1. Phasing in Macromolecular Crystallography
What do we do with it now?

2. Phasing in Macromolecular Crystallography
How do we get from spots on a screen to a pretty picture of our protein?
BY CALCULATING AMPLITUDES AND PHASES!!
Heidi’s Gun4 Data and Stucture

3. Why Do We Care So Much About Phases?

4. Diffraction by X-rays h = 1 k = 2 l = 0 Bragg’s Law nλ = 2dsinθ k0
SOURCE
DETECTOR Θ k
h = 1
k = 2
l = 0 y x z
dhkl
Bragg’s Law
nλ = 2dsinθ

5. Diffraction by X-rays
(h,k,l) = (2,1,0) k k0
SOURCE
Crystal

6. The Phase Problem
A PHASE DIFFERENCE!! k0 Φ k

7. Addition of Waves f1 + f2 + f3 Fs

8. The Phase Problem
imaginary axis
The resulting wave that reaches the detector has a particular phase and amplitude that results from the addition of individual scattering factors from all the atoms in the unit cell, which each have their own phase and amplitude.
Fhkl f3
Φhkl
real axis f2 f1
Fs = V ∑ fj[cos2π(hx + ky + lz) + isin2π(hx + ky + lz)] N
j=1

9. The Phase Problem The Phase Problem:
Each reflection we measure during the diffraction experiment tells us the amplitude of a particular Fs, but not its phase.

10. Addition of Waves f1 + f2 + f3 Fs

11. Solving the Phase Problem
How do we figure out the phase of Fs?
Combine Fs (or FP here) with a wave of known phase (fH) to get a new resultant wave (FPH). FP
(unknown phase) + fH
(known phase)
FPH
(unknown phase)
FPH = FP + fH

12. Solving the Phase Problem: Harker Constructions
imaginary axis
Hurray!! We’ve solved the phase problem!! fH
real axis
FPH FP
Well, sort of… Now we actually have two phases to choose from.
FPH = FP + fH

13. Solving the Phase Problem
imaginary axis
FPH = FP + fH
How do we actually figure out the amplitude and phase of fH? fH
real axis
FPH FP
Amplitude: determined during the diffraction experiment
Phase: determine the real space (x,y,z) location of “H” through the use of the Patterson Function

14. The Patterson Method
The Patterson function is similar to the electron density equation in that they both use amplitudes measured during the diffraction experiment.
However, Patterson functions do not require phases to be computed, just the amplitudes of the (h,k,l) reflections
P(u,v,w) = 1/V ∑ |Fhkl|2cos2π(hu + kv + lw)
h,k,l

15. The Patterson Method
The Patterson map provides a map of interatomic vectors within the unit cell and the peaks in a Patterson map are proportional to the electron density at a particular position.
1-2
1-3
2-1
2-3
3-1
3-2
PATTERSON SPACE 1 2 3
REAL SPACE

16. The Patterson Method
The Patterson function is especially useful because of two important features:
The magnitude of P(u,v,w) is proportional to the product of the atomic numbers of the atoms at the ends of the vector u = (u,v,w).
The symmetry within a unit cell is imposed on the peaks in a Patterson map. This means that symmetry related atoms will also have a peak in the Patterson map.

17. Harker Sections (-x,y,-z) (x,y,z)
180°
(-x,y,-z)
(x,y,z) x z
Should have a peak on the v=0 Harker section of the Patterson map.
(x,y,z) - (-x,y,-z)
= (2x,0,2z) = (u,v,w)
x = u/2, y = 0, z = w/2

18. Harker Sections v = 0 Harker Section (2x,0,2z) = (u,v,w)
x = u/2, y = 0, z = w/2

19. Solving the Phase Problem
imaginary axis
FPH = FP + fH
How do we actually figure out the amplitude and phase of fH? fH
real axis
FPH FP
Amplitude: determined during the diffraction experiment
Phase: determine the real space (x,y,z) location of “H” through the use of the Patterson Function

20. Methods for Determining Phases
Isomorphous Replacement
Single Isomorphous Replacement (SIR)
Multiple Isomorphous Replacement (MIR)
Anomalous Dispersion
Single Wave-Length Anomalous Dispersion (SAD)
Multiple Wave-Length Anomalous Dispersion (MAD)
Molecular Replacement
Direct Methods

21. Isomorphous Replacement
The Goal: Modify our crystal by having it bind a heavy atom.
Why? A heavy-atom derivative of our crystal will create a change in the intensity of observed reflections relative to our native crystal.

