makes exactly the correct angle with the reflecting planes. To overcome disorder in a real crystal, the crystal is rotated through this reflecting position and the diffracted beam capturedRotating the crystal through 1-2°, only a small number of reflections is recorded so they can be measured separately and accurately.

detectorIncident X-ray

xtal

2 max

Moving the detector back improves accuracy (reduces background) but reduces the range observable (hence lower resolution)

Compromise needed to maximise diffraction data obtained.

Quality of intensity data

To assess the effect of compromises in measurement a measure of quality is needed.Often use R- statistic which measures accuracy as a ratio (averaged over large number of cases):

Mean difference between values which should be the sameMean magnitude of measured values

Rsym measures agreement between reflections required to be equal by symmetry (eg. 2-fold axis // b requires (h, k, l) = (-h, k, -l)

Rmerge measures agreement between multiple measurements of the same reflection during the course of data collection

Both R-factors have statistical problems and may understate the true accuracy

Typically, Rsym is between 0.05 – 0.20

Important that all measured data are included in calculating R

Resolution limit

Completeness and Redundancy of data

Intensities are measured to a fixed limit of resolution, usually as high as the crystal will allow. Higher resolution gives more detailed, accurate electron-density maps.Use Rsym as guide for cut-off

There is always a limit to the resolution obtainable. Resolution is expressed ito interplanar spacing d = /2 sin , corresponding to the maximum observed 2 values ( = 1.542 Å for Cu)

Resolution refers to being able to distinguish individual parts of a molecule: small molecule structures are determined to a “resolution” of 0.8 – 1.0 Å (can see each atom); proteins seldom better than 1.5 Å.

Completeness = number of unique reflections measured

total number of unique reflections

Total # of intensity measurementsTotal # of unique reflections measured

Redundancy =

Collecting the diffraction pattern

Collection of X-ray diffraction data gives a list of “reflections”:Indices h, k, l Intensity (I) std deviation on I

Relative scale

• Angular position (2) depends on dimensions of crystal lattice only• Intensity of reflection depends only on nature & arrangement of atoms within unit cell

To deconvolute the intensities of reflections into crystal structure, we must consider the combination of scattered waves the intensity scattered at any angle can be calculated from the sum of the waves scattered from different atoms (ie. Each diffracted beam contains information of the entire atomic structure of the crystal)

Structure FactorA value termed the structure factor and it is usually denoted F describes the phase and amplitude of a diffraction spot at the hkl position, representing the total X-ray wave scattered by the crystal in a given direction

The value is termed so as it depends on the arrangement (structure) of the atoms in the unit cell, or more accurately it is a function of the electron density distribution in the cell.

The structure factor of a reflection at position hkl is defined as:

Equation

Fhkl = |Fhkl|exp [ihkl)]

where |Fhkl| is the structure factor amplitude and hkl) is its phase.

The Fourier Transform• Each diffraction spot is the result of interference of all X-

rays with the same diffraction angle emerging from all atoms

• For a typical protein crystal, myoglobin, each of the 20, 000 diffracted beams that have been measured contains scattered x-rays from each of the ~1500 atoms in the molecule

• To extract information about the individual atoms from such a system requires considerable computation. The mathematical tool used to handle such problems is called a Fourier Transform

From a diffraction pattern…to electron density

• Diffraction pattern and the object diffracting the waves are related through a mathematical operation: the fourier transform

• Structure factors in reciprocal space :- correspond to fourier transform of electron density distribution

• * Fourier transfrom is reversible! Can use computers to get a picture of electron density

Fourier analysis of scattering density

• The parts of the crystal that scatter the X-rays are the electron clouds around the bonds and atoms– electron density function (x,y,z)

• Any periodic function can be represented as a sum of cosine functions, with different amplitudes Fh and phases h

• So determining the Fh and h for all h is equivalent to determining (x)

(x) = Fh cos(360º h x + h)h=-∞h=∞ h can

be any angle between 0 and 360º

Electron Density and Structure Factor

It can be shown mathematically that the |Fhkl| (Amplitude) and hkl (Phase) are exactly the values needed to reconstruct electron density

1.Graphical representation of waves2. Algebraic representation3. Vectorial representation4. Exponential representation (Complex numbers)

Superposition of waves

1.Graphical representation of waves

When radiation is scattered with preservation of phase relationships, the amplitude of the resulting beam in any direction is the sum of the individual waves scattered in that direction, taking into account their relative phases. The phase may be calculated by seeing where the maximum in the resultant wave occurs, relative to some arbitrary origin.

Not convenient with computers, so need an analytical approach