Skip to content

NMR Theory

Scientific background for the NMR observables generated by synth-pdb.

Chemical Shifts

The chemical shift δ of a nucleus (¹H, ¹³C, ¹⁵N) reflects its electronic environment. In proteins, backbone chemical shifts are highly sensitive to secondary structure:

  • α-helix: Cα shifts upfield by ~3 ppm relative to random coil
  • β-sheet: Cα shifts downfield by ~2 ppm relative to random coil

Predictors available (see --shift-predictor):

Method Reference Accuracy (Cα RMSD)
SHIFTX2 Han et al. (2011), J Biomol NMR 50:43 ~0.44 ppm
SPARTA+ / empirical Shen & Bax (2010), J Biomol NMR 48:13 ~0.55 ppm

J-Couplings

Scalar (J) couplings are transmitted through chemical bonds. The three-bond \({}^3J(\text{HN,HA})\) coupling is related to the backbone \(\phi\) dihedral angle by the Karplus equation (Karplus, 1963, J Am Chem Soc 85:2870):

\[{}^3J(\phi) = A\cos^2(\phi - \theta) + B\cos(\phi - \theta) + C\]

Typical secondary structure values: - α-helix (\(\phi \approx -57°\)): \({}^3J \approx 3.9\)–5.0 Hz (< 6 Hz) - β-sheet (\(\phi \approx -120°\)): \({}^3J \approx 8\)–10 Hz (> 7.5 Hz)

Residual Dipolar Couplings

Physical Origin

In solution NMR, isotropic molecular tumbling averages the direct magnetic dipole–dipole interaction between two nuclear spins to exactly zero. This is why solution spectra are sharp compared to solid-state spectra.

When a protein is dissolved in an anisotropic medium (a dilute suspension of rod-like liquid crystals, filamentous bacteriophage particles, or a strained polyacrylamide gel), molecules develop a slight preferred orientation. The dipole–dipole interaction no longer averages to zero — a small residual coupling remains: the Residual Dipolar Coupling (RDC).

"We show that the liquid-crystalline medium provides weak alignment of the solute without significantly perturbing the solution structure." — Tjandra & Bax (1997), Science 278:1111

The RDC Formula

For a backbone N–H bond vector, the RDC value is (Tjandra & Bax, 1997):

\[D(\theta, \phi) = D_a \left[(3\cos^2\theta - 1) + \frac{3}{2}R\,\sin^2\theta\,\cos 2\phi\right]\]
Symbol Meaning
\(D_a\) Axial component of the alignment tensor (Hz); typical: 5–25 Hz
\(R\) Rhombicity of the tensor, \(0 \leq R \leq 2/3\); \(R=0\) = axially symmetric
\(\theta\) Polar angle of the N–H vector with the tensor principal (\(Z\)) axis
\(\phi\) Azimuthal angle of the N–H vector in the tensor XY plane

Extreme Values

Orientation \(D\) (with \(D_a=10\) Hz, \(R=0.5\))
N–H along Z-axis (\(\theta=0\)) \(+2D_a = +20\) Hz (maximum)
N–H along X-axis (\(\theta=90°, \phi=0°\)) \(D_a(-1 + 1.5R) = -2.5\) Hz
N–H along Y-axis (\(\theta=90°, \phi=90°\)) \(D_a(-1 - 1.5R) = -17.5\) Hz (minimum)

Negative RDCs are perfectly physical: they indicate the N–H bond is nearly perpendicular to the alignment axis.

Why RDCs Are Powerful

RDCs encode the orientation of bond vectors relative to a common global frame. This is fundamentally different from NOE distance restraints:

  • NOEs → local distance information (secondary structure)
  • RDCs → global orientational information (tertiary fold topology)

The two observables are therefore complementary. Combining them dramatically improves NMR structure accuracy. A landmark study demonstrated that RDC-constrained calculations refined a 100-ns MD ensemble of HIV-1 protease to within 0.4 Å RMSD of the crystal structure without any additional NOEs (Bewley & Clore, 2000).

Alignment Media

Common alignment media and representative \(D_a\) ranges:

Medium \(D_a\) (Hz) Reference
Dilute bicelle liquid crystals 5–25 Tjandra & Bax (1997)
Filamentous bacteriophage (Pf1) 8–20 Hansen et al. (2000)
Stretched polyacrylamide gels 3–15 Tycko et al. (2000)

Proline Special Case

Proline has no backbone amide proton because its side-chain \(\delta\)-carbon bonds back to the backbone nitrogen, forming a five-membered pyrrolidine ring and making N a secondary (tertiary) amine. Consequently:

  • No backbone N–H bond → no \({}^1D_\text{NH}\) RDC
  • No backbone NH cross-peak in ¹H–¹⁵N HSQC spectra
  • calculate_rdcs automatically excludes PRO residues

Key References

  1. Tjandra, N. & Bax, A. (1997). Direct measurement of distances and angles in biomolecules by NMR in a dilute liquid crystalline medium. Science, 278, 1111–1114. DOI: 10.1126/science.278.5340.1111

  2. Prestegard, J.H., Al-Hashimi, H.M. & Tolman, J.R. (2000). NMR structures of biomolecules using field oriented media and residual dipolar couplings. Q Rev Biophys, 33, 371–424. DOI: 10.1017/S0033583500003656

  3. Bax, A. & Grishaev, A. (2005). Weak alignment NMR: a hawk-eyed view of biomolecular structure. Curr Opin Struct Biol, 15, 563–570. DOI: 10.1016/j.sbi.2005.08.006

  4. Han, B. et al. (2011). SHIFTX2: significantly improved protein chemical shift prediction. J Biomol NMR, 50, 43–57. DOI: 10.1007/s10858-011-9478-4

  5. Shen, Y. & Bax, A. (2010). SPARTA+: a modest improvement in empirical NMR chemical shift prediction by means of an artificial neural network. J Biomol NMR, 48, 13–22. DOI: 10.1007/s10858-010-9433-9

  6. Bewley, C.A. & Clore, G.M. (2000). Determination of the relative orientation of the two halves of the domain-swapped dimer of HIV-1 protease using residual dipolar couplings. J Am Chem Soc, 122, 6009. DOI: 10.1021/ja000635g

  7. Karplus, M. (1963). Vicinal proton coupling in nuclear magnetic resonance. J Am Chem Soc, 85, 2870–2871. DOI: 10.1021/ja00901a059