Protein Strucutre Determination

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Protein Strucutre Determination

• lectire 6 -- Ewald sphere, data collection,
  scaling

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Scattering factor of an atom
An atom is a spherically symmetrical cloud of
electron density which is densest in the center.
By integrating over the electron cloud, we get
the Fourier transform of the atom.
If we define r to be a vector relative to the
center of the atom, then f(S) can be thought of
as a single wave coming from the center of the
atom.
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Scattering from atoms and other centro-symmetric
objects...
... has phase equal to either 0 or 180
The imaginary part cancels out because
sin(2pSr) sin(2pS(r))

net sine part is zero
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Temperature factor, B
The sharper the electron density distribution,
the broader the scattering factor. The
temperature factor, B, modifies the scattering
factor by spreading out the electron density.
Correction factor for atomic scattering factor
A
B0.
B10.
B20.
B30.
2sinq
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Fourier transform as a sum over all atoms
One wave for each atom. The amplitude of the
scattering factor f(g) depends on how many
electrons that atom has. Each f(g) is positive
and real (i.e. not complex).Using Miller indeces
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What reflections can we see without moving the
beam?
crystal
X-ray source
beam
X-ray detector
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Behavior of S versus 2q? with the beam fixed
crystal
2q 0.
beam
s
s0
2q 30
2q 90
2q 170
For a given direction of the incoming x-rays, the
Ewald sphere is the set of possible scattering
vectors S given s0. The radius of the sphere is
1/l (reciprocal units!!)
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The Ewald Sphere
is defined as the sphere of radius 1/l that is
the locus of possible scattering vectors (S) when
the beam (s0) is fixed.
s0
Crystal position defines the coordinate axes.
This sphere is in scattering space (reciprocal
space).
If a scattered wave has S on the Ewald sphere,
it is visible on the film/detector.
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Moving the beam moves the Ewald Sphere
For a given direction of the incoming x-rays, the
set of possible scattering vectors S is the
surface of a sphere of radius 1/l, passing
through the crystallographic origin.
s0
Keeping the crystal fixed, we rotate the X-ray
source. The Ewald Sphere moves in parallel with
the X-ray source. The new set of S vectors
describe the phase vs. direction of scatter for
that position of the source.
s0
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Moving the beam. Crystal fixed.
s0
s0
By moving the X-ray source relative to the
crystal, we can sample every possible S
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The visible part of reciprocal space
limit 2/l(q180)
The set of all vectors S (red), given all
possible directions of the beam (black arrows),
is called reciprocal space.
Remember in this view, the crystal is fixed
(center of image, where the X-rays are pointed).
In real life, we find it easier to move the
crystal, not the source. It doesnt matter which
one you move, the crystal or the source. The
results are the same.
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Determining the space group
(1) Find the high symmetry axes using precession
photos. (2) Are the axes... all 90 apart?
orthorhombic, tetragonal, or cubic 90, 90 and
120? trigonal or hexagonal 90, 90 and ?90?
monoclinic none-of-the-above? triclinic, P1
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Precession camera geometry
Seeing one plane of the reciprocal lattice at a
time
By using a screen, all but one lattice plane is
masked out.
If the angle of the beam with the a axis is w,
then the correct angle setting for the screen is
?.
a
s0
?
Ewald sphere
This method NOT used for data collection. Too
slow.
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Precession photography
A precession photograph contains one complete
reciprocal plane on one film.
a
?
for examplebc plane at h0.
path of beam relative to crystal
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Precession photograph
note systematic absences, relative space of
spots, angle between axes.
0-level.
What crystal form is this?
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Determining the space group
(3)If trigonal of hexagonal, take h01 precession
photo. 6-fold symmetry P6n 6m symmetry
P6n2n2 3-fold symmetry P3, P31, P32 3m
symmetry P3n21 or P3n12 If orthorhombic or
tetragonal. Are cell dimensions.. a ? b?c?
orthorhombic ab?c? tetragonal abc?
cubic
p
p
p
q
p
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Systematic absences
(4)To get screw axis n, look at systematic
absences. examples
orthorhombic odd h reflections missing in h00
line --gt 21 trigonal only l3n reflections
present in 00l line--gt 31 or 32 hexagonal only
l2n reflections present in 00l
line--gt63 tetragonal only l4n reflections
present in 00l line --gt 41 or 43
Enantiomeric space groups (P41 and P43) cant be
distinguished until the phases have been solved.
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Data collection
Measuring the intensity (amplitude squared) of
each reflection. Output of data collection,
thousands of reflections, each with 5
parameters h k l F sigma
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Diffractometers yesterday and today
Counter moves in 2?. Crystal moves in 3 angles
?,?, and ?
Single photon counter (photo multiplier tube)
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Collecting data on photographic film
still widely used!
oscillation image
Raw images are scanned into digital images. Each
image has three angle associated with it (?,? and
?). A series of films, each with a different ?
angle, are collected and digitized.
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Today Image plates
Image plates are ultra-sensitive, reusable films.
Data collection is done the same way as for
photographic film.
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Area detectors
Position sensitive X-ray detectors give a 3D
image of each spot, where fild or image plates
give 2D images.
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Moving the crystal is like moving the Ewald sphere
s0
s0
By moving the X-ray source relative to the
crystal, we can sample every possible S
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data collection
frame 1
10
a
5
b
h0
4
visible part of transform
k0
Ewald sphere
-4
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data collection
frame 4
10
a
5
As the crystal is rotates, the reciprocal lattice
rotates.
b
h0
4
k0
Ewald sphere
-4
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Try this...
How far can you rotate the crystal before two
spots fall on top of each other on the film?

