Rotation Search

X-PLOR performs conventional rotation searches in Patterson space. The real-space Patterson search method of Huber (1985) is employed. The stationary Patterson map P2 is computed from the observed intensities by fast Fourier transformation on the specified grid. The Patterson map P1 to be rotated is computed from either the observed intensities (self-rotation search) or from a search model (cross-rotation search). Note that the rotation search routine takes the layout of the map P1 as it was written using the xrefin map statement (see Section 16.1). In most cases, the layout of P1 will look like this:
 XMIN=-45.0 XMAX=45.0 
 YMIN=-45.0 YMAX=45.0 
 ZMIN=0.0 ZMAX=30.0
where one makes use of the centrosymmetry and 30.0 is the maximum vector length. Patterson vectors are selected according to the value of the Patterson map at the particular grid point (THREshold) and according to the distance from the grid point to the origin (RANGe). Of this selected subset of Patterson vectors, only the strongest NPEAKs Patterson vectors are chosen for the rotation search.

The selected Patterson vectors are rotated using Eulerian angles ($\theta_1$, $\theta_2$, $\theta_3$) as defined by Rossmann and Blow (1962), pseudo-orthogonal Eulerian angles as defined by Lattman (1985), or spherical polar angles ( $\psi,\phi,\kappa$) (see Section 2.4 for the conventions). The Lattman angles are related to the Eulerian angles by

$\displaystyle \theta_{+}$ $\textstyle =$ $\displaystyle \theta_1 + \theta_3$ (19.1)
$\displaystyle \theta_{-}$ $\textstyle =$ $\displaystyle \theta_1 - \theta_3$  
$\displaystyle \theta_2$ $\textstyle =$ $\displaystyle \theta_2$  

The $\theta_2$ angles are sampled at a constant interval $\Delta$. Following Lattman (1985), the variable interval for $\theta_{+}$ is given by $\Delta / \cos (\theta_2/2)$, and the variable interval for $\theta_{-}$ is given by $\Delta / \sin (\theta_2/
2)$. For both Eulerian angles and spherical polar angles, the sampling interval is constant ($\Delta$) for the three angular variables. The rotation search should be restricted to the asymmetric unit of the rotation function (Rao, Jih, and Hartsuck, 1980). Using the convention in the Rao et al. paper, the Patterson symmetry of the rotated map $P_1$ is “1", and that of $P_2$ is the Patterson symmetry of the space group.

The values of the Patterson map P2 at the positions of the rotated Patterson vectors of map P1 are computed by linear eight-point interpolation (Nordman, 1980).

For each sampled orientation $\Omega$ the product function

\end{displaymath} (19.2)

between the rotated vectors of P1 and the interpolated values of the Patterson map P2 is computed.

Upon completion of the rotation function, all sampled grid points are sorted with respect to their product correlation value, and a simple peak search is carried out using the matrix metric defined in Brünger (1990). For two given rotation matrices $\Omega^1, \Omega^2$, the metric is defined as

m (\Omega^1,\Omega^2) = \min_{s=1,n_s} \;
\sqrt{ {\rm Tr} ... O}_s \Omega^2 )
( \Omega^1- {\cal O}_s \Omega^2 )^t \} }
\end{displaymath} (19.3)

where $n_s$ is the number of symmetry operators of the space group of the crystal and ${\cal O}_s$ is the rotational part of the symmetry operator $s$. This definition applies to the default option SELF=FALSE. If SELF=TRUE is specified, the metric is defined as
m (\Omega^1,\Omega^2) = \min_{s=1,n_s, s'=1,n_s} \;
... O}_s')
( \Omega^1- {\cal O}_s \Omega^2 {\cal O}_s')^t \} },
\end{displaymath} (19.4)

i.e., the metric assumes the crystal symmetry for both Patterson maps P1 and P2. In the case of space group $P_1$, the metric $m$ is related to a rotation angle $\kappa$, which defines a rotation around a certain axis $a$:
m = \sqrt{4 ( 1 - \cos{\kappa} ) }
\end{displaymath} (19.5)

Application of this rotation around $a$ transforms the matrix $\Omega^1$ into $\Omega^2$.

Two RF grid points are considered as being in the same cluster if the corresponding rotation matrices yield $m(\Omega^1,\Omega^2) < {\epsilon}$. For example, if $\epsilon$ is set to 0.25, matrices belong to the same cluster if they can be transformed by a rotation of 10$^{\circ}$ or less around a certain axis. The incorporation of crystallographic symmetry in Eq. 19.3 ensures that clusters of grid points at the boundaries of the asymmetric unit of the RF are treated properly.

This peak search removes grid points that are close to grid points with larger RF values. It is not a true peak search, but rather reduces the number of points to be checked by subsequent analysis. The reduced set of the highest grid points is written to a specified file. For example, if $\epsilon$ is set to 0.25, the file will contain grid points that are mutually different by at least 10$^{\circ}$. This file will be read by the $PC$-refinement procedure described below. The value for $\epsilon$ should be chosen to be less than the radius of convergence of the $PC$-refinement (around 10$^{\circ}$). It should be noted that the peak search procedure maps the grid points into an asymmetric unit of the rotation function.

Xplor-NIH 2023-11-10