22. Isomorphous Replacement
Why do we use heavy atoms?
Because they have lots of electrons and scatter x-rays more strongly.
NH = number of heavy atoms
fH = heavy atom scattering power
NN = number of native atoms
fN = avg. scattering power for native atoms

23. Isomorphous Replacement
imaginary axis
imaginary axis FP FP f3
Φhkl
Φhkl
real axis
real axis f2 f1
FPH f1 f2 f3 fH
NATIVE
HEAVY-ATOM DERIVATIVE

24. Isomorphous Replacement
imaginary axis FP
Φhkl
real axis
FPH f1 f2 f3 fH
HEAVY-ATOM DERIVATIVE

25. Isomorphous Replacement
Why do we care about intensity differences in the observed diffraction pattern?
The differences in intensity (ΔF = |FPH| – |FP|) can be used as coefficients for the Patterson Function.
This is useful because our Patterson Maps and Harker Sections are now giving us information about the locations of our heavy-atom derivatives. THIS IS REALLY IMPORTANT!!
P(u,v,w) = 1/V ∑ |ΔF|2cos2π(hu + kv + lw)
h,k,l

26. Heavy Atoms in Isomorphous Replacement
Common Heavy Metals
Platinum Potassium Thiocyanate
Gold Cyanide
Potassium Tetrachloro Platinate
Thimerasol
Amino Acid Ligands
His (pH>7), Lys (pH>9)
Cys, His (pH>7), Lys (pH>9)
Cystines, His (pH>7)
Cys, His (pH>7)
And many more…
How do we make derivatives of our crystal?
Trial and error: add a small amount of the metal reagent ( mM) to the crystallization condition and soak the crystal (seconds, minutes, hours, days)

27. Detecting Derivatives
Unfortunately, like most things in crystallography, just about everything is a variable when trying to get a derivative.
So how do we know when we’ve gotten a derivative?
Answer: Look for differences in spot intensity between native and potential derivative crystals, especially at low resolution where heavy atom differences will be strongest

28. Detecting Derivatives
It must take a long time to find a derivative!!!
Fortunately, we don’t have to collect a full dataset for each derivative
Scaling a few images of a derivative dataset against a native dataset is enough to detect differences in intensity

29. Detecting Derivatives
First things first: A derivative crystal should be different than the native crystal.

30. Detecting Derivatives
Use Scalepack to scale a native dataset against a few images (~3°) of a potential derivative
When derivative and native crystals are scaled together, the Χ2 value should be >1.
Assuming the cell parameters look good,
If Χ2 is…
~50, probably not a derivative (non-isomorphous, wrong indexing, or way too many substitutions)
~10, good chance that you have a derivative
~2-5, well…I would keep looking for derivatives, but keep this one in mind
1, scales well with native so you’ve probably got a native crystal (no heavy-atom substitution)

31. Isomorphous Replacement: The Good and the Bad
Can be quick since data collection can be done at home
Works well for proteins purified from sources other than E. coli (e.g. yeast)
Don’t need high quality data (low resolution is okay)
The Bad
Getting a derivative in the first place is not trivial
Getting a derivative crystal that’s isomorphous can be difficult
Data quality can be poor

32. The Reality of Isomorphous Replacement
imaginary axis FP
Φhkl fH
real axis
FPH
FPH = FP + fH

33. The Reality of Isomorphous Replacement
imaginary axis
In reality, there’s some error in our measurement of FPH and the location of the heavy atoms FP
Φhkl fH
real axis
FPH,obs ε
FPH,calc
ε = FPH,obs – FPH,calc
ε = lack of closure
FPH ≈ FP + fH

34. The Reality of Isomorphous Replacment
Harker Construction of a single reflection from an SIR experiment
Phase Probability
P(α) = exp{-ε2(α)/2E2}
Best Phase
αbest = ∫αP(α)dα
Figure of Merit
m = ∫P(α)exp(iα)dα/∫P(α)dα
m = 1, no phase error
m = 0.5, ~60° phase error
m = 0, all phases equally probably
The phases have a minimum error when the best phase (alpha_best), i.e. the centroid of the phase distribution, is used instead of the most probably phase. The quality of the phases is indicated by the figure of merit, m. A value of 1 for m indicates no phase error, while a value of 0 means that all phases are equally probable.
Minimize the phase error by using the centroid of the phase distribution (Best Phase).

35. The Reality of Isomorphous Replacement
By using multiple heavy atom derivatives, we can get a better estimate of the correct phase
This is Multiple Isomorphous Replacement (MIR)

36. The Reality of Isomorphous Replacment
SIR
MIR
EMPHASIZE THAT THESE ARE STILL VERY IDEALIZED AND THAT IN REALITY THE PHASES THAT WE WOULD NORMALLY GET ARE GOING TO BE PRETTY POOR BECAUSE THE CIRCLES WILL STILL NOT OVERLAP PERFECTLY. MODERN METHODS USE MAXIMUM LIKLIEHOOD METHODS OF GETTING THE BEST PHASES.

37. Anomalous Dispersion Techniques
What is Anomalous Dispersion?
A phenomenon which occurs when electrons absorb and reemit X-rays having an energy close to that of the electron’s nuclear binding energy.
What’s the result of this absorption?
The x-ray’s reemitted have the same wavelength as the incident radiation, but now are phase shifted by 90°.

