1. Methods: X-ray Crystallography
Biochemistry 4000
Dr. Ute Kothe

2. 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

3. 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

4. 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

5. 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

6. Description of Crystalsb γ
Unit cell:
characterized by edge length and angles
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

7. 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

8. 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)

9. 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)!

10. 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

11. 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)

12. 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.
To obtain enough information, diffraction photographs are taken at various different angles of the crystal in the X-ray beam.
X-ray diffraction photograph
of Sperm Whale Myoglobin crystal

13. 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

14. 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 l
Electron Density
Intensity
Phase
Missing Information to get electron density!

15. Molecular Replacement
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)

16. 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

17. 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

18. 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?

19. 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)

20. Resolution Limited by:
Quality of crystal (Conformational inhomogenities, Mobility of individual atoms)
Number of unique reflections used for structure determination (at least four times more than atoms required)
Typically not by wavelength
5Å - a helices roughly visible
3Å - peptide chain visible, side chains if sequence known
2Å - conformations of side chains visible

21. 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 < Ų
Reasons for higher B factor:
Thermal motion of atom
Different conformations of side chain, i.e. lower occupancy at specific position
Protein disordered, e.g. In surface exposed loops

22. Other Important Parameters
Protein backbone should have only allowed conformations (as described in Ramachandran plot)
Completenes of structure (should be >90%)

23. Parameters in Publication
Rasmussen et al., Science 2007

24. 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

25. Native Conformation?
Is protein structure determined from crystal the same as
in vivo solution structure?
Most likely:
Crystals contain 40 – 60 % solvent, i.e. the protein is in simialr environment as in cell
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.
Often X-ray structures and solution NMR structures superimpose well.
Many enzymes are active in crystalline state
BUT: You never know! Exceptions are possible!