🧬 Chapter 4.2 — NMR Restraints

(How NMR data becomes 3D protein structure)

To determine a protein structure by solution NMR, we need many conformational restraints. These restraints translate experimental NMR observables into geometric information (distances and angles).

Important idea:

NMR does not directly give you a structure — it gives you parameters that must be converted into restraints.

Sometimes the relationship is physical (exact equation). Sometimes it is statistical (derived from known structures).

This chapter focuses on two major types of restraints:

1. 📏 Distance restraints (from NOEs)
2. 🔄 Dihedral angle restraints (from J-couplings and chemical shifts)

📏 4.2.1 Distance Restraints (NOEs)

🔹 The Core Principle

The classical NMR structure determination method relies on NOEs (Nuclear Overhauser Effects) between nearby hydrogen atoms.

NOEs connect hydrogen atoms that are:

• Less than ~5–6 Å apart
• Possibly far apart in sequence
• But close in 3D space

👉 This is crucial for folding information.

💡 What Produces an NOE?

NOEs arise from:

• Dipolar interaction between nuclear spins
• Magnetization transfer
• Brownian motion in solution

In a NOESY spectrum:

• Each cross-peak corresponds to a pair of nearby hydrogens
• The intensity (volume V) of the cross-peak relates to distance

📐 Distance–Volume Relationship

Under the two-spin approximation:

V propto r^{-6} f( au_c)

Key idea:

• The NOE intensity scales with the inverse sixth power of distance
• Small changes in distance cause large intensity differences

However:

• Proteins are dynamic
• Distances fluctuate
• We measure averages

So instead of exact distances, we derive:

Upper distance limits (upper bounds)

Lower bounds are rarely used because:

• Internal motion reduces NOE intensity
• That would falsely suggest larger distances
• This introduces bias

Upper bounds are safer:

• Motion only reduces information content
• But does not bias structure

📊 Calibration Curves

Since intensity scaling varies between experiments, a calibration curve is used:

V = rac{k}{b^6}

Where:

• k = scaling constant
• b = upper distance limit

On Figure 4.5 (page 2):

• Experimental volumes vs real distances
• Shows inverse relationship
• Continuous line = calibration curve

Sometimes alternative exponents (4 or 5) are used to account for dynamics.

🧪 Practical Classification of NOEs

Instead of exact calibration, NOEs are often grouped:

Simple but effective.

🔍 NOE Categories (Sequence-Based)

NOEs are classified based on sequence separation:

⭐ Long-range NOEs are the most important

They tell you that two distant sequence parts are close in space.

🧬 Secondary Structure Signatures

α-Helices:

• Characteristic medium-range NOE pattern
• Easy to detect experimentally

β-Sheets:

• Need:
  • Medium-range NOEs (within strands)
  • Long-range NOEs (between strands)
• More complex to interpret

🔄 4.2.2 Dihedral Angle Restraints

Instead of distances, we now restrain angles.

We define:

Allowed ranges of dihedral angles

These ranges come from:

1. Scalar couplings (³J)
2. Chemical shifts

🔬 1️⃣ Dihedral Angles from ³J Couplings

What is a ³J coupling?

• Scalar coupling between atoms separated by three bonds
• Depends on dihedral angle

📐 The Karplus Equation

^3J( heta) = Acos^2 heta + Bcos heta + C

This relates:

• Measured coupling
• To dihedral angle θ

Important:

• θ is not always exactly φ, ψ, or χ
• Often differs by a fixed offset (See Figure 4.6 on page 3)

📊 Parameterization

Table 4.2 (page 3) gives A, B, C values for:

• φ angle couplings
• ψ angle couplings
• χ₁ side chain angle couplings

Figure 4.7 shows:

• Coupling vs φ angle curves
• Sinusoidal behavior

🔁 Important: Averaging

Just like NOEs:

³J couplings are averaged over conformations Because proteins are dynamic.

📌 Using ³J in Structure Calculations

Two approaches:

1️⃣ Convert to allowed angle ranges

Compare measured J value to Karplus curve.

Possible outcomes:

• Up to four allowed angle regions
• Reduce using Ramachandran constraints

2️⃣ Directly include J in target function

V_J = w_J sum (J_ - J_)^2

Adds energetic penalty for mismatch.

🧠 2️⃣ Dihedral Angles from Chemical Shifts

Chemical shifts contain secondary structure information.

📘 CSI Method (Chemical Shift Index)

Basic idea:

• Compare experimental shifts
• With random coil database values
• Assign residue as:
  • α-helix
  • β-sheet
  • random coil

Works for:

• ¹H
• ¹³C (more reliable)

📐 Converting CSI to Angle Restraints

Typical angle ranges:

Helix:

• φ ≈ –60° ± 40°
• ψ ≈ –40° ± 40°

β-strand:

• φ ≈ –120° ± 60°
• ψ ≈ 100° ± 80°

These correspond to Ramachandran core regions.

⚠️ Important Cautions

• Apply only when ≥3–4 consecutive residues agree
• Do not restrain isolated residues
• Usually ≤50% of residues get CSI restraints

Overinterpretation is dangerous.

🤖 Advanced Chemical Shift Programs

Programs like:

• PREDITOR
• TALOS+

They:

• Compare stretches of 3 residues
• Match to database of known structures
• Predict:
  • φ, ψ
  • χ₁
  • cis/trans peptide bond

They use:

• Pattern matching
• Sequence correction
• Homology information
• Neural network filtering

They output:

• Predicted angle
• Uncertainty
• Reliability score

👉 Only use high-confidence predictions.

🚨 Very Important Warning

Because these methods interpret chemical shifts directly:

Chemical shift referencing must be extremely accurate.

Small referencing errors → wrong dihedral angles.

🧩 Big Picture

To determine protein structure by NMR:

You combine:

📏 Distance restraints (NOEs)

→ Define spatial proximity

🔄 Dihedral angle restraints (J-couplings + shifts)

→ Define backbone and side-chain geometry

Together they:

• Reduce conformational space
• Define secondary structure
• Define tertiary fold
• Allow calculation of 3D ensemble

🧠 Conceptual Summary