🧲 Protein Mobility & Dynamics from NMR Relaxation

(Based on Lecture 5 Video 6 )

This lecture explains how nuclear magnetic relaxation can be used to study protein motion and flexibility — from whole-molecule tumbling to tiny local fluctuations. The theory is heavy, but the concepts are powerful and very intuitive once broken down.

1️⃣ What Is Relaxation in NMR?

In NMR, we:

1. Excite spins → create a non-equilibrium state
2. Let nature restore equilibrium
3. Measure how fast this happens

That return to equilibrium is called relaxation.

It is:

• A stochastic (random) process
• Occurring at a defined rate
• Responsible for signal decay

There are two main types:

🟢 T1 (Longitudinal Relaxation)

• Restores equilibrium magnetization
• Governs recovery along the magnetic field axis

🔵 T2 (Transverse Relaxation)

• Does not restore equilibrium
• Governs decay of observable signal in the transverse plane

⚠️ Important: Both T1 and T2 lead to disappearance of measurable magnetization.

2️⃣ Why Relaxation Tells Us About Protein Motion

Relaxation depends strongly on atomic mobility.

Proteins are not static objects — they move on many timescales:

🌀 Correlation Time (τc)

Defined as:

Time required to rotate ~1 radian (57°) on average.

• Long τc → slow motion
• Short τc → fast motion

3️⃣ What Motions Can We Actually Measure?

✔ Overall tumbling → YES ✔ Fast internal mobility → YES ✖ Extremely fast (ps) → Hard to detect ✖ Very slow (ms–s) → Rarely measurable (real-time NMR possible but rare)

The most practical measurements:

• T1
• T2
• Heteronuclear NOE (¹H–¹⁵N)

4️⃣ The Model-Free (Lipari–Szabo) Approach

Instead of describing every motion separately, we compress mobility into:

1. τM → overall molecular tumbling
2. τe → internal motion correlation time
3. Order parameter (S²) → rigidity measure

📊 The Order Parameter (S²)

Range: 0 → 1

Interpretation:

• High S² → residue follows whole-protein tumbling
• Low S² → residue has independent internal motion

Typical values:

• Secondary structure → ~0.8–0.9
• Loops → ~0.4–0.6

Most residues fall between 0.5–0.9.

5️⃣ How T1 and T2 Depend on Molecular Size

T1 and T2 depend strongly on tumbling rate.

For small molecules (1–2 kDa):

• T1 shows a minimum
• T2 relatively long

For proteins (larger molecules):

• T1 increases
• T2 decreases

So for proteins:

The ratio T1/T2 increases as the molecule becomes larger or tumbles slower.

⚠️ Only valid for rigid residues (little internal motion).

Since ~80% of residues in a folded protein are structured, this works well.

6️⃣ Example: Detecting Dimerization via Relaxation

A small copper-binding protein was studied:

APO Form (no copper)

• T1 ≈ 0.4 s
• T2 ≈ 0.1 s
• Stable across residues

After Copper Addition

• T1 increased
• T2 decreased
• T1/T2 ratio increased strongly

Interpretation: 👉 Protein formed a dimer

Why?

• Copper needs 3 ligands
• Each protein provides only 2
• So 2 proteins bind one copper

Key point:

• No chemical shift changes
• No NOEs between monomers
• Symmetric complex → looks identical in shifts

But relaxation detects increased size → slower tumbling → dimer formation.

Relaxation can reveal what chemical shifts cannot.

7️⃣ Local Mobility: Heteronuclear NOE (¹H–¹⁵N)

Used to probe residue-specific flexibility.

Usually plotted as:

1 + NOE

Why? Technical definition reasons.

🧬 Example: Calmodulin

Calmodulin Structure:

• Well-folded N-lobe
• Well-folded C-lobe
• Flexible central linker

Superposition:

• N-lobe aligns well
• C-lobe aligns well
• Whole protein does not align → linker is flexible

NOE Results:

• High NOE → rigid lobes
• Low NOE → flexible linker
• N-terminus → highly flexible
• Loop regions → reduced NOE

So NOE maps flexibility along sequence.

8️⃣ Mutation Study: F141L in Calmodulin

Mutation:

• Phe141 → Leu

Result:

• NOE values reduced in opposite region
• Increased flexibility detected
• Very subtle mobility changes measurable

Interesting: Mutation at Phe89 also causes disease.

So: Relaxation can detect small mobility changes linked to pathology.

9️⃣ Calcium Removal Experiment

Without Ca²⁺:

• N-lobe remains folded
• C-lobe:
  • Many residues invisible
  • Remaining residues show reduced NOE

Interpretation: 👉 C-lobe fold destabilized 👉 Increased flexibility 👉 Structural disruption

Relaxation detects unfolding or partial destabilization.

🔟 Big Picture Takeaways

What Relaxation Can Tell You

✔ Overall tumbling rate ✔ Molecular size changes ✔ Dimerization ✔ Residue-specific flexibility ✔ Loop mobility ✔ Mutation-induced changes ✔ Folding stability

What It Cannot Easily Tell You

✖ Ultra-fast picosecond motions ✖ Very slow millisecond motions (hard, rare)

🧠 Conceptual Summary

Protein motion occurs at multiple timescales.

Relaxation translates motion into measurable parameters:

• T1 → energy recovery
• T2 → signal decay
• T1/T2 → molecular size & tumbling
• Heteronuclear NOE → local flexibility
• S² → rigidity index

Rigid residues:

• High S²
• High NOE
• Reflect global tumbling

Flexible residues:

• Lower S²
• Reduced NOE
• Decoupled from global motion

🎯 Why This Is Powerful

Relaxation allows you to detect:

• Dimer formation without chemical shift change
• Subtle mutation effects
• Loop flexibility
• Local unfolding
• Stability differences

It is one of the most sensitive tools for studying protein dynamics in solution.