Introduction to Macromolecular Xray Crystallography

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Introduction to Macromolecular X-ray
Crystallography Biochem 300 Borden Lacy
Print and online resources Introduction to
Macromolecular X-ray Crystallography, by
Alexander McPherson Crystallography Made Crystal
Clear, by Gale Rhodes http//www.usm.maine.edu/rh
odes/CMCC/index.html http//ruppweb.dyndns.org/Xr
ay/101index.html Online tutorial with interactive
applets and quizzes. http//www.ysbl.york.ac.uk/
cowtan/fourier/fourier.html Nice pictures
demonstrating Fourier transforms http//ucxray.be
rkeley.edu/jamesh/movies/ Cool movies
demonstrating key points about diffraction,
resolution, data quality, and refinement.
http//www-structmed.cimr.cam.ac.uk/course.html No
tes from a macromolecular crystallography course
taught in Cambridge
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Overview of X-ray Crystallography
Crystal - Diffraction pattern - Electron
density - Model
Resolution, Fourier transforms, the phase
problem, B-factors, R-factors, R-free
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Diffraction The interference caused by an
object in the path of waves (sound, water,
light, radio, electrons, neutron..) Observable
when object size similar to wavelength.
Object
Visible light 400-700 nm X-rays 0.1-0.2 nm, 1-2
Å
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Can we image a molecule with X-rays?
Not currently.
1) We do not have a lens to focus X-rays.
Measure the direction and strength of the
diffracted X-rays and calculate the image
mathematically.
2) The X-ray scattering from a single molecule is
weak.
Amplify the signal with a crystal - an array of
ordered molecules in identical orientations.
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The wave nature of light
l
f(x) Fcos2p(ux a) f(x) Fsin2p(ux a)
x
F amplitude u frequency a phase
ulc

• f(x) cos 2px
• f(x) 3cos2px
• f(x) cos2p(3x)
• f(x) cos2p(x 1/4)

