= NMR distance restraint
{c}
An \b[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.<Distance restraints in the Xplor-NIH manual>[Distance Restraints; Setup of Distance Restraints, atom selections and bounds]{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 is
$$
E(r)=\begin{cases}
k(l-r)^2&r<l,\\
0&l\leq r\leq u,\\
k(r-u)^2&r>u.
\end{cases}
$$
A candidate distance of4.0angstroms contributes $0.25k\,\mathrm{\mathring A^2}$, 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>.<Distance restraints in the Xplor-NIH manual>[Square-Well Function, selectable exponent and zero-violation interval]{ref}
An <ambiguous NMR distance restraint> may describe several possible atom pairs contributing to one observation. In one inverse-sixth-power sum convention,
$$
r_{\mathrm{eff}}=\left(\sum_{ij}r_{ij}^{-6}\right)^{-1/6}.
$$
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.<Distance restraints in the Xplor-NIH manual>[Choice of Averaging, eqs.20.1 and20.3; ambiguity discussion]{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.
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