4.1 Nuclear resonance #
4.1.1 Spin, field, and the resonance condition #
A nucleus with spin I ≠ 0 is a small magnet. In a field B₀ the resonance frequency is
Spin Echo
ν = |γ| B₀ / 2π
where γ is the magnetogyric ratio of that nucleus. A stronger field means a higher frequency. At the fields used for macromolecules, often about 10–20 T, proton frequencies are hundreds of megahertz: radiofrequency equipment, not optics.
| Nucleus | Spin | Natural abundance | Why it matters here |
|---|---|---|---|
| ¹H | 1/2 | nearly 100% | most sensitive common nucleus; crowded in proteins |
| ¹²C | 0 | about 99% | silent |
| ¹³C | 1/2 | about 1.1% | backbone and side chains; usually enriched |
| ¹⁴N | 1 | nearly 100% | quadrupolar; lines too broad for ordinary protein work |
| ¹⁵N | 1/2 | about 0.4% | amide reporter; usually enriched |
| ¹⁶O | 0 | nearly 100% | silent |
| ³¹P | 1/2 | 100% | nucleic acids, phospholipids, phosphorylated metabolites |
The excess population is only about one nucleus in 10⁵, so NMR is insensitive next to absorbance or fluorescence. Metabolites can be seen at natural abundance. A protein generally needs concentrations toward millimolar, signal averaging, and enrichment in ¹³C and ¹⁵N. Higher field, more sample, and more scans improve sensitivity. A nucleus with I = 0 cannot be helped.
4.1.2 Chemical shift #
Electrons shield the nucleus, so it feels B₀(1 − σ), where the shielding constant σ is a few parts per million. The chemical shift puts that displacement on a field-independent scale:
δ = (ν_sample − ν_ref) / ν_spectrometer × 10⁶
in parts per million, referenced to a standard compound, not to the water peak. Water dominates an aqueous proton spectrum and has to be suppressed. Because δ is a ratio, a methyl at 1 ppm is at 1 ppm on any spectrometer, but the separation in hertz grows with field. Higher field therefore spreads peaks apart relative to their widths and relative to field-independent couplings.
Methyl protons lie near 0–1 ppm, α-protons near 4 ppm, aromatics near 7 ppm, and backbone amides spread over roughly 6–10 ppm in a folded protein. A disordered peptide bunches at random-coil values. Dispersion is a qualitative report that a fold exists, in the same spirit as near-ultraviolet CD, and with the same limit: it is not a structure.
4.1.3 J coupling #
Nuclei connected through a few bonds split each other’s lines. This scalar coupling J is transmitted by bonding electrons, is quoted in hertz, and does not grow with B₀. A three-bond coupling depends on the dihedral angle through the Karplus relation, so backbone J values restrain the angle φ. The relation is periodic: more than one angle can give the same J. J is through-bond. It does not report two groups that merely sit next to each other across an interface. That is the nuclear Overhauser effect.
4.1.4 Why a protein spectrum is crowded #
A protein of 150 residues contains on the order of a thousand protons, and the proton shift range is only about 10 ppm. In one dimension they cannot all have private peaks. Lines also broaden as tumbling slows. The width of a Lorentzian line is about 1/(π T₂), and the transverse relaxation time T₂ shortens as the rotational correlation time grows, with size, with viscosity, and in the cold. Water at 55 mol L⁻¹ would otherwise swamp the receiver. Solvent suppression is a necessity.
Routine assignment from ¹H spectra alone is comfortable only for small chains. Enrichment with ¹⁵N and ¹³C, replacement of non-exchangeable protons by deuterium, and sequences such as TROSY that keep a slowly relaxing component push the practical size into the hundreds of kilodaltons in favourable cases. A spectrum that is one unresolved lump is evidence that this sample, at this field, did not disperse. It is not a structure.
4.2 From a spectrum to dynamics, and MRI #
4.2.1 One-dimensional spectra #
A one-dimensional proton spectrum of a small metabolite can identify it, and the peak area is proportional to the number of nuclei if every nucleus is allowed to relax between scans. A one-dimensional protein spectrum is an envelope. Its dispersion reports folding, and a global movement on adding a ligand reports binding without naming residues. Metabolomics lives in the first regime. Macromolecular NMR does not stay there.
4.2.2 COSY, NOESY, and the HSQC #
A two-dimensional spectrum puts correlations on a plane. A cross-peak means the two frequencies belong to nuclei that interacted during the mixing period. COSY correlates nuclei that share a J coupling, typically across two or three bonds: neighbours in the covalent structure. TOCSY extends that correlation through a whole coupled network, often one side chain. NOESY correlates nuclei through the nuclear Overhauser effect, a through-space dipolar interaction. Intensity falls off as about r⁻⁶ and is useful out to roughly 5–6 Å. Those contacts are structural restraints, not bonds, and a missing peak may be overlap or exchange rather than a long distance. Structural Biology turns a sufficient set of restraints into a model.
