High dimensional Function

For POTEntial=HIGH, a restraining function for stereospecifically assignable proton pairs is employed (Habazettl et al., 1990). This is useful for resolved but unassigned peaks involving methylene protons. The restraining function is given by

$$E_{NOE}^{class} = S E_{highdim}$$ (20.17)

where $S$ is the energy constant specified by the SCALe statement. Supppose that one observes $2n$ distance restraints involving the two protons $a$ and $b$ of a particular unassigned pair; this means that there are a total of $2n$ ASSIgn statements specified for this particular unassigned pair (NCOUnt has to match $2n$). The $E_{highdim}$ function is then the product of two $2n$-dimensional functions, each corresponding to one of the two possible stereospecific assignments:

$$E_{highdim}= (\sum_{i=1}^{n} E_{i}^{ass}(r_{a}^{i}, r_{b}^{i})) (\sum_{i=1}^{n} E_{i}^{ass}(r_{b}^{i}, r_{a}^{i}))$$ (20.18)

$$E_{i}^{ass}(x, y) = C_{i}(f(x, l_{a}^{i}, u_{a}^{i}) + f(y, l_{b}^{i}, u_{b}^{i}) + 1.5 F(y-x, L^{i}, U^{i}))$$ (20.19)

where $r_{a}^{i}$ and $r_{b}^{i}$ are the distances from spin $i$ to the unassigned spins $a$ and $b$, respectively. When the corresponding ASSIgn statements contain multiple atom selections, averaged distances are computed; only center-averaging is allowed in combination with the high dimensional potential. $l_a^i$ and $l_b^i$ are lower bounds for the distances between spin $i$ and spins $a$ and $b$, respectively. $u_a^i$ and $u_b^i$ are upper bounds for the distances between spin $i$ and spins $a$ and $b$, respectively. $L^i$ and $U^i$ are the lower bounds and upper bounds for the distance difference $y-x$, respectively. These upper and lower bounds are specified through a group of $2n$ assignment statements:

$$\begin{eqnarray*} {\tt assign } \quad ( {\rm spin}_1 ) \quad ( {\rm spin}_a ) \... ...}_3 ) \quad ( {\rm spin}_b ) \quad l_b^3 \quad u_b^3 \quad U^3 \end{eqnarray*}$$

...

$$\begin{eqnarray*} {\tt assign } \quad ( {\rm spin}_n ) \quad ( {\rm spin}_a ) ... ...}_n ) \quad ( {\rm spin}_b ) \quad l_b^n \quad u_b^n \quad U^n \end{eqnarray*}$$

where the ${\rm spin}_i$ terms should be understood as atom selections. $C_i$ are constants that are determined internally, such that the barrier height between the two minima of $E_{highdim}$ is given by BHIG. The functional forms of $f$ and $F$ are similar to the soft-square function (Section 20.3.4) with a linear asymptote, except that the switching region is symmetrically arranged:

$$f(x, l, u) = \left \{ \begin{array}{lll} 0 & {\rm for } & l ... ...m}(x-u) & {\rm for} & x > u + {\rm lim} \\ \end{array}\right.$$ (20.20)

$$F(\delta, L, U) = \left \{ \begin{array}{lll} 0 & {\rm for }... ...\rm for} & \delta > U + \sqrt{2} {\rm lim} \end{array}\right.$$

where lim is set internally to 0.1.