Editing Nuclear Overhauser effect (section)

==Two-dimensional NMR ==
[[File:Noesy pulse sequence.png|thumb|upright=1.3|Pulse sequence for the standard two-dimensional NOESY experiment]]
[[File:Noesy.jpg|thumb|upright=1.3|2D NOESY spectrum of codeine]]The motivations for using two-dimensional NMR for measuring NOE's are similar as for other 2-D methods. The maximum resolution is improved by spreading the affected resonances over two dimensions, therefore more peaks are resolved, larger molecules can be observed and more NOE's can be observed in a single measurement. More importantly, when the molecular motion is in the intermediate or slow motional regimes when the NOE is either zero or negative, the steady-state NOE experiment fails to give results that can be related to internuclear distances.<ref name="Claridge" />
[[File:Codeine Structure.svg|thumb|Codeine molecule]]

Nuclear Overhauser Effect Spectroscopy (NOESY) is a 2D NMR spectroscopic method used to identify nuclear spins undergoing cross-relaxation and to measure their cross-relaxation rates. Since <sup>1</sup>H dipole-dipole couplings provide the primary means of cross-relaxation for organic molecules in solution, spins undergoing cross-relaxation are those close to one another in space. Therefore, the cross peaks of a NOESY spectrum indicate which protons are close to each other in space. In this respect, the NOESY experiment differs from the COSY experiment that relies on J-coupling to provide spin-spin correlation, and whose cross peaks indicate which <sup>1</sup>H's are close to which other <sup>1</sup>H's through the chemical bonds of the molecule.

The basic NOESY sequence consists of three 90° pulses. The first pulse creates transverse spin magnetization. The spins precess during the evolution time t<sub>1</sub>, which is incremented during the course of the 2D experiment. The second pulse produces longitudinal magnetization equal to the transverse magnetization component orthogonal to the pulse direction. Thus, the idea is to produce an initial condition for the mixing period &tau;<sub>m</sub>. During the NOE mixing time, magnetization transfer via cross-relaxation can take place. For the basic NOESY experiment, &tau;<sub>m</sub> is kept constant throughout the 2D experiment, but chosen for the optimum cross-relaxation rate and build-up of the NOE. The third pulse creates transverse magnetization from the remaining longitudinal magnetization. Data acquisition begins immediately following the third pulse and the transverse magnetization is observed as a function of the pulse delay time t<sub>2</sub>. The NOESY spectrum is generated by a 2D Fourier transform with respect to t<sub>1</sub> and t<sub>2</sub>. A series of experiments are carried out with increasing mixing times, and the increase in NOE enhancement is followed. The closest protons show the most rapid build-up rates of the NOE.

Inter-proton distances can be determined from unambiguously assigned, well-resolved, high signal-to-noise NOESY spectra by analysis of cross peak intensities. These may be obtained by volume integration and can be converted into estimates of interproton distances. The distance between two atoms <math>i</math> and <math>j</math> can be calculated from the cross-peak volumes <math>V</math> and a scaling constant <math>c</math>

::: <math> r_{\text{NOE}} = \left(\frac{c}{V_{ij}}\right)^{1/6}</math>

where <math>c</math> can be determined based on measurements of known fixed distances. The range of distances can be reported based on known distances and volumes in the spectrum, which gives a mean <math>c</math> and a standard deviation <math>c_{SD}</math>, a measurement of multiple regions in the NOESY spectrum showing no peaks, ''i.e.'' noise <math>V_{\rm err}</math>, and a measurement error <math>m_v</math>. The parameter <math>x</math> is set so that all known distances are within the error bounds. This shows that the lower range of the NOESY volume can be shown

::: <math> r_{\text{NOE lower}} = \left(\frac{c-xc_{SD}}{\frac{1}{m_v}V_{ij}+V_{\rm err}}\right)^{1/6}</math>

and that the upper bound is

::: <math> r_{\text{NOE higher}} = \left(\frac{c+xc_{SD}}{\frac{1}{m_v}V_{ij}-V_{\rm err}}\right)^{1/6}</math>

Such fixed distances depend on the system studied. For example, locked nucleic acids have many atoms whose distance varies very little in the sugar, which allows estimation of the glycosidic torsion angles, which allowed NMR to benchmark LNA molecular dynamics predictions.<ref>{{cite journal|author=David E. Condon|author2=Ilyas Yildirim|author3=Scott D. Kennedy|author4=Brendan C. Mort|author5=Ryszard Kierzek|author6=Douglas H. Turner|date=December 2013|title=Optimization of an AMBER Force Field for the Artificial Nucleic Acid, LNA, and Benchmarking with NMR of L(CAAU)|journal=J. Phys. Chem. B|volume=118|issue=5|pages=1216–1228|doi=10.1021/jp408909t|pmid=24377321|pmc=3917691}}</ref>  RNAs, however, have sugars that are much more conformationally flexible, and require wider estimations of low and high bounds.<ref name="Condon2015">{{cite journal|vauthors=Condon DE, Kennedy SD, Mort BC, Kierzek R, Yildirim I, Turner DH|date=June 2015|title=Stacking in RNA: NMR of Four Tetramers Benchmark Molecular Dynamics|journal=Journal of Chemical Theory and Computation|volume=11|issue=6|pages=2729–2742|doi=10.1021/ct501025q|pmc=4463549|pmid=26082675}}</ref>

In protein structural characterization, NOEs are used to create constraints on intramolecular distances. In this method, each proton pair is considered in isolation and NOESY cross peak intensities are compared with a reference cross peak from a proton pair of fixed distance, such as a geminal methylene proton pair or aromatic ring protons. This simple approach is reasonably insensitive to the effects of [[spin diffusion]] or non-uniform correlation times and can usually lead to definition of the global fold of the protein, provided a sufficiently large number of NOEs have been identified. NOESY cross peaks can be classified as strong, medium or weak and can be translated into upper distance restraints of around 2.5, 3.5 and 5.0 Å, respectively. Such constraints can then be used in molecular mechanics optimizations to provide a picture of the solution state conformation of the protein.<ref>{{Cite journal
|last1=Braun
|first1=W.
|last2=Gō
|first2=N.
|year=1985
|title=Calculation of Protein Conformations by Proton-Proton Distance Constraints A New Efficient Algorithm
|journal=[[J. Mol. Biol.]]
|volume=186
|issue=3
|pages=611–626
|doi=10.1016/0022-2836(85)90134-2|pmid=2419572
}}
</ref>  Full structure determination relies on a variety of NMR experiments and optimization methods utilizing both chemical shift and NOESY constraints.