🧬 Lecture 8 — Protein Crystallography

⭐ Electron density, Patterson maps, Molecular Replacement & SAD phasing

This lecture is SUPER important because it explains how we actually solve the phase problem and build models in macromolecular crystallography.

Think of the workflow like this:

👉 Diffraction → Intensities → Phases → Electron density → Model building

This lecture focuses mainly on how to get phases.

🔵 The Patterson Function — Solving the phase problem indirectly

💡 What is the Patterson function?

The Patterson function is:

A convolution of the electron density with itself → giving an interatomic vector map.

Mathematically it is calculated using only measured intensities |F|², so:

✅ Phases are NOT needed This is extremely powerful.

So even though we cannot calculate electron density directly (because we lack phases), we can calculate a Patterson map.

🧠 What does a Patterson map show?

It shows:

👉 Vectors between atoms (distance + direction) Not the actual atom positions.

Key properties:

• Large origin peak (all atoms overlap with themselves)
• Number of peaks = N^2 - N + 1
• Has inversion symmetry P(u,v,w)=P(-u,-v,-w)

This means:

➡️ Peaks always come in pairs ➡️ Map is more complex than electron density

🖼️ Image-only slides explanation (2-atom and 3-atom Patterson)

📍 Page 4 diagrams

These diagrams show:

• Real atom positions in the unit cell
• All vectors between atoms drawn
• Patterson map showing peaks at vector positions

For:

🔹 2 atoms

Vectors:

• self vectors (origin)
• vector 1→2
• vector 2→1

Total peaks:

2^2 - 2 +1 = 3

🔹 3 atoms

Now many more vectors appear:

• 1→2, 2→1
• 1→3, 3→1
• 2→3, 3→2

Total peaks:

3^2 - 3 +1 = 7

🧠 Important takeaway:

Patterson maps become extremely crowded for macromolecules → difficult interpretation.

🟡 Difference Patterson — Finding heavy atoms

To solve phases via isomorphous replacement, we need:

👉 Positions of heavy atoms

But we cannot directly calculate their structure factors.

Instead we use:

Delta F_ = |F_| - |F_P|

A Patterson using ΔFiso² approximates heavy-atom Patterson (half-scale).

⚖️ Vector weights (example bromobenzene)

Peak height depends on electron number product:

• H–H → 1
• C–C → 36
• Br–Br → 1225

Thus:

⭐ Heavy atoms dominate Patterson maps → makes them easier to locate.

🟢 Harker Sections — Using symmetry to simplify Patterson

Symmetry operations create special planes in Patterson space called:

👉 Harker sections

Example shown:

• Space group P212121
• Harker sections at
  • u = ½
  • v = ½
  • w = ½

These restrict possible heavy-atom coordinates → makes solving easier.

🧠 Example heavy atom solution (pages 8–9)

By analysing peaks in:

• v = ½ section
• w = ½ section

Heavy atom positions determined:

• (±0.17, ±0.2, −0.1)
• (±0.08, 0.0, ±0.15)

Important crystallographic ideas here:

✅ Origin can be shifted ✅ Symmetry operations generate equivalent solutions ✅ Hand ambiguity possible

🔴 Molecular Replacement (MR)

💡 Concept

Instead of heavy atoms, use:

👉 A known homologous structure (search model)

We rotate and translate this model in the unit cell until:

➡️ Calculated diffraction matches observed diffraction.

Then phases come from:

F_

🧠 Why Patterson is useful for MR

📍 Image explanation (page 11)

The diagram shows:

• Intramolecular vectors → depend only on orientation
• Intermolecular vectors → depend on position

Thus:

⭐ First step = rotation search ⭐ Second step = translation search

Also:

• Choosing a Patterson integration radius helps isolate intramolecular vectors.

🟣 Rotation Function — Finding orientation

Rotation function compares:

• Observed Patterson
• Model Patterson (at different rotations)

Best overlap → correct orientation.

Parameters affecting success:

• Resolution
• Model quality
• B-factor falloff
• Integration radius
• Unit cell box size

If radius too large:

❗ intermolecular vectors contaminate signal.

🟠 Translation Function — Finding position

Now we fix orientation and move model around.

Translation affects:

👉 Intermolecular vectors only

We compare predicted Patterson with observed → best match gives position.

The image slide shows:

• Model shifting inside unit cell
• Intermolecular vector patterns changing.

🔵 Anomalous Scattering & SAD

💡 Atomic scattering becomes complex near absorption edge

Atomic scattering factor:

f = f_0 + f' + i f''

• f′ = dispersive correction
• f″ = anomalous signal

🖼️ Image explanation — absorption edge (page 15)

Graph shows:

• XANES region (sharp features)
• EXAFS oscillations
• Large jump in absorption at edge

This corresponds to:

⭐ Maximum anomalous signal.

📊 Optimal data collection wavelengths

1. Peak → maximum f″ (strong anomalous signal)
2. Inflection → minimum f′
3. High-energy remote → f′ ~ 0
4. Low-energy remote → both small

These allow:

• SAD
• MAD
• dispersive phasing

🔴 SAD Phasing

SAD uses:

F^+ eq F^-

Because anomalous scatterers break Friedel symmetry.

But:

❗ There is phase ambiguity → two possible solutions.

🟢 Phase Refinement / Density Modification

To improve maps:

Methods:

• 🔁 NCS averaging
• 🌊 Solvent flattening / flipping
• 📈 Histogram matching

The slide image shows:

👉 messy density → clearer helical features after modification.

🟡 Summary — How phases are obtained

Main approaches:

1️⃣ Molecular replacement 2️⃣ Isomorphous replacement (heavy atoms) 3️⃣ Anomalous scattering (SAD/MAD) 4️⃣ Dispersive differences

🔵 Types of electron density maps

Experimental maps

• MIR / MAD / SAD Fourier maps

Model-based maps

• Fo−Fc difference map
  • positive peaks → missing atoms
  • negative peaks → wrong atoms
• 2Fo−Fc map → main refinement map
• 3Fo−2Fc map → shows new features more clearly.

⭐ BIG conceptual takeaways for exams

✅ Patterson = vector map → no phases needed ✅ Heavy atoms dominate Patterson peaks ✅ Harker sections reduce dimensionality ✅ MR uses orientation (rotation function) then position (translation function) ✅ SAD uses anomalous differences → phase ambiguity ✅ Density modification crucial for usable maps