Regardless of the complexity of the experimental design that produced them, the intensities of signals in a nuclear magnetic resonance (NMR) spectrum depend on the populations of molecules in the various allowed nuclear spin energy states immediately before the analyzing RF pulse is applied. In the simplest case, these populations are the Boltzmann populations, characteristic of the system at thermal equilibrium. Under such conditions, the relative intensities of signals from various chemically different groups of molecule spins are proportional to the relative number of spins of each type. But in a variety of circumstances, including an experimental design more complex than a single RF pulse, the number of molecules in some or all of the permitted nuclear energy states may differ from those present at thermal equilibrium. In this event, the intensity of a signal observed in the spectrum will be changed. The change in signal intensity is called a nuclear Overhauser effect, or NOE. There are several ways of expressing the NOE quantitatively. Most commonly used (1) is the equation
where fj{S} indicates the NOE on the signal(s) from spin I when there is a perturbation of level populations associated with spin S. I 0 is the normal intensity of the signal(s) for spin I observed from a sample at thermal equilibrium before the analyzing RF pulse, and Ip is the intensity of the same signal when there has been a perturbation of spin S before the analyzing pulse. The value of f {S} depends on the gyromagnetic ratios of spins I and S, how these spins move in the sample, the strength of the magnetic field used for the NMR experiment, and the details of how the energy level populations associated with spin S are perturbed during the course of an experiment. Importantly, f {S} depends on the distance between spins I and S. Values of f^S} ranging from 0.5 to -1.0 are possible when both I and S are protons. The first value corresponds to a 50% enhancement of the intensity associated with spin I, and the latter represents the disappearance of the signal intensity for this spin.
A change in NMR signal intensity can arise in molecular situations where two or more spins interact strongly with each other, such that perturbation of the level populations of one of the spins by RF pulses or by some other way ultimately results in the perturbation of the level populations corresponding to the other spin. It should be apparent that the NOE is strongly related to relaxation, those natural processes in any sample that tend to return the populations of the spin energy levels to their Boltzmann (thermal) equilibrium values if these level populations have been altered in some way. (See NMR (Nuclear Magnetic Resonance).) Consequently, the formation and disappearance of a NOE will be time-dependent.
The power of experiments that produce NOEs lies in that the bulk of the interactions that lead to relaxation are strongly dependent on the distance between interacting spins. Thus, observation of NOEs can provide internuclear distance data that ultimately can be used to deduce information about the three-dimensional structures of biopolymers. The distance dependence of the NOE is strong. In the case of 1H- 1H interactions, NOEs are typically observed only when the distance between the spins is 0.22-0.55 nm, with the effect observed at 0.55 nm being less than 1% of that when the internuclear distance is 0.22 nm.
When molecules move slowly in the sample, an additional phenomenon complicates the NOE experiment. "Slow motion" conditions in this context arise when molecules have masses in excess of about 5 kDa or when solutions are highly viscous. The complication is called spin diffusion; it occurs because the magnetization transfers or level population alterations mentioned above become rapid between all protons. Under these conditions, magnetization that originates with one spin of the molecule may be transferred not only to a nearby proton but also to more distant protons. Thus, distance information that is implicit in the NOE may be obscured.
It is also possible to create NOEs because of interactions between various coherences that can be formed as a result of the application of RF pulses to the sample. These experiments, called rotating-frame NOEs, or ROEs, exhibit a different dependence on the details of molecular motions than the NOEs described above. The NOE and ROE experimental results are thus complementary. The formation of the ROE is also time-dependent and must compete with the decay of spin coherence by transverse relaxation. (See also NOESY Spectrum, ROESY Spectrum, Distance Geometry, Simulated Annealing.)
