Methods:
X-ray Crystallography
Biochemistry 4000
Dr. Ute Kothe
Why X-rays?
Optics:
Limit of resolution ~ l/2
Distance between atoms: 0.1 – 0.3 nm
Wavelenght of X-rays: 0.1 – 10 nm
 Allows structure determination
at atomic resolution
Problem:
No lenses available for X-rays
(as for visible light in microscope)
Remember:
Wavelength, Amplitude, Phase
Overview of steps
Growing a Crystal
Collecting X-ray diffraction pattern
Solving of Phase Problem
Calculate Electron Density Map
Constructing a Structural Model
Refining the Structural Model
Growth of Crystals - Background
Why?
Diffraction pattern of single molecule too weak for detection
=> sum of several diffraction patterns of identically oriented
molecules with identical conformation
Porcine Elastase
Supersaturated Solution: Concentration higher than intrinsic solubility S0
Growth of Crystals - Approaches
Hanging Drop Method:
Empirical Approach:
Test hundreds of conditions
Parameters:
• pH
• salt type & concentration
• precipitant
(ammonium sulfate, polyethylene
glycol, 2-methyl-2,4-dimethyl-pentane
diol (MPD), propanol)
• temperature (2 – 30°C)
• protein concentration
(1 – 100 mg/ml)
• additional substances
(substrates, inhibitors, cofactors etc.) Alternative:
Sitting Drop
Microdialysis
Description of Crystals
a
Unit cell:
γ
characterized by edge length and angles
b can contain several molecules
is repeated multiple times along the axis of the crystal
Assymetric Unit: minimal part
of the unit cell which is related
to other parts by defined
symmetry operations
Space Groups
Crystal can be
described
by shape of unit
cells (called lattice
type, see figure) and
symmetry type of
asymmetric unit
 65 different space
groups possible for
natural biomolecules
Example:
P212121
Diffraction and Interference
Huygen’s principle
of diffraction:
Each point in front of
a wavefront acts as a
point of progapation
for a new wavelet.
b) Constructive interference of scattered waves
=> Amplitude = 2E
c) Destructive interference of scattered waves
=> Amplitude = 0 (wave 180° out of phase)
Bragg’s Law
Requirement for
constructive interference:
2 d sin θ = n l
n: integer values (0, 1, 2, ...)
 If angle θ and wavelenght
l are known, the distance d
between planes can be
determined.
Simplification: reflection at planes (one dimension: distance d of planes)!
Diffraction in 3D: Miller Indices
Van Laue Conditions replace Bragg’s Law
(too much to explain in Bchm 4000)
Instead of “n”, three Miller Indices (h, k, l) define the
integer number of wavelenght that result in an observed
reflection froma 3D crystal.
Example: 2 points in space
1D reflection
2D reflection
More Diffraction Patterns
2D example:
Each diffraction
spot corresponds
to one set of
Miller indices (h,
k, l).
Friedel’s law:
I(h,k,l) = I(-h,-k,-l)
http://www.esc.cam.ac.uk/teaching/teaching.html
Diffraction Photograph
Each spot contains information
from all atoms!
“Pattern of spots” (diffraction
pattern) allows determination of
crystal parameters (size of unit
cell, symmetry within unit cell).
Structural information has to de
derived (see below) from
difference in spot intensities.
X-ray diffraction photograph
of Sperm Whale Myoglobin crystal
 To obtain enough information,
diffraction photographs are taken
at various different angles of the
crystal in the X-ray beam.
The Structure Factor: F(S)
How to get to an electron density map from the intensities of
reflections?
• Intrinsic amplitude of scattered X-ray from each type of atom
depends on its electron density r(xyz) (described by atomic
scattering factor f)
• f is used to describe scattering vector for each atom for one
reflection (with Miller indices h,k,l)
• Sum of atomic scattering vectors is molecular scattering factor F,
also called structure factor (can be calculated for a given
structure):
F(hkl) = ∫r(xyz) e-2pi(hx+ky+lz)
Fourier Transformation
The Phase Problem
Continued: How to get to an electron density map from the intensities
of reflections?
