📏 Protein Structure from NMR: Distances as Structural Information

This lecture focuses on how we extract distances between atoms from NMR data — and why those distances are crucial for determining protein structure.

The main hero of this story: NOEs (Nuclear Overhauser Effects) — or more precisely, NOESY cross-peak intensities .

🧭 1. Structural Information from NMR

NMR provides three major types of structural information:

• Dihedral angles (local backbone geometry)
• Distances (mostly short-range)
• Orientations

This lecture focuses entirely on distances, especially those obtained from NOESY spectra .

🔬 2. What Is an NOE (Really)?

Strictly speaking, what we call “NOEs” are:

Cross-peak intensities in NOESY spectra

Each cross peak corresponds to one distance between two atoms, usually two hydrogens .

Example:

A NOESY spectrum of hen egg white lysozyme (144 amino acids) contains hundreds or thousands of cross peaks — each representing a distance .

That’s enormous structural information.

🧩 3. The Big Challenge: Assignment

Even if you measure a cross peak, you must answer:

Which two atoms does this distance belong to?

This is why resonance assignment is essential.

• Once chemical shifts are assigned,
• Each cross peak can be mapped to two specific atoms.

But:

• Some regions are crowded
• Multiple atoms may share similar chemical shifts
• Ambiguity arises

3D NOESY helps reduce ambiguity by separating peaks along a heteronucleus dimension (¹⁵N or ¹³C) .

🧱 4. Why Distances Define Structure

Imagine a protein as a long flexible chain.

If you measure a distance between two residues far apart in sequence but close in space:

➡ You just created a structural constraint.

Repeat this hundreds or thousands of times:

• 100 amino acid protein
• 10–20 NOEs per residue
• ~1000–2000 distances total

Example: A 68-residue protein was solved using 993 distance constraints .

Distances define the structure like a molecular spider web.

📊 5. Types of NOEs (Extremely Important)

Not all distances are equally useful.

1️⃣ Intra-residual (same residue)

• Local information only
• Side chain or backbone confirmation
• ❌ No tertiary structure info

2️⃣ Sequential (|i – j| = 1)

• Between neighboring residues
• Define secondary structure
• ❌ Not tertiary structure

3️⃣ Medium-range (|i – j| = 2–4)

• Common in α-helices
• Define secondary structure
• Still mostly local

4️⃣ Long-range (|i – j| > 4)

• Define tertiary structure
• Essential for 3D fold
• Can even define quaternary structure

These are gold.

📦 6. 2D vs 3D NOESY

Small proteins:

• 2D NOESY is possible

Larger proteins:

• Signal overlap becomes disastrous
• You must use 3D NOESY

3D NOESY principle:

• Two proton dimensions
• Third dimension: heteronucleus (¹⁵N or ¹³C)

Typical setup:

• ¹⁵N-edited NOESY (HN–H)
• ¹³C aliphatic
• ¹³C aromatic

There are even:

• 4D NOESY (rare)
• Carbon–carbon NOEs (possible but weak, due to smaller γ)

📐 7. How Do We Convert NOE → Distance?

Theoretically:

ext{NOE intensity} propto rac{1}{r^6}

But in practice:

• Mobility affects NOE
• Side chains are more flexible
• Cross-peak volumes vary over orders of magnitude

So we do not calculate exact distances.

Instead:

We define an upper distance limit

Example: If intensity corresponds to ≤ 4.2 Å

We write: r le 4.2 ext{ Å}

Not: r = 4.2 ext{ Å}

Important empirical fact:

For side chains, using r⁻⁴ instead of r⁻⁶ often works better .

This is one of those practical NMR realities.

🧬 8. NOEs Define Secondary Structure

🌀 In α-Helices

Characteristic NOEs:

• HN(i) – Hα(i–3)
• HN(i) – Hα(i–4)
• Sequential HN–HN
• Weak HN–Hα

🧵 In β-Sheets

Characteristic NOEs:

• Strong HN–Hα sequential
• HN–Hα across strands
• Hα–Hα across strands

📈 9. Sequence Plot Visualization

A sequence plot shows:

• X-axis: residue number
• Bars indicating NOE types

Patterns reveal:

• Helices
• Sheets
• Turns

Older articles frequently used these plots.

🚫 10. Hydrogen Bonds: Should We Include Them?

Tempting idea: If we know it's a helix → we know hydrogen bonds must exist.

But:

❌ You should NOT include hydrogen bonds unless experimentally detected.

Why? Because hydrogen bonds alone can artificially force almost any structure .

Exception: If detected via scalar coupling across H-bond:

• NH–C=O
• N–C Then they can be included.

🧲 11. Paramagnetic Relaxation Enhancement (PRE)

Another source of distance information.

Principle:

Unpaired electrons (paramagnetic centers) enhance relaxation.

Measure:

• Relaxation rate without spin label
• Relaxation rate with spin label
• Difference = PRE

PRE ∝ 1/r⁶

Why PRE is Powerful

• Detects distances up to 25–30 Å
• Much longer range than NOEs

That’s huge for defining global fold.

🧪 12. Introducing Spin Labels

Most proteins are not naturally paramagnetic.

So we:

1. Engineer a cysteine
2. Attach a spin label
3. Often use a nitroxyl radical
4. Forms disulfide bond

Nitroxyl radicals:

• Stable for weeks/months
• Widely used

In membrane proteins (hard systems): PRE can provide critical long-range distances .

🏗 Final Big Picture

To determine a protein structure:

You collect:

• Hundreds to thousands of NOE-derived upper distance limits
• Possibly PRE long-range distances
• Possibly dihedral angle constraints (next lecture)

Then feed everything into a structure calculation program.

The structure emerges as the one that satisfies:

ext{All distance constraints simultaneously}

It’s like solving a massive 3D geometric puzzle.

🔑 Key Takeaways

• Each NOESY cross peak = one interatomic distance
• We assign peaks using resonance assignments
• Distances are upper limits, not exact values
• Long-range NOEs define tertiary structure
• 3D NOESY is essential for larger proteins
• PRE gives long-range (25–30 Å) constraints
• Hydrogen bonds should not be assumed unless measured
• ~1000+ distances can define a small protein structure