1. 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
Online tutorial with interactive applets and quizzes.
Nice pictures demonstrating Fourier transforms
Cool movies demonstrating key points about diffraction, resolution, data quality, and refinement.
Notes from a macromolecular crystallography course taught in Cambridge

2. Overview of X-ray Crystallography
Crystal -> Diffraction pattern -> Electron density -> Model
Resolution, Fourier transforms, the ‘phase problem’, B-factors,
R-factors, R-free …

3. 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: nm
X-rays: nm, 1-2 Å

5. 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.

7. Varying intensity
Reciprocal space
Resolution
What is the physical basis for the pattern you see?
How do you get electron density from this information?

8. The wave nature of light
f(x) = Fcos2π(ux + a)
f(x) = Fsin2π(ux + a) x
F = amplitude
u = frequency
a = phase
ul=c
f(x) = cos 2πx
f(x) = 3cos2πx
f(x) = cos2π(3x)
f(x) = cos2π(x + 1/4)

9. Interference of two waves
Wave 1 + Wave 2
Wave 1
Wave 2
In-phase
Out -of-phase

10. Bragg’s Law
Sin q = AB/d
AB = d sin q
AB + BC = 2d sin q
nl = 2d sin q

11. nl = 2d sin q

14. We are sampling the continuous transform at specific points determined by the periodic lattice.
The lattice determines the spacing in the diffraction pattern.
The intensity of the spots contains the information about the lattice content.
Each individual spot on the diffraction pattern contains information about your entire molecule.

15. Diffraction pattern The intensity of each spot contains
Practically:
Assign a coordinate (h, k, l)
and intensity (I) to every spot in
the diffraction pattern—
Index and Integrate.
Ihkl , shkl
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.

17. 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

18. Calculating an electron density function from the diffraction pattern
F(h) = Fcos2π(uh+ a)
F(h) = Fsin2π(uh + a)
r(x) = ∫ F(h)e-2πi(hx)dh
F = amplitude
u = frequency
a = phase
Experimental measurements:
Ihkl, shkl
Fhkl ~ √Ihkl

19. Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods

20. 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

21. 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.

22. Native
Fnat
Heavy atom derivative
Fderiv

23. The steps of the isomorphous replacement method

24. 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).

25. Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods

26. 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

27. 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.

28. 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.

29. Overcoming the Phase Problem
Heavy Atom Methods (Isomorphous Replacement)
Anomalous Scattering Methods
Molecular Replacement Methods
Direct Methods

30. 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.

31. 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.

32. 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

33. Electron density maps

34. Are phases important?
Duck intensities
and cat phases

35. Does molecular replacement introduce model bias?
Cat intensities with
Manx phases

36. An iterative cycle of phase improvement
Building
Refinement
Solvent flattening
NCS averaging

37. Model building
Interactive graphics programs allow for the creation of a ‘PDB’ file.
Atom type, x, y, z, Occupancy, B-factor

38. The PDB File: ATOM 1 N GLU A 27 41.211 44.533 94.570 1.00 85.98
ATOM CA GLU A
ATOM C GLU A
ATOM O GLU A
ATOM CB GLU A
ATOM CG GLU A
ATOM CD GLU A
ATOM OE1 GLU A
ATOM OE2 GLU A
ATOM N ARG A
ATOM CA ARG A
ATOM C ARG A
ATOM O ARG A
ATOM CB ARG A
ATOM CG ARG A
ATOM CD ARG A
ATOM NE ARG A
ATOM CZ ARG A
ATOM NH1 ARG A
ATOM NH2 ARG A .

39. 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.

40. 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.

41. R = Monitoring refinement S||Fobs| - |Fcalc|| S|Fobs|
Rfree: an R-factor calculated from a test set that has not been used
in refinement.