A ¹⁵N-HSQC correlates each amide proton with its nitrogen. There is approximately one backbone peak per residue that has an amide proton. Prolines are absent. Asparagine, glutamine, and tryptophan side chains add peaks. A ligand that moves a subset of peaks has changed those environments. The residues are often near the site and are sometimes responding to a conformational change elsewhere. A chemical-shift map is a response, not automatically a contact surface.
4.2.3 Relaxation #
T₁ is how fast magnetisation along B₀ regrows. T₂ is how fast transverse magnetisation loses phase, and it sets the linewidth. Both depend on tumbling. Picosecond-to-nanosecond motions are read from T₁, T₂, and the heteronuclear NOE. Microsecond-to-millisecond motions, often the scale of binding and catalysis, add to transverse relaxation or, if exchange is slow, split a peak in two. Amide hydrogen exchange with D₂O reports protection by hydrogen bonds and burial on a slower scale still. A protein can be rigid on a picosecond scale, mobile at a loop on a millisecond scale, and stable against exchange for hours. “Dynamic” is incomplete until the timescale is named. Chemical exchange also broadens lines, so a broad line is not automatically a large mass.
4.2.4 MRI is the same resonance used differently #
Magnetic resonance imaging uses the proton resonance of tissue water. A field gradient makes frequency depend on position, so a frequency becomes a coordinate, used for slice selection, phase encoding, and frequency encoding. Contrast comes from water content and from local T₁ and T₂. Functional MRI adds the BOLD effect: deoxyhaemoglobin is paramagnetic and alters the field nearby, so a change in blood oxygenation changes the water signal. Neurobiology owns what that vascular proxy says about neural activity. BOLD is not a spectrum of haemoglobin and not a spike train.
MRI and a protein NOESY share the resonance condition and almost nothing else. One encodes the position of bulk water in a subject. The other encodes through-space distances in a concentrated macromolecule. A clinical scanner does not produce a protein structure.
4.3 EPR #
4.3.1 Unpaired electrons #
The electron magnetic moment is about 660 times a proton’s. The resonance condition is
hν = g μ_B B₀
where μ_B is the Bohr magneton and g is about 2 for a free electron, shifted by the local electronic structure. At ordinary laboratory fields the frequency is microwave, often near 9 GHz, and the spectrum is plotted against field at fixed frequency. Hyperfine coupling to a nucleus splits the line. A nitroxide, with the unpaired electron next to ¹⁴N (spin 1), gives three lines.
Most proteins are EPR-silent because every electron is paired. The silence is an advantage: a signal belongs to a radical, a paramagnetic metal, or an added label. A spin trap converts a short-lived radical into a longer-lived adduct so that the spectrum can report that a radical of a certain type was formed. It does not report a protein structure.
4.3.2 Spin labels and distances #
Site-directed spin labelling attaches a small paramagnetic group, most often a nitroxide, at a chosen cysteine. The lineshape reports probe mobility: a packed label gives a broad spectrum, a flexible one a narrow spectrum. The quantity is motion of the probe, related to but not identical with motion of the backbone.
Two labels interact magnetically. Continuous-wave EPR sees that interaction only below about 2 nm. Pulsed dipolar methods (DEER, or PELDOR) measure about 2–8 nm in favourable cases, overlapping FRET and extending a little further. The distance is between the unpaired electrons on their tethers, not between α-carbons, and the tether conformations broaden it. The caveat is the same kind as κ² in FRET and antibody linkage in localisation microscopy: the number is about the labelled molecule until the tether is modelled.
4.4 Key terms, summary, and self-check #
Key terms. Magnetogyric ratio; chemical shift; J coupling; T₁ and T₂; COSY; NOESY; HSQC; TROSY; BOLD; g-factor; spin label; DEER.
Summary. NMR detects nuclei with spin. Chemical shift reports environment, J reports through-bond neighbours, and the NOE reports through-space neighbours out to about 6 Å. Large proteins give crowded, broad spectra that are unpacked with isotopes and extra dimensions. MRI uses the same resonance on tissue water and encodes position. EPR detects only unpaired electrons, so most proteins are silent, and pairs of spin labels report distances of a few nanometres.
Self-check.
1. Why is ¹²C invisible, and why are protein samples often enriched in ¹³C and ¹⁵N?
2. What does a COSY cross-peak report, what does a NOESY cross-peak report, and which falls off as about r⁻⁶?
3. Why do lines of a large protein broaden, and why does a higher field still help a crowded spectrum?
4. What is the source of the MRI signal, and why is a BOLD image not a structure of haemoglobin?
5. Why is a typical protein EPR-silent, and how does the pulsed spin-label distance window compare with FRET?
