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/~rhodes/CMCC/index.html
http://ruppweb.dyndns.org/Xray/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.berkeley.edu/~jamesh/movies/
Cool movies demonstrating key points about diffraction, resolution, data quality, and refinement.
http://www-structmed.cimr.cam.ac.uk/course.html
Notes from a macromolecular crystallography course taught in Cambridge
Overview of X-ray Crystallography
Crystal -> Diffraction pattern -> Electron density -> Model
Resolution, Fourier transforms, the ‘phase problem’, B-factors,
R-factors, R-free …
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
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.
The wave nature of light
l
f(x) = Fcos2π(ux + a)
f(x) = Fsin2π(ux + a)
F = amplitude
u = frequency
a = phase
a)
b)
c)
d)
f(x) = cos 2πx
f(x) = 3cos2πx
f(x) = cos2π(3x)
f(x) = cos2π(x + 1/4)
x
ul=c
Interference of two waves
Wave 1 + Wave 2
Wave 1
Wave 2
In-phase
Out -of-phase
Bragg’s Law
Sin q = AB/d
AB = d sin q
AB + BC = 2d sin q
nl = 2d sin q
nl = 2d sin q
Diffraction pattern
The intensity of each spot contains
information about the entire molecule.
The spacing of the spots is due to
the size and symmetry of your lattice.
Practically:
Assign a coordinate (h, k, l)
and intensity (I) to every spot in
the diffraction pattern—
Index and Integrate.
Ihkl , shkl
Fourier transform:
F(h)= ∫ f(x)e2πi(hx)dx
where units of h are reciprocals of the units of x
Reversible!
f(x)= ∫ F(h)e-2πi(hx)dh
Calculating an electron density function from the diffraction pattern
r(x) = ∫ F(h)e-2πi(hx)dh
Experimental measurements:
Ihkl, shkl
Fhkl ~ √Ihkl
F(h) = Fcos2π(uh+ a)
F(h) = Fsin2π(uh + a)
F = amplitude
u = frequency
a = phase
Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods
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
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.
Native
Fnat
Heavy atom derivative
Fderiv
The steps of the isomorphous replacement method
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).
Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods
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
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.
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.
Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods
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.
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.
Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods
F = amplitude
u = frequency
a = phase
FT
r(x,y,z)
electron density
Electron density maps
Are phases important?
Duck intensities
and cat phases
Does molecular replacement introduce model bias?
Cat intensities with
Manx phases
An iterative cycle of phase improvement
Building
Refinement
Solvent flattening
NCS averaging
Model building
Interactive graphics programs allow for the creation of a ‘PDB’ file.
Atom type, x, y, z, Occupancy, B-factor
The PDB File:
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
ATOM
.
.
.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N
CA
C
O
CB
CG
CD
OE1
OE2
N
CA
C
O
CB
CG
CD
NE
CZ
NH1
NH2
GLU
GLU
GLU
GLU
GLU
GLU
GLU
GLU
GLU
ARG
ARG
ARG
ARG
ARG
ARG
ARG
ARG
ARG
ARG
ARG
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
A
27
27
27
27
27
27
27
27
27
28
28
28
28
28
28
28
28
28
28
28
41.211
42.250
42.601
43.691
41.725
42.804
43.628
44.194
43.713
41.662
41.839
41.380
42.184
41.035
39.564
38.845
37.423
36.945
37.771
35.634
44.533 94.570
44.748 95.621
43.408 96.271
42.865 96.065
45.720 96.687
46.349 97.563
47.387 96.817
47.051 95.754
48.540 97.296
42.882 97.053
41.607 97.739
40.458 96.835
39.619 96.424
41.607 99.045
41.944 98.851
42.152 100.169
42.439 99.980
43.413 99.208
44.208 98.537
43.598 99.111
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
85.98
86.10
85.99
85.71
86.36
86.44
86.98
87.40
87.02
85.65
85.29
85.31
85.09
84.62
84.07
84.00
84.27
84.53
83.83
84.38
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.
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.
Monitoring refinement
R
S||Fobs| - |Fcalc||
=
S|Fobs|
Rfree: an R-factor calculated from a test set that has not been used
in refinement.