🧬 Protein Crystallography — Full Fun Summary

🧊 1. Crystal basics — Unit cell & symmetry

📦 What is a unit cell?

A crystal = repeating box (unit cell) containing atoms arranged periodically.

Defined by:

• Edge lengths → a, b, c
• Angles → α, β, γ

These 6 parameters completely describe crystal geometry.

👉 This is the foundation of diffraction — the periodic arrangement is what produces diffraction spots.

🔷 Crystal systems & lattices (Image slide explanation)

On the slide (p.3), you see:

• 7 crystal systems
  • cubic
  • tetragonal
  • hexagonal
  • orthorhombic
  • rhombohedral
  • monoclinic
  • triclinic

Each system has different symmetry constraints.

There are also:

• 14 Bravais lattices → ways atoms can be arranged inside the unit cell.

🔁 Symmetry operations in biocrystallography

Proteins crystals often contain symmetry like:

• translation
• rotation
• screw axis (rotation + translation)

Example:

• 4₁ screw axis → rotate 90° + move along axis.

This symmetry reduces how much structure must be solved experimentally.

🧩 Space groups

Combination of:

• 14 lattices
• 32 point groups

→ gives 230 space groups

BUT:

👉 Only 65 are common for biomolecules.

📌 What defines a crystal structure?

To describe one we need:

1. atomic structure of asymmetric unit
2. unit cell parameters
3. space group symmetry

Image slide shows:

• asymmetric unit repeated → forms whole crystal.

💥 2. Diffraction theory — Bragg’s Law & Miller indices

✨ Bragg’s law

Condition for diffraction:

2d sin heta = nlambda

Meaning:

• waves scattered from parallel planes interfere constructively.

Thus:

• diffraction spots correspond to specific lattice plane spacing (resolution).

🔢 Miller indices (hkl)

They describe which planes diffract.

Example:

• (1 2 2) → plane cuts axes differently.

These indices define the position of each reflection.

🌍 3. Reciprocal space & Ewald sphere (Image explanation)

The Ewald construction slide shows:

• a sphere of radius 1/λ in reciprocal space.

Diffraction occurs when:

👉 reciprocal lattice points intersect the Ewald sphere.

Thus:

• rotating the crystal → more reflections appear.

📸 Diffraction patterns (image slide)

You saw:

• central intense beam
• surrounding discrete spots

These spots are:

👉 structure factor amplitudes

They are the experimental data.

🔬 4. Structure factors — heart of crystallography

Each reflection corresponds to:

F(hkl) = sum f_i e^{2pi i(hx_i + ky_i + lz_i)}

Meaning:

• all atoms contribute to each diffraction spot.

Important:

• structure factor is complex
  • amplitude
  • phase

But:

👉 Experiment measures only intensity = amplitude²

→ Phase is lost → PHASE PROBLEM

🌊 Diffraction intuition (image slides explained)

Slides show:

Diffraction from:

• single electron → blurry spot
• molecule → transform pattern
• lattice → discrete reciprocal lattice

Crystal diffraction =

👉 molecular transform sampled by reciprocal lattice

This explains why diffraction is:

• spotty
• structured
• not continuous.

🧮 5. Electron density maps (Fourier transform)

Key relationship:

• Diffraction = Fourier transform of electron density
• Electron density = inverse Fourier transform of diffraction

Electron density equation:

[

ho(x,y,z) = sum |F_| e^{iphi_} e^{-2pi i(hx+ky+lz)} ]

Interpretation:

• each reflection contributes a density wave.

🌊 Density wave slides (p.14-16)

Images show:

• waves with different:
  • h,k values
  • phases

When summed:

👉 produce localized density peaks → atoms.

Beautiful concept:

Structure = interference of many density waves.

❗ 6. The Phase Problem (BIG deal)

We measure:

• intensity → gives amplitude

But not:

• phase

Thus cannot directly reconstruct density.

Slide shows cartoon emphasizing this is the main challenge in crystallography.

