1.
Lecture 3 Structure Factors and electron density

2.
Review ‐ identify miller planes
Green planes ‐
Blue planes ‐
Magenta planes
Green arrow
Magenta arrow
Brown arrow

3.
slide 3
1. Arrows on the figure correspond to dhkl
2. Arrows also are used to represent the reciprocal scattering vector s.
The direction of s is identical to dhkl with length 1/dhkl.
k b
a
‐b
‐a
1,1,0
1,0,0
1,0,0

4.
What defines + or ‐ miller planes?
(using the 1, 1, 0 plane, +/‐ definitions define the scattering vector direction)

5.
b
a
‐b
‐a
1,‐1,0 1,1,0
‐1,1,0
Relationship between miller planes and the reciprocal axis
‐1,‐1,0
Reciprocal Lattice
Crystal lattice
1,1,0
‐1,‐1,0
Where is the x‐ray beam for this experiment?
Where is the c‐axis in this figure?
Index remaining spots in the diffraction image on the right

6.
File of structure factor amplitudes
experimental data
(Fobs)
6
slide
PDB file atm res. Chn res# x y z occ. B
Fcalc(calculated from the atoms in the pdf file)
Each Fcalc is a summation of all atoms in pdb.
Structure Factor Equation
Crystallographic R‐Factor

7.
slide 7
A. Diffraction Data (Fobs, but no phases α)
B. Determine Phases (Fobs collected, but no phases)
Molecular Replacement (MR, requires coordinate file of related structure)
Heavy‐atom methods (SIR, MIR)
Anomalous dispersion (SAD, MAD)
C. Calculate an electron density map, based on phases above .
Build a protein model (atom type, x, y, z, B) into the map.
[electron density provides a bootstrap method for building partial models ]
D. Refine protein model.
Optimize the positions of the atoms ( x, y, z, and B) in the unit cell.
This is performed by mimimizing the difference between Fobs and Fcalc. The values
of Fcalc are derived from the atom positions of the protein model in the unit cell.
The Structure Factor equation and the Electron Density equation play important roles in all
of the steps of structure determination by X‐ray crystallography.
Steps in X‐ray structure determination

8.
Electron density equation (Density , ρ (e‐/Ǻ3) at position x,y,z in the unit cell)
|F(hkl)| structure factor amplitude (scalar) measured in the diffraction experiment.
α (phase) in radians. The “phase” of a hkl reflection. Lost during the diffraction
experiment
Thus, F(hkl) is a vector with amplitude |F(hkl)| and phase α(hkl).
h k l are the miller indices!!!
x y z fractional coordinates of the unit cell. Sampling from 0‐1 in each direction.
electron density (ρ(xyz)) is calculated by summing all structure factors at each xyz
(sampling grid) position in the unit cell. Thus, ALL reflections contribute to the electron
density at each place in the cell.
F hkl, α
FT

9.
h F α
The electron density in the unit cells of a
crystal is a periodic wave function. It may
be decomposed into its component waves .
Each component wave is described by a
structure factor amplitude (|F|)and
Phase (α).
In an x‐ray diffraction experiment, |F| is
measured, but α is lost.
To derive the lost phases and ultimately
electron density, must identify
atom positions (black dots), which can be
used to calculate the structure factors.
(e.g. |F| and α)
108°
144°
144°
162°
234°
Structure
factors
Highly schematic figure!!

10.
The Isotropic Atomic Scattering Factor
B=8π2<u>2
1. X‐ray scattering is proportional to the # of
electrons in the atom.
a. Hg is called a “heavy atom”. Lots of
electrons!
2. Scattering falls off with resolution (e.g.
resolution dependent).
3. Scattering falls off with increasing movement
of the atom. The B‐factor (refined for each
atom , found in pdb files) models the
movement of atoms in crystals by the
equation:
atomic number
# electrons
Where u is the mean atomic displacement of
an atom. Thus, a B‐factor of 10 corresponds
to a mean atomic displacement of 0.36Å

11.
Isotropic scattering factors in electrons found in International tables Vol. 4. (Table 2.2)
The Isotropic Atomic Scattering Factor
Scattering factor (f) plot for oxygen vs.
resolution at B=0Å2 and B=10Å2
B=0 Å2 resolution fall off only
B=10 Å2 motion and resolution fall off only

12.
STRUCTURE FACTOR summation of scattering vectors of individual atoms in unit cell
h k l 1 2 0
F is the summation of scattering vectors for each atom
in the unitcell.
The magnitude of f is dependent on the scattering
atom.
The orientation of the vector is dependent on the
x, y, z position of the atom in the unit cell.
F

