Proton Chemical Shift Restraints

X-PLOR has the facilities to refine against nonexchangable proton chemical shifts. The predicted chemical shift is calculated for each proton as the sum of four terms. The terms are the random coil, ring-current, magnetic anisotropy and electric-field chemical shifts.

The ring-current is calculated as

$$\sigma_{ring} = iBG({\bf r}),$$ (24.1)

where $i$ is a ring-current intensity factor, $B$ is a proportionality constant related to the susceptibility of the ring, $G$ is a spatial term and ${\bf r}$ is the vector from the observed proton to the ring.

The magnetic anisotropy term represents the sum of the anisotropies arising from the C'-O and C'-N bonds of the backbone and the side-chain functional groups of Asp, Glu, Asn, and Gln.

$$\sigma_{ani}= \sum_j (3r_{X(j)H})^{-3} [\Delta \chi_{1,j} (3 \cos^2 \theta -1 ) + \Delta \chi_{2,j} (3 \cos^2 \phi -1)] ,$$ (24.2)

where the sum is over functional group; $R_{X(j)H}$ is the distance from the proton to the center of anisotropy $X$ along the C-O or C-N bond $j$; $\Delta\chi_{1,j}$ and $\Delta\chi_{2,j}$ are constants that depend on the bond type $j$; $\theta$ is the angle between the $X(j)H$ vector and the normal to the plane defined by the atoms of the bond of interest plus a third atom; and $\phi$ is the angle between the $X(j)H$ vector and the axis located in the plane of the three atoms that is perpendicular to the $z$ axis defined by the bond of interest.

Finally, the electric-field chemical shifts is calculated by

$$\sigma_E=\sum_i (Q_i/r_{iH}^2) \cos \theta_i,$$ (24.3)

where the sum is over all heavy atoms, $\theta_i$ is the angle between the $X_i$-H and C-H vectors, $Q_i$ is the charge on atom $X_i$, and $r_{iH}$ is the distance from $X_i$ to H.

The potential is calculated as $E_{PROT}=\sum_i K_{PROT}(\sigma_{calc,i} - \sigma_{obs,i})^2$. For more information, see Kuszewski, Gronenborn, and Clore (1995).

For non-stereoassigned protons, there is a method of calculating the potential to account for both possible assignments. For details, see Kuszewski, Gronenborn, and Clore (1996a).