1. Diffraction intensity
2. Patterson map
Lecture 6
2-1-2006
Calculation of the electron density r(x,y,z)
We already know:
This is equivalent to:
Because:
F is a Fourier synthesis of r
The reverse is also true:
or
r is a Fourier synthesis of F
This two Fourier syntheses form the mathematic foundation for all crystallographic
calculation
Physical meaning of Fourier synthesis
Scattering --First Fourier Transform
Focusing ----Second Fourier Transform
An simple example of Fourier transform
f=2
+
By superimposing three cosine waves, we have
f=3
=
+
If we have more higher frequency terms, we have
f=5
Symmetry in diffraction pattern
• A diffraction pattern possesses at least the same
or higher symmetry than the crystal
– Remember that reciprocal lattice rotates with the
crystal itself.
• All diffraction patterns have a center of
symmetry
I(h,k,l) = I(-h, -k, -l)
These two reflections are called a Friedel pair.
• From diffraction intensity, we should be able to
determine space group symmetry in most cases
(symmetry of diffraction pattern, systematic
absences)
Symmetry in diffraction pattern: a center of symmetry
Therefore,
Symmetry in diffraction pattern: a center of symmetry
- electron density r is a real quantity
Because
The implication is:
Which means r is real, as expected for a physical quantity.
Space groups with screw axes, glide planes, F, C, I centered lattices will
have systematic absences.
Effect of the size of the unit cell on the diffraction
intensity
V is the volume of unit cell
Vcr is the size of crystal
l is wavelength
Experimental data look like this …
h
k
l
F
0
1
2
25
1
0
3
14
-1
2
3
14
-2
1
2
11
1
2
1
10
3
-2
0
23
3
1
3
19
a
h,k,l are miller indices,
F is structure factor amplitude
No phase information.
Intensity statistics: Rmerge
• Only a small number of reflections are collected on each photograph
• To construct the reciprocal space, we need to merge reflections from
different frames.
• Measurement error is quantitatively described by Rmerge
R
=
I (hkl) -
/
I (hkl)
Where hkl are miller indices of a particular reflection, and i is the ith
measurement of that reflection
• Rmerge varies from a few percent in low resolution ranges to over
thirty percent in high resolution ranges
Intensity statistics: what are the important quantities
Intensity statistics: Wilson plot
•
•
After data merging, the first thing to do is to calculate the Wilson plot
Wilson provide an estimation of temperature factor
Because
we have
If ri ≠ rj, 2pi(ri-rj)S will vary from 0 to 2p, resulting in 0 value for I(abs, S)
Therefore,
and we know
then we have
With some adjustments, then we have
Intensity statistics: Wilson plot
•
•
After data merging, the first thing to do is to calculate the Wilson plot
Wilson provide an estimation of temperature factor
Slope => B factor
Y intercept => absolute scale C
However, for 2pi(ri-rj)S to
vary from 0 to 2p, S has to be
at least 1/3Å-1
What is phase problem?
Electron density can simply be calculated as:
However, we can only measure I(hkl), and we can only get |F(hkl)|=sqrt(I).
The phase information a(hkl) is missing.
Structure determination is all about phase determination.
Methods for phase determination
Molecular replacement (MR)
• Only one native crystal is needed. One homologous
structure must be available. Quick and simple.
Multiple isomorphous replacement (MIR)
• Use heavy atom to perturb diffraction intensities
Hg Pt
• At least one native crystal and two crystals soaked in
U
two heavy atom solutions must be available.
• Need no information about the unknown structure.
Multiwavelength anomalous dispersion (MAD)
Se
(S  Se)
• As X-ray wavelength approaching absorption edge,
anomalous scattering occurs – Friedel pairs are no
longer equal in intensities.
• Only one crystal is needed but multiple data sets must
be collected at three different wavelengths.
• Se-Met protein is usually needed. Excellent electron
density map.
The first crystal structure of a protein molecule
• 1962: Max Ferdinand Perutz and Sir John Cowdery
Kendrew win the Nobel Prize in Chemistry for their
studies on the structures of globlular proteins.
• The structure of myoglobin was solved by MIR.
(Max Perutz, 1914-2002)
Patterson function
In 1934, Patterson published a paper suggesting that the Patterson function:
giving rise to a map showing interatomic vectors (not individual atomic
positions, though)
The three variables u, v, w vary
from 0 to1 within the crystal unit
cell.
No phase information is needed
for Patterson synthesis
Pattern map shows intermolecular vectors
There are a total of N2 vectors, of
which N are located at the origin,
and N2-N are distributed throughout
the volume of the unit cell.
If atom i contains Zi electrons and
atom j contains Zj electrons, the
corresponding vector rij will have a
weight proportional to ZiZj.
Patterson map has a center of
symmetry.
The space group of patterson map
may be derived from crystal space
group by adding a center of
symmetry and losing translational
elements associated with screw
axes or glide planes.
Pattern map shows intermolecular vectors
By definition:
Because:
we have:
Pattern map shows intermolecular vectors another example
Patterson maps often have serious overlapping of
peaks
1-D
1. Peaks are generally broad
2. N2 –N peaks in total
2-D
Simple structures can be solved from Patterson
maps
Space group Pm: (x,y,z) and (x,-y,z)
There will be a set of vectors between symmetry-related atoms at:
(0,2y, 0) in Patterson map.
Simple structures can be solved from Patterson
maps
• Space group symmetries usually give rise
to concentrated vector points in the forms
of Harker lines or Harker planes
• Example: space group P21
– General positions are: (x,y,z), (-x, y+1/2, -z)
– Interatomic vector for symmetry-related
atoms is: (2x, ½, 2z)
– There will be a lot of vector on v=1/2 section
in Patterson map
– v=1/2 is called Harker section
– From Harker peaks, we can obtain x and z
coordinates for every atom
Patterson map in protein crystallography
• Locate heavy atom positions for multiple
isomorphous replacement method or multiwavelength anomalous dispersion method
• Rotation and translation function searches for
molecular replacement method