🧩 1. Structure Quality Considerations Before Dynamics

Before discussing dynamics, the chapter emphasizes important structural validation aspects:

🔁 The Third Backbone Dihedral Angle (ω)

• The ω angle (peptide bond) is usually fixed to 180° (trans) in NMR structure determination.
• Only when experimental evidence supports it is it set to 0° (cis).
• Fixing ω creates very narrow structural distributions.
• Some refinement protocols allow slight deviations from 180°.

👉 Important: Over-constraining ω artificially reduces structural variability.

🚫 Interatomic Bumps (Close Contacts)

• These occur when nonbonded atoms are closer than allowed by van der Waals radii.
• This is energetically very unfavorable.
• Structure validation tools:
  • WHAT-IF
  • MolProbity

👉 A good structure should have very few steric clashes.

🔗 Hydrogen Bonding Quality

Hydrogen bonds are crucial structural stabilizers.

How can NMR detect them?

1. Through-hydrogen-bond J-coupling → Rarely measured.
2. Indirect methods (more common):
   • Amide H/D exchange rates
   • Temperature dependence of chemical shifts

⚠ Limitations:

• Only the donor is identified.
• Many donors (Ser, Thr, Tyr hydroxyls) are invisible in NMR.
• The complete hydrogen bond network cannot be determined experimentally.

👉 Therefore, force-field refinement is critical.

Special backbone hydrogen bond potentials improve structure generation.

Quality indicators:

• Number of unsatisfied donors/acceptors
• Computed hydrogen-bonding energy

🌊 2. Protein Dynamics: The Big Picture

Proteins are ensembles of interconverting states, not static objects.

⏳ Timescales of Motion

Energy barriers determine exchange rates:

🌀 Global Tumbling

Proteins rotate in solution.

Correlation time ( au_m ):

• Derived from Stokes–Einstein–Debye equation
• Typically several to hundreds of ns

Key concept:

• Motions faster than τm → internal fast motions
• Motions slower than τm → collective conformational exchange

The figure on page 2 (Fig. 4.13) beautifully summarizes this hierarchy.

📡 3. NMR Observables Affected by Dynamics

NMR reports on dynamics via multiple observables.

🧪 Chemical Shift and Exchange

Chemical shifts are sensitive to environment.

For two-state exchange (A ⇌ B):

🟢 Fast exchange (kex >> Δν)

• One averaged peak
• Position = population-weighted average

🔴 Slow exchange (kex << Δν)

• Two separate peaks
• Intensities reflect populations

🟡 Intermediate exchange

• One averaged peak
• Line broadening (exchange broadening)
• May become undetectable

📌 Peak width at half height: T_2^{-1} = pi Delta u_{1/2}

Exchange affects T₂, not T₁.

Figure 4.14 (page 3) clearly illustrates these regimes.

📉 4. Relaxation and Linewidth

Linewidth (T₂) depends on:

1. Exchange broadening
2. Tumbling correlation time (τm)
3. Fast internal motions

T₂ increases almost linearly with τm (Fig. 4.15).

🔄 T₁ Relaxation

T₁ depends on:

• Tumbling
• Motions faster than tumbling

It does not depend on slow conformational exchange.

📏 Distance Averaging in Exchange

For dipolar-coupled nuclei:

If exchange is:

• Fast on chemical shift timescale
• Slower than tumbling

Relaxation depends on ⟨r⁻⁶⟩.

If exchange is faster than tumbling: Relaxation depends on ⟨r⁻³⟩².

For NH bonds:

• r is fixed → distinction irrelevant.

🔁 NOE and Dynamics

Steady-state NOE depends on:

• T₁
• Dipolar interaction correlation time
• τm
• Fast internal motion

🧲 RDCs and Dynamics

Residual Dipolar Couplings report on:

• Orientation of bond vectors
• Motions from ps → ms

Internal motion averages vector orientations → averaged RDC values.

Thus: 👉 RDCs encode dynamic information.

🧠 5. NMR Experiments for Dynamics

Dynamic information comes from relaxation measurements.

Relaxation arises from:

• Fluctuating magnetic fields
• Caused by molecular motion
• Frequency dependence described by spectral density J(ω)

J(ω) is the Fourier transform of time correlation functions.

🧬 6. ¹⁵N Relaxation Experiments

The backbone ¹H–¹⁵N amide bond is ideal because:

1. Present in all non-Pro residues
2. Approximates two-spin system
3. Easy to measure in HSQC

Measured parameters:

• ¹⁵N T₁
• ¹⁵N T₂
• ¹H–¹⁵N NOE

They probe:

• ps–ns motions
• μs–ms exchange

🧮 7. Extracting Tumbling Correlation Time

For isotropic tumbling:

au_m = rac{1}{2 u_N} sqrt{rac{6T_1}{T_2} - 7}

But proteins are often:

• Nonspherical
• Anisotropically rotating

Thus T₁/T₂ may vary per residue even without internal motion.

🧩 8. Model-Free Formalism (Lipari–Szabo)

Goal: Separate:

• Overall tumbling
• Internal motion

Assumption: Internal and overall motions are separable:

C(t) = C_0(t) C_i(t)

Basic Model-Free Parameters

Three parameters:

1. τm – global tumbling
2. S² – order parameter
3. τe – internal motion timescale

📌 Order Parameter S²

Measures amplitude of motion:

• S² = 1 → completely rigid
• S² = 0 → fully flexible
• Typical backbone: 0.7–0.95

Represents angular restriction of bond vector.

Spectral Density (Basic Model)

[ J( u) = rac{2}{5} left( rac{S^2 au_m}{1+ u^2 au_m^2}

• rac{(1-S^2) au}{1+ u^2 au^2} ight) ]

with:

au^{-1} = au_m^{-1} + au_e^{-1}

🧠 9. Extended Model-Free (Clore)

Accounts for:

• Fast internal motion (τf)
• Slow internal motion (τs)

With separate order parameters:

• S²f
• S²s

Total: S^2 = S_f^2 S_s^2

Up to six fitting parameters:

• τm
• S²f
• S²s
• τf
• τs
• Rex

⚡ 10. Conformational Exchange (Rex)

Rex contributes to T₂ only.

Captures:

• μs–ms exchange
• Chemical shift modulation

Exchange affects:

• T₂
• Not T₁

🎯 Final Conceptual Takeaways

This chapter builds a powerful conceptual framework:

Proteins are dynamic ensembles.

NMR observables encode motion through:

• Chemical shift averaging
• Line broadening
• Relaxation rates
• NOE values
• RDC averaging

Timescale Mapping:

(Figure 4.13 visually summarizes this hierarchy.)

🧠 Big Insight

Structure determination is incomplete without dynamics.

Relaxation analysis allows:

• Quantifying flexibility (S²)
• Extracting timescales (τe)
• Detecting conformational exchange (Rex)
• Understanding protein function through motion