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Strict NCS

Coordinate

is related by NCS to input coordinate

according to

$\begin{displaymath} X' = {\cal R} X + \vec{T} \end{displaymath}$

(18.3)

where ${\cal R}$ is a 3x3 rotation matrix and $\vec{T}$ is a translation vector.

The ${\cal R}$ matrices and the $\vec{T}$ vectors are entered using the XNCSrelation and NCSRelation statements. The first 3 input vectors correspond to the ROWS of ${\cal R}$ , and the 4th vector is $\vec{T}$ . It is not necessary to input the identity transformation, since the program sets it by default, but the user may do so for completeness.

The NCS-related interprotomer nonbonded contacts can be turned on by turning on the packing energies PVDW and PELE. Under this option, these energies are for NCS-related molecules only; the lattice energies are automatically shut off.

Many users will have the NCS relations defined in a skew frame (an auxiliary frame) in which the NCS relations have simple forms (e.g., pure rotations). The SKEW option allows a user to specify the relation between this frame and the orthogonal Å frame using a rotation matrix and translation vector. Coordinates in the skew frame are also given in Å. In this case, XNCSrelation and NCSRelation statements take the NCS relations in the skew frame. To use this option, SKEW must be given before XNCSrelation and NCSRelation.

For defining the skew frame, the algebraic conventions of Bricogne's (1976) skewing and averaging method have been adopted, except that the reference frame is the orthogonal frame used by XPLOR. This corresponds to Bricogne's ijk frame. The matrix ${\cal P}$ is Bricogne's “transition matrix." The vector $\vec{O}$ gives the coordinates of the origin of the skew frame in the reference frame, and therefore corresponds to Bricogne's ${\cal Q}^{-1} \vec{O}'$ . Using the following conventions:

$\vec{X}_{1}$: = input coordinate in reference orthogonal Å frame
$\vec{X}'_{1}$: = NCS-related coordinate in orthogonal Å frame
$\vec{X}_{2}$: = input coordinate in skew frame
$\vec{X}'_{2}$: = NCS-related coordinate in skew frame
${\cal P}$: = skew (transition) matrix
${\vec{O}}$: = ${\cal Q}^{-1} \vec{O}'$ = origin of skew frame in the reference frame (in Bricogne's package, $\vec{O}'$ is in fractional crystallographic coordinates; ${\cal Q}^{-1}$ is the cell orthogonalization matrix)
${\cal U}$: = rotation matrix relating $\vec{X}_2$ and $\vec{X}'_{2}$ (in skew frame)
$\vec{V}$: = translation vector relating $\vec{X}_{2}$ and $\vec{X}'_{2}$

one obtains the following relationships:

$\begin{displaymath} \vec{X}_{1} = {\cal P} \vec{X}_{2} + {\cal Q}^{-1} \vec{O'} \end{displaymath}$

(18.4)

$\begin{displaymath} \vec{X}_{2} = {\cal P}^{-1} \vec{X}_{1} - {\cal P}^{-1} {\cal Q}^{-1} \vec{O'} \end{displaymath}$

(18.5)

$\begin{displaymath} \vec{X'}_{2} = {\cal U} \vec{X}_{2} + \vec{V} = {\cal U} {\... ...} {\cal P}^{-1} {\cal Q}^{-1} \vec{O'} \quad + \quad \vec{V}. \end{displaymath}$

(18.6)

$\begin{displaymath} \vec{X'}_{1} = {\cal P} \vec{X'}_{2} + {\cal Q}^{-1} \vec{O'}, \end{displaymath}$

(18.7)

$\begin{displaymath} \vec{X'}_{1} = {\cal P} {\cal U} {\cal P}^{-1} \vec{X}_{1} ... ...}^{-1} \vec{O'} + {\cal P} \vec{V} + {\cal Q}^{-1} \vec{O'} \end{displaymath}$

(18.8)

$\begin{displaymath} {\cal R} = {\cal P} {\cal U} {\cal P}^{-1} \end{displaymath}$

(18.9)

$\begin{displaymath} \vec{T} = -{\cal P} {\cal U} {\cal P}^{-1} {\cal Q}^{-1} \vec{O'} + {\cal P} \vec{V} + {\cal Q}^{-1} \vec{O'} \end{displaymath}$

(18.10)

Users wishing to convert their Bricogne GENERATE mode 1 input to the XPLOR input should note the following:

${\cal P}$ is input as ROWS, not columns as in GENERATE.
The skew vector is input after the skew matrix, and is given in orthogonal Å.
It is not necessary to input the Identity NCS operation.
The NCS rotation matrices are input as rows, not columns.

Subsections

Xplor-NIH 2024-09-13