Lecture 5 Book Chapter 4.5

Protein structure

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🧩 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?

Through-hydrogen-bond J-coupling → Rarely measured.
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:

Motion Type	Timescale
Local librations	picoseconds
Side chain motions	ps–ns
Global tumbling	ns
Conformational exchange	μs–ms
Large rearrangements	seconds

🌀 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:

Exchange broadening
Tumbling correlation time (τm)
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:

Present in all non-Pro residues
Approximates two-spin system
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:

τm – global tumbling
S² – order parameter
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:

Timescale	Technique
ps–ns	¹⁵N T₁, T₂, NOE
μs–ms	Rex, CPMG
ms–s	EXSY
ps–ms	RDCs

(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

Quiz

Score: 0/30 (0%)

Q0. In standard NMR structure determination, the peptide bond ω angle is typically fixed to which value unless experimental evidence suggests otherwise?

0°

90°

120°

180°

Q1. Interatomic bumps in protein structures refer to:

Hydrogen bonds shorter than 2 Å

Nonbonded atoms closer than allowed by van der Waals radii

Covalent bond distortions

Disulfide bond mispairing

Q2. Which of the following is rarely used to directly detect hydrogen bonds by NMR?

H/D exchange rates

Temperature dependence of chemical shifts

Through-hydrogen-bond J-coupling

NOE measurements

Q3. In the fast exchange regime on the chemical shift timescale, how many peaks are observed for a nucleus exchanging between two states?

Two separate peaks

One averaged peak

No peak

Three peaks

Q4. In the slow exchange regime, the relative intensities of the two peaks reflect:

Exchange rate only

Chemical shift difference only

Relative populations of states

Temperature only

Q5. The full peak width at half height (Δν1/2) is directly related to which relaxation time?

τm

τe

Q6. Which relaxation time is NOT affected by slow conformational exchange?

Rex contribution to T2

Linewidth

Q7. The tumbling correlation time τm for proteins typically lies in which range?

Femtoseconds

Picoseconds only

Several to hundreds of nanoseconds

Seconds

Q8. In the Lipari–Szabo model-free approach, the order parameter S² primarily reports on:

Exchange rate

Chemical shift difference

Amplitude of internal motion

Hydrogen bond strength

Q9. What does S² = 1 indicate?

Fully flexible bond vector

Complete conformational exchange

Rigid bond vector

Slow tumbling

Q10. The spectral density function J(ν) is mathematically related to:

Fourier transform of time-correlation functions

Laplace transform of chemical shifts

Derivative of T1

Van der Waals energy

Q11. Which NMR experiment set is most commonly used to characterize backbone dynamics?

NOESY and TOCSY

COSY and HSQC

15N T1, 15N T2, and 1H–15N NOE

EXSY only

Q12. Residual Dipolar Couplings (RDCs) primarily report on:

Chemical shift differences

Bond vector orientation

Exchange rates only

Hydrogen bond energy

Q13. In the extended model-free formalism, S² is defined as:

S²f + S²s

S²f − S²s

S²f × S²s

S²f / S²s

Q14. Rex contributes directly to which relaxation parameter?

T1 only

T2 only

NOE only

τm only

Q15. Fixing the ω angle to 180° in NMR structure determination results in very narrow structural distributions.

True

False

Q16. Hydrogen bond networks can be fully determined by NMR data alone.

True

False

Q17. In intermediate exchange, the resonance linewidth increases due to exchange broadening.

True

False

Q18. T1 relaxation depends on molecular tumbling and motions faster than tumbling.

True

False

Q19. The 1H–15N amide bond is absent in proline residues.

True

False

Q20. Anisotropic rotational diffusion can make the T1/T2 ratio site-specific even without internal motion.

True

False

Q21. In the basic model-free formalism, τe describes overall tumbling.

True

False

Q22. If S² = 0, the bond vector samples all possible orientations.

True

False

Q23. Exchange occurring on a micro- to millisecond timescale contributes to Rex.

True

False

Q24. The NOE depends on T1 and dipolar interaction correlation times.

True

False

Q25. The spectral density function is independent of frequency.

True

False

Q26. Exchange faster than tumbling can change how distance averaging affects relaxation rates.

True

False

Q27. Model-free analysis assumes separability of internal and overall motions.

True

False

Q28. In slow exchange, only one averaged resonance is observed.

True

False

Q29. RDC values encode dynamic information because internal motions average bond vector orientations.

True

False