NOESY and ROESY

Introduction

Through-bond experiments reveal how atoms are connected, but they cannot tell a cis ring junction from a trans one or show how a chain folds back on itself. NOESY and ROESY answer spatial questions. Their cross-peaks connect protons that are close in space, usually within about 5 Å, regardless of how many bonds separate them. Reading these maps well requires attention to molecular size, mixing time and several look-alike signals that are not proximity at all.

Core explanation

The NOESY sequence uses three 90° pulses. The first creates transverse magnetisation, which evolves during t₁ and is labelled by its frequency. The second pulse returns this frequency-labelled magnetisation to the z axis. During the mixing time τm, longitudinal magnetisation is exchanged between nearby protons by dipolar cross-relaxation, the same process behind the one-dimensional NOE. The third pulse converts the result into observable signal. A cross-peak at (δA, δB) shows that magnetisation originally on A reached B during τm.

Cross-relaxation rates fall steeply with distance, approximately as r⁻⁶. Halving the distance increases the initial rate by a factor of 64, so strong peaks usually mean short distances, typically 2–3 Å, and peaks fade beyond about 5 Å. In the initial-rate regime , where τm is short enough that intensity grows linearly, cross-peak volumes can be calibrated against a known reference distance, such as geminal CH₂ protons at about 1.78 Å, giving approximate distances from (V ref ÷ V)^(1/6) × r ref.

The sign and size of the NOE depend on the correlation time τc relative to the Larmor frequency ω₀. Small molecules (below about 600 Da in organic solvents at 400–600 MHz) tumble fast, ω₀τc ≪ 1, and give a positive NOE: NOESY cross-peaks of opposite sign to the diagonal. Large molecules such as proteins tumble slowly, ω₀τc ≫ 1, and give a negative NOE: cross-peaks of the same sign as the diagonal. Near ω₀τc ≈ 1.1, typically molecules of about 700–1500 Da, the NOE passes through zero and NOESY may show nothing at all.

ROESY solves this problem. It records cross-relaxation in the rotating frame, under a spin-lock. The rotating-frame NOE (ROE) is positive for all correlation times, so ROESY works for mid-sized molecules. Its cross-peaks always have opposite sign to the diagonal. The spin-lock, however, can also cause TOCSY-type transfer between coupled protons, which gives peaks of the same sign as the diagonal and can cancel or mimic ROEs. Offset effects make ROE intensities less straightforward to quantify.

Three hazards dominate interpretation. Chemical exchange gives cross-peaks between two sites that interconvert during τm, such as rotamers or an OH exchanging with water; in NOESY of small molecules and in ROESY, exchange peaks have the same sign as the diagonal, which distinguishes them from NOE peaks. Spin diffusion in large molecules or with long τm relays magnetisation A → B → C, giving an A–C peak even if A and C are far apart. Zero-quantum artefacts between coupled protons in NOESY give antiphase distortions near the diagonal. Typical mixing times are 300–800 ms for small molecules and 50–150 ms for proteins, the shorter values reducing spin diffusion.

Step-by-step reasoning

1. Estimate molecular mass and choose NOESY (small or large) or ROESY (intermediate). 2. Phase the diagonal and note the sign of each cross-peak. 3. Discard same-sign peaks in small-molecule spectra as exchange or TOCSY transfer. 4. Compare cross-peaks with predicted short distances in candidate structures. 5. Build a consistent model from several contacts rather than one.

Visual explanation

Picture a phase-sensitive NOESY of a small molecule in two colours: the diagonal is red, NOE cross-peaks are blue, and a single red off-diagonal pair between two rotamer signals marks exchange rather than proximity.

Real-world analogy

NOESY is like mapping who sits near whom by noting who catches colds from whom over a week. Close neighbours pass them on quickly; given enough time, colds also spread indirectly through a third person, which is spin diffusion. The analogy omits the dependence on tumbling rate.

Real-world example

Solution structures of proteins deposited in the Protein Data Bank are built largely from thousands of NOESY-derived distance restraints. For natural products, ROESY is commonly used to establish relative configuration of stereocentres in molecules around 700–1200 Da.

Why?

Why do small and large molecules give NOEs of opposite sign? Cross-relaxation reflects the balance of zero-quantum and double-quantum relaxation pathways. Fast tumbling favours the double-quantum pathway (positive NOE); slow tumbling favours the zero-quantum pathway (negative NOE).

Common misconception

"No NOESY cross-peak proves the protons are far apart." For a mid-sized molecule the NOE may be near zero for every pair. Overlap, weak signals or too short a mixing time can also hide real contacts; ROESY should be tried.

Worked example

Question: A cross-peak volume for a proton pair is 1/8 of that for a geminal CH₂ pair at 1.78 Å, both in the initial-rate regime. Estimate the distance.

Reasoning: r = r ref × (V ref ÷ V)^(1/6) = 1.78 × 8^(1/6). 8^(1/6) = 2^(1/2) ≈ 1.41. r ≈ 1.78 × 1.41 ≈ 2.5 Å.

Answer: About 2.5 Å, a close contact, with uncertainty of perhaps ±10–20% from motion and calibration.

Quick check

1. Why is ROESY preferred over NOESY for a molecule of about 1000 Da? Answer: Its NOE may be near zero because ω₀τc ≈ 1, while the ROE stays positive at all tumbling rates.

Exam focus

Describe NOESY and ROESY as through-space experiments with approximately r⁻⁶ dependence. Link NOE sign to correlation time. Use sign to separate exchange from NOE peaks, and name spin diffusion as the risk of long mixing times.

Advanced insight

Quantitative distances require a full relaxation matrix analysis when spin diffusion is significant, since each peak depends on the whole network of nearby protons. Transferred NOE experiments exploit the negative NOE of a ligand bound to a large protein to determine the ligand's bound conformation from signals of the free ligand in fast exchange.

Summary

NOESY and ROESY record dipolar cross-relaxation during a mixing time, giving cross-peaks between protons within about 5 Å. Intensity varies roughly as r⁻⁶. NOE sign depends on tumbling, vanishing for mid-sized molecules, where ROESY is used. Exchange peaks, spin diffusion and TOCSY or zero-quantum artefacts must be recognised before drawing structural conclusions.

Practice questions

1. What sign do NOESY cross-peaks of a protein have relative to the diagonal? Answer: The same sign, because slow tumbling gives a negative NOE. 2. In a small-molecule NOESY, a cross-peak has the same sign as the diagonal. What is the most likely origin? Answer: Chemical exchange between two interconverting sites, rather than a through-space NOE. 3. Why are shorter mixing times used for proteins than for small molecules? Answer: Proteins cross-relax quickly, and long mixing times allow spin diffusion to create misleading indirect peaks. 4. By what factor does the initial NOE build-up rate change if a distance doubles? Answer: It falls by 2⁶ = 64 times.