13.
f1, f2, and f3 are the scattering contribution of 3 different atoms in the unit cell of a crystal.
All atoms contribute to the measured intensity.
The phase (α) requires knowledge of the positions of atoms in the unitcell.
F (hkl) corresponds to a waves scattered from all atoms in the crystal unit cell for a given set
of miller planes h, k , l.
STRUCTURE FACTOR EQUATION
Summation over all atoms (j) in the unit cell for each h, k, l

14.
STRUCTURE FACTOR EQUATION (algebraic calculation)
It easy to break the structure factor equation down into scattering components along real
(A, where A=f1cos α ) and imaginary axes (B, where B = B=f1sinα ).
The summation of all A and B components (for each atom, see equations below) results in
the structure factor amplitue |F|, which is equal to the SQRT (A2+B2) with resultant
phase (α) = Atan2(B,A), or tan‐1 B/A. Normally the phase is converted from radians to
degrees.
A
B

15.
STRUCTURE FACTOR Diagrams – Graphical representation of structure factors.
A=1, B=0 A=0.71, B=0.71
A=|F(hkl)|cosα
B=|F(hkl)|sinα
α=tan‐1 (B/A)
|Fhkl| = SQRT(A2 + B2)
I = A2 + B2
If |F| = 1
A
B
α

16.
hkl = 200
F=12
α = 0°
hkl = 200
F=12
α = 72°
Note relation of e‐ density wave (green) relative to miller planes (grey)
Phase diagram (left) and e‐ density and miller planes (right)

17.
Friedels law F(hkl) = F*(hkl) Thus, F(hkl) and F(‐h‐k‐l) have equal amplitudes
but opposite phases (no anomalous scattering)
hkl planes 0.65
0.4
0.2
0.1
h reflection
‐h reflection
a 36°
72°
144°
234°
‐36°
‐72°
F(h)
F(‐h)
f 2*pi h x1 2*pi*(h*x1) f*cos(2*pi*(hx)) A f*sin (2*pi*(hx)) B F=SQRT(A+B) phase
6.00 6.28 1 0.10 0.63 4.85 4.85 3.53 3.53 6.00 36
6.00 6.28 ‐1 0.10 ‐0.63 4.85 4.85 ‐3.53 ‐3.53 6.00 ‐36
Assume atoms are carbons ( 6 electrons)

18.
How do we get phases?

19.
Experimental Data h, k , l, I and sigma I (Native Dataset used for refinement and
deposited with the final coordinates in the
pdb file.)
Experimental Data for Phasing (generally not deposited).
1. Prior protein model for molecular replacement phasing x,y,z coordinates of model
2. Heavy‐atom experimental data (isomorphous replacement) h, k , l, I, sig.I
Intensity differences between native and HA data
used to find x,y,z positions of heavy‐atoms.
heavy atom phases used as initial estimates of protein phases
3. Anomalous diffraction data h, k, l, I+, sig.I+ , ‐I, sig.‐I
Intensity differences between I+ and I‐ (Friedel pairs)
used to find x,y,z positions of anomalous scattering atoms.
anomalous scatter phases used as initial estimates of protein phases.

20.
Possible problem: No appropriate model structure or
structural model too different
Possible problem: heavy atoms (e.g. Hg) don’t bind OR don’t
bind specifically. Heavy‐atoms change the unit cell – non‐
isomorphous, which prevents finding the positions.
Generally, requires labeling of the protein (SeMet). Great if molecule
can be produced in e. coli. Need anomolous scatter e.g Fe protein
Positive….100% incorporation of SeMet

21.
Collect an X‐ray diffraction data set.
Calculate structure factor amplitudes for Known protein structure (Pmodel)
However, don’t know where this model is in the unit cell…
Molecular Replacement
1. Finds the correct rotation of Pmodel relative to Pcrystal (Pc).
2. Finds the correct translation of Pmodel in Pcrystal.
3. This is effective when Pmodel is structurally similar to Pcrystal. The breakdown
occurs somewhere around ~35% or less sequence identity.
model
Mo
d
c
c
c
c
c
c
c
c
#1 #2

22.
Fph Fp Fh
FPH = FP + FH
For SIR or MIR Phasing Method
Dataset of
Native
crystal
Dataset of Native
Crystals derivatized
With heavy‐atom
Hg, or U, or Pb, or Pt
(look at scattering
factor table)
Measure the intensity (amplitude) differences between Heavy‐atom dataset (Fph) and
Native dataset (Fp).
Subtraction of Fph – Fp is an estimate of the contribution of the heavy‐atom to the diffraction
pattern.
Find the phase of the heavy atom (Fh). Need to find the xyz position of the atom: Patterson Map.
Estimate of
Heavy‐atom
contribution

