NMR distance restraint

An NMR distance restraint is a restriction on distances between selected atoms used in a structural calculation from nuclear magnetic resonance data. It often specifies an allowed interval or upper bound, with a penalty for a candidate structure violating it. Distances inferred from the nuclear Overhauser effect, including NOESY cross-peak information, are important examples. The restraint combines experimental interpretation with atom assignments and a model; it is not simply an exact ruler reading.^[ref]

For an original interval example, suppose a proton pair has allowed separation $l=2.0$ to $u=3.5$ angstroms. A flat-bottom quadratic penalty isA candidate distance of4.0angstroms contributes $0.25k{\mathring{\mathrm{A}}}^2$, whereas3.0angstroms contributes zero. Here $k$ has units of energy per squared length; the penalty is a fitting objective, not a measured pair interaction energy. Many restraints and stereochemical terms together constrain a protein structure.^[ref]

An ambiguous NMR distance restraint may describe several possible atom pairs contributing to one observation. In one inverse-sixth-power sum convention,For original candidate distances2 and4angstroms, this gives about1.995angstroms. Dividing the sum by2 before taking the inverse sixth root instead gives about2.239angstroms: an average and a sum are different models. An ambiguous assignment also does not assert that every candidate pair separately satisfies the nominal bound.^[ref]

Cross-peak overlap, nuclear spin diffusion and conformational averaging affect interpretation. A structural calculation therefore retains assignment and averaging information alongside its distance bounds; an absent or weak peak alone need not establish a large separation.

An ambiguous NMR distance restraint represents an observation compatible with several atom-pair assignments rather than one uniquely identified pair. It combines candidate contributions according to a specified rule during an NMR distance restraint calculation. A single overlapping NOESY feature can motivate such a representation; ambiguity concerns which nuclei generated the observation.^[ref]

For an inverse-sixth-power sum model with one molecular copy,where $\mathcal{A}$ contains the candidate pairs. An upper bound applies to this effective quantity, not independently to every $r_{ij}$. Xplor-NIH's optional OR syntax lists alternative pairs and uses sum averaging; it avoids adding unwanted cross-combinations of broad atom selections.^[ref]

For an original example, suppose the possible assignments are pair $(a,b)$ at3angstroms and pair $(c,d)$ at8angstroms. ThenA3.5angstrom upper bound is compatible with this sum even though one candidate pair is8angstroms apart. Enforcing a separate3.5angstrom limit on both candidates would assert stronger information than the ambiguous observation supplies.

The pair list matters too: these alternatives need not include $(a,d)$ or $(c,b)$. Selecting all atoms from $\{a,c\}$ against all atoms from $\{b,d\}$ would introduce those extra combinations and change the calculation. An explicit assignment representation preserves the intended alternatives.^[ref]

Sum normalization, molecular symmetry and the distinction between assignment ambiguity and conformational averaging must remain part of the restraint model. Reducing an effective distance during refinement does not by itself prove that a particular candidate assignment is the uniquely correct one.