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Lecture 7 Video 5

Protein structure
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🌟 Diffraction Basics β€” Full Conceptual Summary

This lecture introduces the fundamental physics behind X-ray diffraction, which is essential for understanding how we determine protein structures.

🦴 Historical Start β€” First X-ray Image

One of the first X-ray transmission images showed a human hand.

Key ideas:

  • X-rays interact with electrons
  • Bone scatters more than soft tissue β†’ because bone contains calcium phosphate (many electrons).
  • Materials can even be identified by scattering strength (e.g., distinguishing gold vs cubic zirconia).

πŸ‘‰ This shows that scattering intensity depends strongly on electron number.

🌈 Electromagnetic Waves β€” Why X-rays?

Electromagnetic radiation spans from radio waves β†’ visible light β†’ X-rays β†’ gamma.

Important relationships:

  • Shorter wavelength β†’ higher energy
  • To see atoms β†’ need wavelength ~ 1 Γ… (angstrom) β†’ This is why hard X-rays are used in crystallography.

Wave properties:

  • Electric field vector βŸ‚ Magnetic field vector βŸ‚ Direction of propagation
  • The electric field interacts strongly with electrons (magnetic interaction is negligible).

πŸ‘‰ Therefore diffraction theory mainly considers electric field–electron interaction.

βš›οΈ How X-rays Scatter β€” Quantum Picture

A common misunderstanding: ❌ Not two beams reflecting from crystal planes.

βœ” Correct picture:

  • A single photon travels through the crystal
  • Usually β†’ nothing happens (~99%)
  • Sometimes β†’ photon induces coherent oscillation of electrons
  • Photon β€œdisappears” (virtual process) and a new photon appears in a scattered direction.

Diffraction pattern formation:

  • Many individual photon scattering events accumulate
  • Detectors can count single photons β†’ build pattern gradually.

πŸ‘‰ Diffraction is fundamentally quantum mechanical + probabilistic.

🌊 Interference β€” Adding Waves

Phase matters!

  • If waves are 180Β° out of phase β†’ destructive interference β†’ zero signal.
  • If in phase β†’ constructive interference β†’ strong signal.

This determines where diffraction spots appear.

🧠 Complex Numbers Make Wave Addition Easier

Instead of adding sine waves point-by-point:

We use complex vectors (Argand diagram):

  • Vector length = amplitude
  • Angle = phase

Resultant wave: F_3 = F_1 + F_2

Intensity: I propto |F|^2

This becomes extremely important later in structure factor calculations.

πŸ“ Wave Function Parameters

General 1D wave:

F(x) = A cos(2pi Hx + alpha)

Where:

  • A = amplitude
  • H = frequency
  • Ξ± = phase shift

Changing these gives:

  • Higher amplitude β†’ taller peaks
  • Higher frequency β†’ more oscillations
  • Phase shift β†’ wave moves left/right

πŸ”¬ Fourier Series β€” Key Concept for Crystallography

Jean-Baptiste Fourier showed:

Any periodic function can be described as a sum of simple waves.

Connection to proteins:

  • Diffraction waves = simple sine waves
  • Electron density of protein = complex target function

Thus:

⭐ Electron density = Fourier sum of diffracted X-rays

This is the mathematical foundation of structure determination.

🎯 Scattering by a Single Atom

Important vectors:

  • Incoming wavevector β†’ Sβ‚€
  • Scattered wavevector β†’ S₁
  • Scattering vector β†’ mathbf{S} = mathbf{S_1} - mathbf{S_0}

As scattering angle increases:

  • Magnitude of S increases
  • Corresponds to higher resolution information

⚑ Thomson Scattering β€” Intensity Dependence

Key conclusions:

  • Scattering mainly from electrons (not nuclei)
  • Strongest scattering:
    • Forward direction
    • Backward direction
  • Weakest scattering:
    • Perpendicular direction

Also:

  • Elastic scattering β†’ energy conserved
  • Only direction changes.

🧩 Atomic Scattering Factor

Describes how strongly an atom scatters.

Important trends:

  • At zero angle β†’ equals number of electrons
  • With increasing angle β†’ scattering intensity falls off

Example:

  • Fe (26 electrons) scatters more than:
  • S (16) β†’ O (8) β†’ C (6) β†’ H (1)

πŸ‘‰ Heavy atoms give stronger diffraction signal.

🌑️ B-factor (Atomic Displacement Factor)

Atoms are not fixed β€” they vibrate.

This causes:

  • Blurring of electron density
  • Loss of high-resolution reflections

B-factor meaning:

  • Low B β†’ rigid, well-defined position (protein core)
  • High B β†’ flexible, mobile (protein surface)

Typical values:

  • 2 – 200 Γ…Β²

If B too high:

  • Density averages out β†’ atom may be invisible in map.

πŸ“‰ B-factor vs Resolution

Higher B-factor:

  • Faster decay of scattering at high angles
  • Limits achievable resolution

Thus: ⭐ Overall protein B-factor correlates with diffraction quality.

🧠 BIG PICTURE β€” Why This Lecture Matters

This lecture builds the physics + math foundation for:

  • Why diffraction patterns form
  • How intensity relates to electrons
  • How waves interfere
  • Why Fourier transforms reconstruct structures
  • Why flexible atoms are hard to see
  • Why heavy atoms help phasing
  • Why resolution depends on motion

πŸ‘‰ These ideas directly lead into:

  • Structure factors
  • Electron density maps
  • Model building
  • Refinement

Quiz

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