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Interference of two waves
Wave 1 Wave 2
Wave 1
Wave 2
In-phase
Out -of-phase
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Braggs Law
Sin q AB/d AB d sin q AB BC 2d sin q
nl 2d sin q
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nl 2d sin q
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Diffraction pattern
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Fourier transform
F(h) ? f(x)e2pi(hx)dx
where units of h are reciprocals of the units of x
Reversible!
f(x) ? F(h)e-2pi(hx)dh
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Calculating an electron density function from the
diffraction pattern
F(h) Fcos2p(uh a) F(h) Fsin2p(uh a)
r(x) ? F(h)e-2pi(hx)dh
F amplitude u frequency a phase
Experimental measurements
Ihkl, shkl
Fhkl vIhkl
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Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous
Replacement) Anomalous Scattering
Methods Molecular Replacement Methods Direct
Methods
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Heavy Atom Methods (Isomorphous Replacement)
The unknown phase of a wave of measurable
amplitude can be determined by beating it
against a reference wave of known phase and
amplitude.
Combined Wave
Unknown
Reference
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Generation of a reference wave Max Perutz
showed 1950 that a reference wave could be
created through the binding of heavy atoms.
Heavy atoms are electron-rich. If you can
specifically incorporate a heavy atom into your
crystal without destroying it, you can use the
resulting scatter as your reference wave.
Crystals are 50 solvent. Reactive heavy atom
compounds can enter by diffusion.
Derivatized crystals need to be isomorphous to
the native.
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Native Fnat
Heavy atom derivative Fderiv
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The steps of the isomorphous replacement method
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Heavy Atom Methods (Isomorphous Replacement)
The unknown phase of a wave of measurable
amplitude can be determined by beating it
against a reference wave of known phase and
amplitude.
FPH
FP
FH and aH
Can use the reference wave to infer aP. Will be
either of two possibilities.
To distinguish you need a second reference wave.
Therefore, the technique is referred to as
Multiple Isomorphous Replacement (MIR).
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Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous
Replacement) Anomalous Scattering
Methods Molecular Replacement Methods Direct
Methods
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Anomalous scattering
Incident X-rays can resonate with atomic
electrons to result in absorption and
re-emission of X-rays.
Results in measurable differences in
amplitude Fhkl ? F-h-k-l
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Advances for anomalous scattering methods
Use of synchrotron radiation allows one to tune
the wavelength of the X-ray beam to the
absorption edge of the heavy atom.
Incorporation of seleno-methionine into protein
crystals.
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Anomalous scattering/dispersion in practice
Anomalous differences can improve the phases in a
MIR experiment (MIRAS) or resolve the phase
ambiguity from a single derivative allowing for
SIRAS.
Measuring anomalous differences at 2 or more
wavelengths around the absorption edge
Multiple-wavelength anomalous dispersion
(MAD). Advantage All data can be collected from
a single crystal.
Single-wavelength anomalous dispersion (SAD)
methods can work if additional phase information
can be obtained from density modification.
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Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous
Replacement) Anomalous Scattering
Methods Molecular Replacement Methods Direct
Methods
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Molecular Replacement
If a model of your molecule (or a structural
homolog) exists, initial phases can be
calculated by putting the known model into the
unit cell of your new molecule.
1- Compute the diffraction pattern for your
model. 2- Use Patterson methods to compare the
calculated and measured diffraction
patterns. 3- Use the rotational and translational
relationships to orient the model in your
unit cell. 4- Use the coordinates to
calculate phases for the measured amplitudes. 5-
Cycles of model building and refinement to remove
phase bias.
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Direct Methods
Ab initio methods for solving the phase problem
either by finding mathematical relationships
among certain phase combinations or by
generating phases at random.
Typically requires high resolution (1 Å) and a
small number of atoms.
Can be helpful in locating large numbers of
seleno-methionines for a MAD/SAD experiment.
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Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous
Replacement) Anomalous Scattering
Methods Molecular Replacement Methods Direct
Methods
FT
F amplitude u frequency a phase
r(x,y,z) electron density
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Electron density maps
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Are phases important?
Duck intensities and cat phases
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Does molecular replacement introduce model bias?
Cat intensities with Manx phases
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An iterative cycle of phase improvement
Building Refinement
Solvent flattening NCS averaging
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Model building
Interactive graphics programs allow for the
creation of a PDB file. Atom type, x, y, z,
Occupancy, B-factor
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The PDB File
ATOM 1 N GLU A 27 41.211 44.533
94.570 1.00 85.98 ATOM 2 CA GLU A 27
42.250 44.748 95.621 1.00 86.10 ATOM 3
C GLU A 27 42.601 43.408 96.271 1.00
85.99 ATOM 4 O GLU A 27 43.691
42.865 96.065 1.00 85.71 ATOM 5 CB GLU A
27 41.725 45.720 96.687 1.00 86.36 ATOM
6 CG GLU A 27 42.804 46.349 97.563
1.00 86.44 ATOM 7 CD GLU A 27 43.628
47.387 96.817 1.00 86.98 ATOM 8 OE1 GLU
A 27 44.194 47.051 95.754 1.00
87.40 ATOM 9 OE2 GLU A 27 43.713
48.540 97.296 1.00 87.02 ATOM 10 N ARG A
28 41.662 42.882 97.053 1.00 85.65 ATOM
11 CA ARG A 28 41.839 41.607 97.739
1.00 85.29 ATOM 12 C ARG A 28 41.380
40.458 96.835 1.00 85.31 ATOM 13 O ARG
A 28 42.184 39.619 96.424 1.00
85.09 ATOM 14 CB ARG A 28 41.035
41.607 99.045 1.00 84.62 ATOM 15 CG ARG A
28 39.564 41.944 98.851 1.00 84.07 ATOM
16 CD ARG A 28 38.845 42.152 100.169
1.00 84.00 ATOM 17 NE ARG A 28 37.423
42.439 99.980 1.00 84.27 ATOM 18 CZ ARG
A 28 36.945 43.413 99.208 1.00
84.53 ATOM 19 NH1 ARG A 28 37.771
44.208 98.537 1.00 83.83 ATOM 20 NH2 ARG A
28 35.634 43.598 99.111 1.00 84.38 . . .
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Occupancy
What fraction of the molecules have an atom at
this x,y,z position?
B-factor
How much does the atom oscillate around the x,y,z
position?
Can refine for the whole molecule, individual
sidechains, or individual atoms. With
sufficient data anisotropic B-factors can be
refined.
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Refinement
Least -squares refinement

• S whkl (Fo - Fc)2hkl

Apply constraints (ex. set occupancy 1) and
restraints (ex. specify a range of values for
bond lengths and angles) Energetic refinements
include restraints on conformational energies,
H-bonds, etc.
Refinement with molecular dynamics An energetic
minimization in which the agreement between
measured and calculated data is included as an
energy term. Simulated annealing often increases
the radius of convergence.
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Monitoring refinement
Rfree an R-factor calculated from a test set
that has not been used in refinement.