• Draw an Ewald sphere and a lattice.
• Put your pencil on a hkl position that is on the
  sphere.
• Rotate sphere and pencil together until pencil
  hits the next reciprocal lattice plane.
• How far did you rotate?

10
a
5
b
h0
4
k0
Ewald sphere
-4
For this exercise to work in reality, the lattice
and sphere must be drawn to scale. Ewald sphere
radius 1/?
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X-ray diffractometer with area detector
Axis of two-theta arm
The detector (or film) sits on a Two-theta arm
that can swing out, away from the beam to collect
high-resolution data.
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Schematic diffractometer
beamstop
crystal
X-ray source
beam
2? detector setting
low-resolution limit
goniostat
X-ray detector
high-resolution limit
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The film is a window on the Ewald sphere
All lattice points that fall on the Ewald sphere
are on (meaning photons are being scattered
that direction), but most of those reflections
are off camera (they end up on the laboratory
wall). The detector or film collects a window of
the Ewald sphere.
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As the crystal moves, the window moves
...relative to the reciprocal lattice.
path of detector through reciprocal lattice
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One sweep through reciprocal space collects a
donut-shaped volume of data
This is the volume of reciprocal space that has
been seen by the detector.
s0
? axis machine axis, center of donut
Red circle shows the low-resolution limit for
this detector position. A low-angle setting of
the detector would be necessary to collect the
low resolution data.
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Multiple???sweeps are usually necessary
Gray area volume of reciprocol space that has
been seen by the Ewald sphere (and thus, the
detector).
Intersecting donuts of data add up to the whole
Unique Set or more. Most reflections have
multiple copies.
Completeness what fraction of the unique set
has been collected, at least once. (Number of
reflections depends on resolution cutoff.)
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Data reduction
hkl F s 200 99.0 0.2210 65.1 0.3201 78.5
0.2220 6.3 0.1221 19.9 0.2222 88.1 0.2 ...

• indexing
• background estimation
• integration of spots
• merging of partials
• scaling
• merging of syms

raw images
reflection data Structure factors
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Data reduction

• indexing finding the location of each
  reciprocol lattice point HKL
• background estimation like subtracting the
  baseline, in 2D
• integration of spots intensity is proportional
  to F2
• merging of partials One reflection may be split
  between two films.
• Scaling If there is significant decay, then
  data is scaled in blocks of time.
• Averaging of syms Symmetry-related reflections
  are averaged

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Indexing the data
A reciprocal lattice is initialized using the
known cell dimensions. Spots are predicted to be
at the places where the lattice intersects the
Ewald sphere. A systematic search (rotation of
the lattice) is done until the predictions match
the observations.
Small refinements in the beam position might be
required.
When the solution is found, every spot has an
index (hkl).
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Calibrating the film, or detector.
For photographic film, or any type of X-ray
counter, a calibration curve has been
pre-calculated. The pixels are counted,
multiplied by I from the calibration curve, to
get I(hkl) for each spot.
absorbance
I
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Merging partials
If films were switched while a spot was on the
Ewald sphere, both copies (partials are summed
together to get I(hkl).
?153.0
?153.5
First half of spot hits the Ewald sphere.
Other half of spot passes through.
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How can partials exist?
Reflections are points in reciprocal space,
right? Wrong. Reflections have size and shape in
all three S directions (a, b, c), because the
crystal lattice is not perfect and
infinite. Reflections have additional shape in
the laboratory dimensions (x,y on the film or
detector), because the beam is not infinitely
small and the crystal is not infinitely small.
beam
The spot shape is partly the shape of the
intersection of the beam and crystal.
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Spot profiles in 3D-1
Profile of the average spot, summed over all
spots with similar x,y,2?,?
Intensity of each spot (blue) is the summed only
within the spot profile limits (grey).
This prevents counting spurious data like this.
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Scaling within a dataset

• Reflections may have errors in amplitude within a
  dataset because
• Xray intensity varied.
• Film/detector sensitivity varied.
• Crystal orientation/ cross section varied with
  w.
• Crystal decayed over time.
• Exposure time varied.
• Background radiation varied.
• Scaling assumes
• (1) Symmetry-related reflections have the same
  amplitude
• (2) Reflections that were collected together are
  scaled together (i.e. applied the same scale
  factor).
• Quality of the data set
• Should be lt 2

a sym op
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Example Structure Factor file
data_r1pkqsf ------------------------------------
------------------------ _audit.revision_id
1_0 _audit.creation_date 2003-07-15 _audit.u
pdate_record 'Initial release' loop_ _refln
.wavelength_id _refln.crystal_id _refln.index_h _r
efln.index_k _refln.index_l _refln.F_meas_au _refl
n.F_meas_sigma_au _refln.status 1 1 -39 0
26 70.300 34.700 0 1 1 -39 0
27 158.300 25.740 0 1 1 -39
1 1 156.000 15.800 0 1 1 -39
1 25 54.100 23.690 0 1 1
-39 1 26 201.400 11.450 0 1
1 -39 2 25 151.900 11.970 0
1 1 -39 3 22 202.800 22.730
0 1 1 -38 0 26 75.900 37.400
0
Structure Factors are deposited in the PDB
(www.rcsb.org) along with the atomic coordinates.