🧊 Cryo-EM & Negative Staining — Full Theoretical Summary

🔬 Theoretical Resolution in Electron Microscopy

📏 Nyquist limit and pixel size

A very important concept in cryo-EM is that the detector pixel size determines the theoretical resolution.

• If pixel size = 1 Å
• Then theoretical resolution = 2 × pixel size = 2 Å

This comes from the Nyquist sampling theorem — you must sample at least twice per feature size.

👉 Meaning:

• To resolve a feature of size d, you need pixels ≤ d/2

So:

• Smaller pixel size → higher theoretical resolution
• Achieved by increasing magnification

This is a detector sampling limit, not a physical electron wavelength limit (electron wavelength is picometer → much smaller).

⚠️ Why you often do NOT reach theoretical resolution

Even if microscope settings are perfect:

Sample quality determines real resolution

Important factors:

• Protein flexibility → blurring during averaging
• Structural heterogeneity
• Ice thickness
• Preferred orientation
• Beam damage

Thus:

Resolution is ALWAYS limited by the weakest link — usually the sample.

📊 How resolution is measured — Fourier Shell Correlation (FSC)

Principle

1. Split particles randomly into two independent halves
2. Reconstruct two independent maps
3. Compare similarity at different spatial frequencies

• Correlation = 1 → identical maps
• Correlation decreases at higher resolution

Resolution cutoff

Cryo-EM convention:

Resolution = frequency where FSC = 0.143

This empirical threshold was derived by comparing cryo-EM maps with crystallographic maps.

📡 Frequency space intuition

• Center of Fourier transform → low spatial frequency → low resolution (large features)
• Edge → high spatial frequency → high resolution (fine detail)

Signal fades into noise at high frequency → defines resolution limit.

🎯 CTF Fit — Contrast Transfer Function

What is CTF?

When imaging in cryo-EM:

• The microscope optics modulate spatial frequencies
• This produces oscillating rings in Fourier space

These rings describe the Contrast Transfer Function (CTF).

Why CTF fitting is important

By fitting theoretical CTF to experimental image:

You determine:

• Defocus value
• Phase distortions
• Frequency transfer properties

Accurate CTF estimation is essential for high-resolution reconstruction.

If CTF is wrong:

• High-frequency information is misinterpreted
• Resolution decreases dramatically

⚫ Negative Staining — Amplitude Contrast

How contrast is generated

Heavy metal salts (e.g. uranyl acetate) surround the protein.

Electrons interacting with heavy atoms:

• Scatter to high angles or backwards
• Do not reach detector → dark regions

Electrons passing through protein:

• Mostly transmitted
• Bright signal

Result:

Protein appears bright on dark background → “negative image”.

Key properties

Advantages:

• Strong contrast
• Easy particle picking
• Good for shape determination

Disadvantages:

• Low resolution (~20 Å)
• Possible structural distortion
• Surface flattening on carbon film

❄️ Cryo-EM — Phase Contrast

What happens physically

• Most electrons pass straight through sample (direct beam)
• Small fraction is elastically scattered

These two beams:

Interfere as waves at the detector.

This wave interference between direct beam and scattered beam generates phase contrast.

Important:

• Contrast is weaker than negative stain
• But preserves native structure → enables atomic resolution

Why underfocus is used

Cryo-EM intentionally uses defocus:

• Enhances phase contrast
• Produces CTF oscillations
• Improves visibility of particles

🧱 From 2D Projections to 3D Structure

What images represent

Each particle image is:

A 2D projection of a 3D object

Different orientations → different projection shapes.

🧩 2D Classification

Steps:

1. Automatically pick particles (includes junk)
2. Sort particles into classes based on similarity
3. Average within each class → class averages

Benefits:

• Improves signal-to-noise ratio
• Removes contaminants
• Selects good particles for reconstruction

🧠 3D Reconstruction — Fourier principle

Key theoretical idea:

• Fourier transform of each 2D projection = slice through 3D Fourier space
• Different projections share common lines (common axes)

Thus:

• Align projections pairwise via common axes
• Fill 3D Fourier space
• Inverse transform → 3D density map

This is the basis of single-particle reconstruction.

⚠️ Preferred orientation problem

If particles freeze mainly in one orientation:

• Missing angular information
• Reconstruction becomes impossible or anisotropic

Solution:

• Modify grid conditions
• Change detergent / support film
• Tilt data collection

⚡ Electrostatic Potential Map vs Electron Density Map

Very important conceptual difference.

X-ray crystallography

• X-rays scatter from electron clouds
• Map = electron density

Cryo-EM

• Electrons scatter from electrostatic potential
• Includes nuclear charge + electron distribution

Thus:

Cryo-EM map = electrostatic potential map

Not strictly the same as electron density.

This affects:

• Interpretation of hydrogen atoms
• Charge distribution visibility
• Refinement models

📌 Key Intuition Summary (Exam-Ready)

• Resolution limit = 2 × pixel size (Nyquist)
• Real resolution limited by sample quality
• FSC 0.143 → resolution cutoff
• CTF fit → determines defocus → critical for high resolution
• Negative stain → amplitude contrast (heavy metal scattering)
• Cryo-EM → phase contrast (wave interference)
• 2D projections → classified → averaged → reconstructed in Fourier space
• Cryo-EM maps electrostatic potential (not pure electron density)