Obtaining Phase Information

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Obtaining Phase Information
Chapter 6

• (Solutions to The Phase Problem)

Aline Gomez Maqueo
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Background

• Where does this come from?

Braggs Law
However, in reality we have, not one atom per
unit cell, but rather multiple atoms
Each with its own contribution the diffraction
pattern (remember the fj)
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Background

• Where does this come from?

There is a contribution from both real and
imaginary components, that is
And we know that
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Background

• Where does this come from?

There is a contribution from both real and
imaginary components, that is
Where,
We can say, that the structure factor is
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Background

• Where does this come from?

There is a contribution from both real and
imaginary components, that is
So,
Can be expressed,
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The Problem

• From last class we learned that the electron
  density r is

As stated earlier we can rewrite the structure
factor F as
Which gives us
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Oh no!! The phase angle of the reflection!
Houston, we have a phase problem!
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How important is the phase?

• It basically has most of the information.
• (Kevin Cowtans Duck and Cat Story)

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• If you superimpose the ducks intensities with
  the cats phase information and do the back
  transform, you get

You lost most of the duck but you can still make
out the shape of the cat quite clearly. The
information in the phase is very important!
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Techniques for determining phase information

• Guessing (Not practical for more complicated
  molecules)
• Heavy-atom method (isomorphous replacement IR)
• Anomalous Scattering (MAD Phasing)
• Molecular Replacement
• Iterative improvement of phases (Chapter 7)

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Heavy-atom Method

• Each atom in the cell contributes to each
  reflection in the diffraction pattern in
  different amounts.
• This method consists in preparing, in addition to
  the regular crystal, one or more heavy-atom
  derivatives.
• To prepare these derivatives, the crystal is
  soaked in a heavy-atom solution (Hg, Pt, Au), and
  in many cases, these atoms bind to one or more
  specific sites in the molecule.
• Isomorphous replacement derivatives and native
  crystals must be isomorphous.

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Heavy-atom Method (cont)

• Derivatives must be high resolution like the
  native.
• The crystallographer must observe changes in the
  relative intensities of pairs of points that
  appeared reversed in the derivatives diffraction
  pattern with respect to the natives.
• The basic idea is to obtain the difference
  between a single reflection amplitude FP in the
  native data and the corresponding reflection
  amplitude in the FHP from the heavy-atom
  derivative.

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• Imagine creating a difference crystal1 in
  real space

(The big red is the heavy atom)
In reciprocal space we can do the same, and end
up with a difference diffraction pattern with the
same number of reflections
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• You might say, sure, but we still dont know the
  phases, but to solve that problem we use a
  Patterson function, which has pretty much the
  same form as the electron density function, but
  it uses the square of the structure factor.

Electron density (1D)
Patterson function (1D)
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Well, the Patterson function doesnt tell us the
phases directly, but what it does tell us is the
vectors between atoms!

• Quoting Randy Read, if there is a peak of
  electron density for atom 1 at position x1 and a
  peak of electron density for atom 2 at position
  x2, then the Patterson map will have peaks at
  positions given by x2-x1 and x1-x2.

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• Now there is a problem, every heavy atom for
  which we solve the Patterson map gives us
  basically two solutions, that derive from solving
  the quadratic equation

See, both vectors give the same FPH when
squared!

• For this reason we need to obtain the Patterson
  map for at least two heavy-atom derivatives. The
  solution that they have in common would be the
  phase solution for our system. However since
  there is no perfect system, it generally
  necessary to have more than that.

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• Once you have solved the Patterson maps for some
  derivatives, you can determine the phases.
• There is, however, a limited number of heavy
  atoms that you can solve for, because the number
  of vectors for your Patterson map is equal to N2,
  where N is the number of atoms!

Patterson maps are very useful not only for
Isomorphous Replacement but for other
crystallography techniques as well!
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Anomalous Scattering

• This technique also takes advantage of the
  properties of heavy atoms and of Friedels Law.
• According to Friedel, every reflection has its
  mate, with the same intensity but with opposite
  phase.

Friedels Law
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• However, some atoms can absorb X-rays at a
  specific wavelength (absorption edge), and
  instead of diffracting normally, their structure
  factor (amplitude and phase) is changed, and the
  Friedel pairs (hkl and -h-k-l) dont have the
  same intensity. This is called anomalous
  scattering.

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• Once again, as with IR, the phase can be obtained
  by Patterson methods.
• MAD, is Multi-wavelength Anomalous Dispersion, it
  works on the same principle described but
  requires you to be able to change the wavelength
  of the X-ray beam. That way you can tune the
  source to the absorption edge of different atoms
  in your molecule, and obtain difference maps
  between the various data sets.
• MAD Phasing is specially useful for proteins
  containing natural heavy atoms such as the iron
  in a heme, or for proteins expressed in E. Coli
  with Selenomethionine.

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• Why dont we use MAD Phasing using the most
  common components of a protein? For example, C.
• First, because there would be too many atoms to
  solve the Patterson Map.
• And second, because, C, O, N, etc, have
  absorption edges outside the wavelength range
  used for X-ray crystallography or outside of the
  X-ray area altogether.

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Molecular Replacement

• This method consists in using phases of similar
  proteins as models.
• Back to Cowtans cat example

Suppose we dont know the structure of Mr. Cat
but we have his diffraction data.
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But we know Mr. Taillesss structure and we can
calculate his fourier transform.
If we use Mr. Cats diffraction data with Mr.
Taillesss phases, we can approximate the shape
and difference between them.
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• Replacement, in MR stands for repositioning,
  relocating which is exactly what you do, by two
  elementary transformations

Rotation
Translation
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• Phasing models can be used with not only known
  structures but theoretical ones.
• In practice, this try and error method can take a
  lot of time and effort.
• To make sure you are not fabricating data you use
  the R-factor

You take two separate sets of data, one for
manipulating and the other for checking that your
model still fits the data (Rcrys and Rfree).
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References

• Read, Randy. The Phase Problem Introduction to
  Phasing Methods Protein Crystallography Course
  http//perch.cimr.cam.ac.uk/course.html
• Kevin Cowtans Book of Fourierhttp//www.ysbl.yo
  rk.ac.uk/cowtan/fourier/fourier.html
• Rupp, Bernharnd. Crystallography 101 an
  introductory course to Crystallographyhttp//www
  -structure.llnl.gov/Xray/101index.html
• Rhodes, Gale. Crystallography Made Crystal Clear,
  Academic Press, 2nd Ed. 2000 pp. 101-132.

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