38. Effects of Anomalous Dispersion
Radiation scattered by an atom is actually composed of two components:
“Normal” Thompson scattering (no phase change relative to incident radiation)
A minor anomalous component phase shifted by π/2
f(λ) = f0 + Δf'(λ) + if''(λ) = f'(λ) + if''(λ)
fH+
fH-
FP+
FP-
FPH+
FPH-
f''
FPH+
FPH-
fH+
fH-
FP+
FP-
Bijvoet Differences
(That’s Bi-foot)
|F+| = |F-|
Friedel’s Law
|F+| = |F-|
Breakdown in Friedel’s Law

39. Wavelength Dependence of Anomalous Dispersion

40. Anomalous Signal Increases With Scattering Angle

41. Using Anomalous Dispersion to Solve the Phase Problem: MAD
P(u,v,w) = 1/V ∑ (|F+| - |F-|)2cos2π(hu + kv + lw)
h,k,l

42. Choosing Appropriate Wavelengths
Peak
(f'')
Remote
Inflection (f')

43. MAD Requirements
Strong anomalous signal (at least 1 Se per 17 kDa of protein)
Tunable x-ray source (synchrotron)
Preferably the ability to measure the absorbance spectrum of your protein
High solvent content
Best data possible- high resolution and low Rmerge

44. Single Wavelength Anamalous Dispersion (SAD)
Solve the phase ambiguity by evaluating the quality of the maps for both solutions using density modification programs.
Basic Requirements:
Strong anomalous signal (usually at least 1 Se per 17 kDa of protein)
As usual, best data possible (resolution and error)
High solvent content (>50%)
Accurate measurement of phasing errors

45. Sulfur Anomalous Phasing

46. Sulfur Anomalous Phasing

47. Sulfur Anomalous Phasing
Ramagopal et al. Acta Cryst. (2003) D59,

48. Detecting Anomalous Signals
Using Scalepack, scale data as normal, except turn on the ANOMALOUS flag (writes out F+ and F- separately in a .sca file).
Rescale this .sca file, but this time with the anomalous flag turned off. This compares F+ and F-.
Examine the X2 values. Presence of an anomalous signal should give X2 > 1 (or could indicate absorption or detector problems). Useful for also examining the resolution cut-off of the anomalous signal.

49. The Good and the Bad of MAD/SAD Structure Determination
Essentially eliminate the isomorphism problems of SIR/MIR. Generally better phases.
Ability to use molecular biology to derivatize your protein (SeMet).
May also be able to use naturally bound anomalous scatters (Zn, Fe, Ca, etc.)
Usually need to go to synchrotron
However, there’s often an anomalous signal with Cu-Kα; could be useful in SIRAS/MIRAS methods.
The potential of sulfur anomalous phasing essentially eliminates the need to derivatize your protein
Method can be limited to especially good data
Due to availability of synchrotron resources, anomalous phasing is the primary method of choice for phasing

50. Calculation of Protein Phases
After solving the real space location of the heavy atom through isomorphous or anomalous difference Patterson’s, determine the protein phases:
Refine the xyz coordinates of the heavy atom through cycles of refining the occupancy and B-factor
Determine the protein phases from the refined heavy atom coordinates and refine these phases
Look at the initial map and see how you did
Then go and talk to Devin to see where to go from here
Programs that can do these steps include PHASES and MLPHARE (available in the CCP4 package)

51. Automated Methods: SOLVE
Fortunately, there are automated processes available to do everything from scaling data, solving the location of heavy atoms, and determining protein phases. One such program is SOLVE.
Works by converting each decision making step into an optimization problem through scoring and ranking of possible solutions
Locate and refine heavy atom sites through difference Patterson’s or direct methods and generate phases. Converts MAD data to pseudo-SIRAS.
Score potential heavy atom sites by four criteria:
Agreement between calculated and observed Patterson maps
Cross-validation of the heavy-atom sites through difference Fourier analysis- delete a site in a solution and recalculate phases
Figure of Merit (m)
Non-randomness of the electron density map- identify solvent and protein regions and score based on connectivity in solvent and protein region.
SOLVE is pretty cool because it actually only finds 1 or 2 sites in the Patterson function, scores solutions, and then analyses each of those solutions based on the four criteria above. Additional sites are added based on difference fourier analysis and solutions are scored. The initial set of solutions are called seeds and sometimes a particular seed will emerge that is especially promising. SOLVE will focus on this seed and try to build it up as quickly as possible. Cycles are repeated until no new sites are found and no significant improvement of the phases are achieved. At the conclusion of the algorithm, electron density maps are generated

52. Tutorial
Use Heidi’s Gun4 MAD data to calculate and analyze Patterson Maps and Harker Sections to determine the real space coordinates of the Selenium within the structure.