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Space group: I222a = 79 Åb = 117 Åc = 129 Å, α = β = γ = 90°Resolution:1.5 Å
Wavelength: 1.54 Å(MAR research Imaging plate)
Crystal parametersUnit cell
a
b
x
y
(010
)
(100)
(010
)
(120)
e.g. α = β = γ = 90°
X-ray diffraction
F (h,k,l)
The relationship between the electron density ρel(x,y,z) and the structure
factors F(hkl) can be described by a Fourier transformation (FT).
This transformation is accurate and in principle complete. If we know the
structure factors (diffraction by electrons) we can calculate the actual real
structure (the density of the electrons in real space).
ρel (x,y,z)
X-ray diffraction
Fourier Transformations
I = | F 2 |
rdersF rsi
Vol
el
vvv vvπρ 2)()( ⋅= ∫
∑∑∑ ++−⋅=h k l
lzkyhxi
EZel ehklF
Vzyx )(2)(
1),,( πρ
F(hkl) = F(hkl) ⋅ei hklϕ
Structure Factor:
Electron Density:
F (h,k,l)
X-ray diffraction
Fourier Transformations
F(hkl) = F(hkl) ⋅ei hklϕ
ϕϕϕsincos ⋅+= ie
i
Vector representation of F
Vector representation of F in complex plane:
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
… a molecule, and its Fourier Transform:
… an atom, and its Fourier Transform:
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
… a crystal, and its Fourier Transform:
→ reciprocal space
… a lattice, and its Fourier Transform:
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
… a duck, and its Fourier Transform:
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
… and a new friend, a Fourier cat: Phases (colors)
Inten-sities
… again, our Fourier Duck:
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
Intensities and phases
The picture that contributed the phases is still visible, whereas the picture which contributed the magnitudes has gone!
Phases contain the bulk of the structural information.
We need the intensities and phases to calculate a realistic picture.
From: Kevin Cowtan's Book of Fourier; http://www.ysbl.york.ac.uk/~cowtan/fourier/fourier.html
I = | F 2 |
F(hkl) = F(hkl) ⋅ei hklϕ
Houston, Houston!
we have a
Phase Problem !
We can measure intensities, but ....
From: G. Rhodes; Crystallography Made Crystal Clear
Phasing Methods
Heavy atom method (MIR, multiple isomorphous replacement)Each atom in the unit cell contributes to all observed reflection intensities. The basic principle of the MIR method is to collect diffraction data of several (multiple) crystals of the same protein, that share the same crystal properties (isomorphous), but differ in a small number of heavy atoms. The experimental approach is normally to soak protein crystals with diluted solutions of heavy metal compounds (e.g. mercury or platinum derivatives), that often bind specifically to certain protein residues. These additional atoms cause a slight perturbation of the diffraction intensities. To achieve a perturbation large enough to measured correctly, the added atoms must diffract strongly, i.e. elements with a high number of electrons (heavy atoms) are used. The differences of the reflection intensities can be used to locate the positions of the heavy atoms within the unit cell, which allows to estimate initial phases.
Anomalous scattering (MAD, multiple wavelength anomalous diffraction)The MAD method is based on the capacity of heavy atoms to absorb X-rays of a specific wavelength. Near its characteristic absorption wavelength, the diffraction intensities of the symmetry related reflections (Friedel pairs, h,k,l and -h,-k,-l) are no longer equal. This effect is called anomalous scattering. The characteristic absorption wavelengths of typical protein atoms (N,C,O) are not in the range of the X-rays used in protein crystallography and therefore are not contributing to anomalous scattering. However, the use of synchrotron X-ray sources with adjustable wavelengths allows to collect diffraction data under conditions where heavy atoms exhibit strong anomalous scattering. In practice, several diffraction data sets are collected from the same protein crystal at different wavelengths. From the small differences between the Friedel pairs, the location of the heavy atoms can be determined and initial phases of the native data are estimated.
Molecular replacement (MR)In some cases is structure to be examined is known to be very similar to an other structure, that has already been solved experimentally. This could be e.g. the same protein from an other organism or a mutant of this protein. In these cases the phases computed from of the known protein structure (phasing model) can be used as initial estimates of the phases of the unknown protein.
∑∑∑ ++−⋅=h k l
lzkyhxi
EZel ehklF
Vzyx )(2)(
1),,( πρ F(hkl) = F(hkl) ⋅ei hklϕ
Phase Problem
From: G. Rhodes; Crystallography Made Crystal Clear
w(hkl) resolution dependent weight factorReflections of the test set T are excluded from the refinement procedure.
Least squares refinement:
1 2 3 4 5 6 71234567890123456789012345678901234567890123456789012345678901234567890ATOM 74 N ASP A 10 12.982 78.264 31.707 1.00 48.50 N ATOM 75 CA ASP A 10 14.137 79.163 31.764 1.00 46.20 C
COLUMNS DATA TYPE FIELD DEFINITION---------------------------------------------------------------------------------1 - 6 Record name "ATOM "7 - 11 Integer serial Atom serial number.
13 - 16 Atom name Atom name.17 Character altLoc Alternate location indicator.18 - 20 Residue name resName Residue name.22 Character chainID Chain identifier.23 - 26 Integer resSeq Residue sequence number.27 AChar iCode Code for insertion of residues.31 - 38 Real(8.3) x Orthogonal coordinates for X in Angstroms.39 - 46 Real(8.3) y Orthogonal coordinates for Y in Angstroms.47 - 54 Real(8.3) z Orthogonal coordinates for Z in Angstroms.55 - 60 Real(6.2) occupancy Occupancy.61 - 66 Real(6.2) tempFactor Temperature factor.73 - 76 LString(4) segID Segment identifier, left-justified.77 - 78 LString(2) element Element symbol, right-justified.79 - 80 LString(2) charge Charge on the atom.
Anatomy of a PDB file: Coordinate Section
jatom
rsin
j
Bs
j eesfhklFvv⋅⋅−
⋅⋅= ∑ π24
1 2
)()(
0
20
40
60
80
residue no.
fj X Z Y occ. B-FactorATOM 1 N THR 1 17.047 14.099 3.625 1.00 13.79ATOM 2 CA THR 1 16.967 12.784 4.338 1.00 10.80ATOM 3 C THR 1 15.685 12.755 5.133 1.00 9.19ATOM 4 O THR 1 15.268 13.825 5.594 1.00 9.85ATOM 5 CB THR 1 18.170 12.703 5.337 1.00 13.02ATOM 6 OG1 THR 1 19.334 12.829 4.463 1.00 15.06ATOM 7 CG2 THR 1 18.150 11.546 6.304 1.00 14.23
Crystallography and Coordinate Transformation Section
Lattice Types & Symmetry
2-dimensional lattice with 2-fold symmetry axes P21 entry in International Tables of Crystallography
Symmetry
NOTE:
Biological macromolecules are
chiral.
Of all 230 possible space groups, only
those 65 without mirror planes and
centers of symmetry are allowed for
protein crystals.
Symmetry
PDB - file biologically active state: tetramer
Symmetry
NOTE: the content of a PDB file does NOT necessarily represent the biologically active oligomeric form of a protein!
X-ray crystallography
IntroductionG. Rhodes. Crystallography made Crystal Clear, Academic Press, San Diego, USA.
Advanced Textbooks:Giacovazzo, H.L. Monaco, D. Viterbo, F. Scordani, G. Gill, G. Zanotti, M. Catti. Fundamentals of Crystallography, International Union of Crystallography, Oxford University Press, Oxford, UK.
J. Drenth, Principles of Protein Crystallography, Springer, New York, USA.