🔓 7. Methods to solve the phase problem

Six major methods:

1. Patterson method
2. Direct methods
3. Isomorphous replacement
4. MAD
5. SAD
6. Molecular replacement

🧪 Historical breakthrough — Heavy atoms

Perutz realized:

• protein atoms scatter weakly
• heavy atoms scatter strongly

Thus:

• adding Hg / Pt etc. changes diffraction → gives phase info.

📐 Argand diagram (image slide explanation)

Vectors:

• FP = protein
• FH = heavy atom
• FPH = combined

By measuring amplitudes we can geometrically estimate phases.

⭕ Harker construction (SIR)

Graphical solution:

• circles represent unknown phase possibilities
• intersection gives phase solutions.

But:

👉 ambiguity remains → solved using MIR or anomalous scattering.

⚗️ How heavy atoms are introduced

Methods:

• SeMet labeling
• metal substitution
• heavy atom soaking
• halides
• noble gas pressurization

Mercury binding to cysteine is very common.

📍 8. Patterson function — locating heavy atoms

Definition:

• convolution of electron density with itself.

Key properties:

• calculated without phases
• gives interatomic vector map
• huge origin peak
• inversion symmetry.

Example slides show:

• 2-atom → 3 peaks
• 3-atom → 7 peaks

General rule:

N^2 - N + 1

Difference Patterson

Use:

Delta F = |F_| - |F_P|

Approximates heavy atom Patterson.

Thus:

👉 locate heavy atom positions → obtain phases.

🔎 9. Molecular replacement (MR)

Used when homologous structure exists.

Steps:

Rotation function

• find orientation of model.

Translation function

• place model in unit cell.

Uses Patterson vectors:

• intramolecular → orientation
• intermolecular → position.

Important factors:

• resolution
• B-factor
• model quality.

🌈 10. Anomalous scattering & SAD

At absorption edge:

• scattering factor becomes complex:

f = f_0 + f' + if''

Useful energies:

• peak → strong anomalous signal
• inflection → dispersive difference
• remote → baseline.

SAD produces:

• phase ambiguity → refined later.

🧠 11. Density modification

Improve phases using:

• solvent flattening
• histogram matching
• NCS averaging.

🗺️ 12. Types of electron density maps

Experimental maps

• from MIR / MAD / SAD.

Model-based maps

• Fo-Fc → error detection
• 2Fo-Fc → model building
• 3Fo-2Fc → weak features.

🏗️ 13. Model building

Goal:

👉 Convert density → atomic model.

Skeleton building (image slide)

Algorithm finds:

• continuous density path → protein backbone.

Baton building (manual)

• place Cα atoms stepwise.

Requires knowledge of:

• helices
• sheets
• side-chain chemistry
• hydrogen bonds
• hydrophobic packing.

Resolution effects (image slide)

• 4 Å → only fold
• 3 Å → side chains
• 2 Å → waters / H-bonds
• 1 Å → full atomic detail.

⚠️ Common model building errors

• wrong fold
• chain direction reversed
• register shift
• wrong side chain rotamers
• incorrect SSE connections.

🤖 Automated building

Programs:

• PHENIX
• ARP/wARP

Pros:

• fast
• objective

Cons:

• difficult regions remain.

🔧 14. Structure refinement

Goal:

ext{Minimize } |F_o - F_c|

Uses:

• restraints on geometry
• least squares / simulated annealing.

B-factor

Represents:

• atomic motion / disorder.

Low B → rigid High B → flexible or poorly defined.

R-factor & Rfree

R = rac{sum |F_ - F_|}{sum F_}

Typical values:

✅ 15. Model validation

Important metrics:

• RSCC > 0.9
• Ramachandran statistics
• bond RMSD
• Wilson B-factor
• completeness
• I/σI > 2
• Rsym < 40% in highest shell.

These are summarized in the final statistics slide.

🎯 BIG conceptual flow (VERY important for exam)

1. Grow crystal
2. Collect diffraction intensities
3. Solve phase problem
4. Calculate electron density
5. Build atomic model
6. Refine
7. Validate

👉 This is the full crystallographic pipeline.