🧬 Lecture 7 — Video 6 Summary

(Protein crystallography: symmetry, diffraction geometry & data statistics)

🔷 1. Recap: Unit Cell — The “Building Block” of Crystals

A unit cell is the smallest repeating box that builds the entire crystal.

It is defined by 6 parameters:

• 3 unit cell lengths:
  • ( a, b, c )
• 3 interaxial angles:
  • ( alpha, eta, gamma )

Crystal Systems (7 total)

They range from least constrained → most constrained:

🔷 2. Bravais Lattices (14 total)

Beyond just corners, lattices can have additional centering:

• Primitive (P) → lattice points only at corners
• Body-centered (I) → extra point in the middle
• Face-centered (F) → points on each face
• Base-centered (C) → points on one pair of faces

These combinations give 14 Bravais lattices.

🔷 3. Symmetry Operations in Protein Crystals

Three major symmetry operations:

🟢 Translation

Move the molecule without rotating.

🔵 Rotation

Rotate around an axis:

• 2-fold
• 3-fold
• 4-fold
• 6-fold

🟣 Screw Axis

Rotation + translation along axis

Example:

• 4₁ screw → rotate 90° and translate ¼ unit cell.

🔷 4. Space Groups — Why Only 65 for Biomolecules?

Mathematically:

• 230 total space groups

But proteins only use 65.

Reason: Chirality

• Proteins contain L-amino acids
• Mirror symmetry or inversion would create D-amino acids
• This cannot happen in real protein crystals

Therefore: 👉 Only non-inverting symmetry space groups are allowed.

🔷 5. What Do You Need to Describe a Crystal Structure?

Three key things:

1. Atomic structure of the asymmetric unit (e.g., one protein molecule)
2. Unit cell dimensions
3. Space group symmetry

By applying symmetry operations to the asymmetric unit → You reconstruct the entire crystal lattice.

📡 6. Diffraction Geometry — Bragg’s Law

When X-rays hit parallel crystal planes:

Constructive interference occurs when:

2dsin heta = lambda

Where:

• d → distance between crystal planes
• θ → Bragg angle
• λ → X-ray wavelength

This condition produces visible diffraction spots.

🔷 Miller Indices (hkl)

They define which plane diffracts.

Example in lecture:

• Plane intersects:
  • a-axis once → h = 1
  • b-axis twice → k = 2
  • c-axis twice → l = 2

So plane = (1 2 2).

🌐 7. The Ewald Sphere — When Diffraction Happens

Important conceptual tool:

• Virtual sphere with radius ( 1/lambda )
• Crystal sits at center
• Reciprocal lattice rotates as crystal rotates

Diffraction condition:

👉 A diffraction spot appears when a reciprocal lattice point touches the Ewald sphere.

As the crystal rotates → different reciprocal lattice layers (“lunes”) intersect → new reflections recorded.

📊 8. Understanding “Table 1” in Crystallography Papers

This table shows data quality statistics.

🧩 Space Group & Unit Cell

Example:

• P2₁2₁2₁ → primitive lattice + three 2-fold screw axes
• Orthorhombic → angles = 90° (often not listed)

Unit cell lengths shown in Ångström.

🎯 Resolution

• Example datasets:
  • 3.4 Å (lower resolution)
  • 1.9 Å (higher resolution → better detail)

“Last shell” = highest resolution range used in statistics.

📉 R-merge

Measures consistency between repeated measurements.

• Lower = better
• Typically increases in last shell (~0.5)

High resolution dataset → usually lower R-merge overall.

🔊 Signal-to-Noise (I/σI)

Rule of thumb:

• Cut data when I/σI ≈ 2

Example:

• Overall: 8.5
• Last shell: 1.6 → borderline

Better dataset:

• Overall: 26
• Last shell: 2.9 → strong data.

✅ Completeness

How many possible reflections were measured.

• Ideal: ~100%
• Even ~80% can still give good electron density maps

But always aim for highest completeness.

🔁 Redundancy (Multiplicity)

How many times each reflection was measured.

• Good target: >3
• Example: ~6 → very good statistics.

⭐ Key Big-Picture Takeaways

• Protein crystals are defined by unit cell + symmetry + asymmetric unit
• Only 65 space groups are possible due to chirality
• Diffraction happens when Bragg’s law is satisfied
• The Ewald sphere explains diffraction geometrically
• Always check Table 1 in papers to judge structure quality:
  • Resolution
  • R-merge
  • I/σI
  • Completeness
  • Redundancy