Fourier Transformation (other way around)
r(xyz) = ∫F(hkl) e2pi(hx+ky+lz)
Magnitude of F is proportional to square root of intensity:
|F(hkl)| ~ √ I(hkl)
Thus:
r(xyz) = 1/NV ∑ ∑ ∑ √I(hkl) e-i(hkl) e2pi(hx+ky+lz)
h
k
Electron Density
l
Intensity
Phase
Missing Information to get electron density!
Molecular Replacement
1. Method to obtain Phase information
• Only possible if structure of a very similar molecule is
known (native vs. Mutant, close homolog), at least 25%
sequence identity
• Calculate structure factors Fcalc for known molecule
• Compare with data set to estimate phase for known
structure (acalc)
• Use this Phase information together with observed
structure factors (Fobs) to calculate first electron density
map
• Refine Structure (see below)
Multiple Isomorphous Replacement
2. Method to obtain Phase information
• Use crystals containing heavy atoms at specific positions
(isomorphous crystals) as well as crystal of native protein
• Heavy atoms strongly diffract
• Difference data set of heavy atom derivative and native crystals
gives structure factors of heavy atoms only = limited data set
• Can be used to determine position and phase of heavy atoms
(as for small molecule crystals, Patterson function)
• Needs to be done for at least 2 heavy atom derivatives to
obtain enough information to estimate phase of native data set
and to solve the structure
Multiple Anomalous Dispersion
3. Method to obtain Phase information
• Use one crystal containing heavy atoms at specific
positions as well as crystal of native protein
• Heavy atoms not only diffract X-rays, but also absorbs X-rays of
certain wavelength
• Using synchroton X-ray source, diffraction data are generated
at different wavelenghts of the heavy-atom crystal
• Different data sets can be used to obtain phase information
similar to multiple isomorphous replacement
Electron Density Map
Computed electron density
map can be represented by
“chicken wire” on computer.
Knowing the primary
sequence of a protein (or
nucleic acid) the atoms can
be placed by hand into the
electron density map.
 Rough, inaccurate
model of protein structure
What about hydrogen atoms?
Structural Refinement
Iterative Process:
• model is subtly changed
• Structure Factors are calculated (Fcalc) and compared by Rcrys
factor with measured Fobs to see how well a model fits the data:
• differences are minimized (minimal R)
• good structure: Rcrys < 20% (0.2)
Problem:
• R factor can be artifically lowered by adding more solvent
• better Rfree: as Rcrys, but calculated with 10% reflections which
have not been used for refinement (unbiased data set)
Resolution
Limited by:
1. Quality of crystal (Conformational inhomogenities, Mobility of
individual atoms)
2. Number of unique reflections used for structure determination
(at least four times more than atoms required)
3. Typically not by wavelength
5Å
3Å
2Å
- a helices roughly visible
- peptide chain visible, side chains if sequence known
- conformations of side chains visible
Temperature Factor (B factor)
• Describes local imprecision in electron density
• Calculated during structure refinement
• Measured in Ų
 The higher the B factor, the more imprecise the atom position.
 Good protein structure: B < 20 - 30 Ų
Reasons for higher B factor:
1. Thermal motion of atom
2. Different conformations of side chain, i.e. lower occupancy at
specific position
3. Protein disordered, e.g. In surface exposed loops
Other Important Parameters
• Protein backbone should
have only allowed
conformations (as described
in Ramachandran plot)
• Completenes of structure
(should be >90%)
Parameters
in
Publication
Rasmussen et al., Science 2007
Protein Data Base
• structure data (x, y, z coordinates of each atom) are submitted to
Protein Data Base
• freely accessible, structure can be downloaded and visualized with
free programs such as RasMol or Swiss PDB Viewer
Native Conformation?
Is protein structure determined from crystal the same as
in vivo solution structure?
Most likely:
1. Crystals contain 40 – 60 % solvent, i.e. the protein is in simialr
environment as in cell
2. Often proteins crystallize in different space groups with different
crystal contacts between proteins, but the determined structure is
the same, i.e. It is independent of crystal packing.
3. Often X-ray structures and solution NMR structures superimpose
well.
4. Many enzymes are active in crystalline state
BUT: You never know! Exceptions are possible!