23.
1. If you define the location of the heavy
atom, you can calculate the vector FH.
With FH you can define the phase of the
protein (αP).
FPH = FP + FH
Remember these are each VECTORS….we only have amplitudes

24.
|Fp|
|Fph|
In SIR, we have 2 circles corresponding to Fp and Fph which are native and
derivative (heavy atom) data sets. The following diagram would be
constructed for each hkl reflection in the dataset.
We want to know the phase (α) of Fp. Because
Fp and Fph are vector summations of atoms
(see earlier slides) we can write the following vector
equation.
α
|Fp|
|Fph|
Fh
Fp + Fh = Fph
This says if we know the vector for Fh, we
Can solve for Fp

25.
positional From FH, get αP
Phaser……………….automate all steps + refinement

26.
Initial Phase estimates Build initial model / refine
More information in the model
than in the starting map.
Calc. new map.

27.
Crystallographer observes
incorrect fits between improved
map and model.
Refinement program cannot
“fix” all problems . Manual
intervention through molecular
graphics
Correct model, refine model,
recalculate electron density
map. R‐factor should be lower,
improved map quality, protein
geometry improved.

28.
Anomalous Scattering http://skuld.bmsc.washington.edu/scatter/

29.
Anomalous scattering factors are determined experimentally by monitoring Fluorescence as
A function of energy (wavelength). Performed at a synchrotron beamlines. Requires a
Tunable x‐ray source.

30.
Selenium, atomic # = 34, Sulfur mimick
Edge keV Å
K 12.6578 0.9795
L-I 1.6539 7.4965
L-II 1.4762 8.3989
L-III 1.4358 8.6352

31.
12595.00 ‐4.793897 0.5074802
12600.00 ‐4.879107 0.5071128
12605.00 ‐4.972221 0.5067459
12610.00 ‐5.074862 0.5063793
12615.00 ‐5.189123 0.5060131
12620.00 ‐5.317906 0.5056473
12625.00 ‐5.465356 0.5052819
12630.00 ‐5.637684 0.5049169
12635.00 ‐5.844824 0.5045523
12640.00 ‐6.104173 0.5041881
12645.00 ‐6.450631 0.5038243
12650.00 ‐6.972532 0.5034610
12655.00 ‐8.055866 0.5030980
12660.00 ‐8.319967 3.846461 peak
12665.00 ‐7.052211 3.843189
12670.00 ‐6.486044 3.839924
12675.00 ‐6.116062 3.836665
12680.00 ‐5.840367 3.833412
Energy Kev f’ f’’
12900.00 ‐3.196559 3.696152 remote
Energy Kev f’ f’’

32.
f”, varies strongly near the absorption edge, becoming
most positive at energies > E.
f” is the component of scattered radiation 90°
out of phase with the normally scattered
component fo
fo
f” f”
fo
f”
fo
f”
fo
f”
fo
E
Se
2
2
"
v
v
g
f B


 when >B
Else, 0
Peak
High energy remote
Low energy remote
Scattering factor f” is maximal at the peak, which corresponds to maximal Fph+ and Fph‐
intensities observed in the diffraction data of Friedel pairs.
SAD data is collected at energies/λ just above the peak.

33.
Friedels law F(hkl) = F*(hkl) Thus, F(hkl) and F(‐h‐k‐l) have equal amplitudes
but opposite phases (no anomalous scattering)
hkl planes 0.65
0.4
0.2
0.1
h reflection
‐h reflection
a 36°
72°
144°
234°
‐36°
‐72°
F(h)
F(‐h)
f 2*pi h x1 2*pi*(h*x1) f*cos(2*pi*(hx)) A f*sin (2*pi*(hx)) B F=SQRT(A+B) phase
6.00 6.28 1 0.10 0.63 4.85 4.85 3.53 3.53 6.00 36
6.00 6.28 ‐1 0.10 ‐0.63 4.85 4.85 ‐3.53 ‐3.53 6.00 ‐36
Assume atoms are carbons ( 6 electrons)

34.
Friedel pairs
At the appropriate X‐ray energy, the white atom
Scatters anomalously and Friedel’s law breaks down. This is
because the phase of the anomalous scattering atom is
advanced by 90 degrees relative to the other atoms (Δf”).
This results in different intensities at I h,k,l and I –h,‐k, ‐l,
which are measured in the x‐ray diffraction experiment. As a result,
FPH+ and FPH‐ are different. These differences can be used to locate the x, y, z
position of the selenium atom, providing phase estimates for the entire protein
structure.
h k l plane
‐h ‐k ‐l plane
Breakdown in Friedel’s law
FPH (hkl)≠FPH (‐h‐k‐l)
when an anomalous
scattering atom is
present
Δf”
Δf”
0.9

35.
MAD data collection statistics