High dimensional Function

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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

$\begin{displaymath} E_{NOE}^{class} = S E_{highdim} \end{displaymath}$

(20.17)

where

is the energy constant specified by the SCALe statement. Supppose that one observes

distance restraints involving the two protons

and

of a particular unassigned pair; this means that there are a total of

ASSIgn statements specified for this particular unassigned pair (NCOUnt has to match

). The $E_{highdim}$ function is then the product of two

-dimensional functions, each corresponding to one of the two possible stereospecific assignments:

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

(20.18)

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

(20.19)

where $r_{a}^{i}$ and $r_{b}^{i}$ are the distances from spin

to the unassigned spins

and

, 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.

and

are lower bounds for the distances between spin

and spins

and

, respectively.

and

are upper bounds for the distances between spin

and spins

and

, respectively.

and

are the lower bounds and upper bounds for the distance difference

, respectively. These upper and lower bounds are specified through a group of

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.

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

and

are similar to the soft-square function (Section 20.3.4) with a linear asymptote, except that the switching region is symmetrically arranged:

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

(20.20)

(20.21)

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

where lim is set internally to 0.1.

Xplor-NIH 2024-09-13