Rotor parameterisation

The natural way to parameterise the rotation axis of rotor frame order model is to use a 3D point, the pivot point, and a unit vector using the spherical angle basis set. This would result in the parameter set

$\begin{subequations}\begin{align} \mathfrak{M}&= \mathfrak{E}_{\textrm{ax}}+ \ma... ... , E_\phi \right\}+ \left\{ p_x, p_y, p_z\right\}. \end{align}\end{subequations}$

However this is an overparameterisation as the point $\left\{\vphantom{ p_x, p_y, p_z}\right.$ p_x, p_y, p_z $\left.\vphantom{ p_x, p_y, p_z}\right\}$ can lie anywhere on the line defined by itself and the unit vector. Due to computational truncation artifacts, this results in the pivot point shooting out to infinity along the line during optimisation.

The minimal set of independent parameters for the rotor model is four. There are many parameterisation of lines in 3D using this minimal set of four parameters, however many of these suffer from singularity problems which can be fatal for optimisation. By using geometry of the model together with information about the molecular system, a parameterisation can be constructed which avoid singularities:

A point of reference is the PDB frame close to the molecule is chosen, in this case the centre of mass (CoM) of the total system.
The point on the line defining the rotation axis closest to the reference point defines an orthogonal system of the rotation axis to the vector of the CoM to closest point. This point can be optimised with the parameters $\mathfrak{p}_1= \left\{ p_x, p_y, p_z\right\}$ with no singularity problems and minimising truncation artifacts in the numerical PCS calculation.
As the rotation axis is perpendicular to the CoM- $\mathfrak{p}_1$ vector, it can be defined using a single angle of rotation ( E_α^ax). The reference starting point for the angle is arbitrarily set to the xy-plane. The rotation of a unit vector about the CoM- $\mathfrak{p}_1$ centred at $\mathfrak{p}_1$ is smooth, hence singularities are also avoided.

The full parameter set for the rotor model implementation is therefore

$\begin{subequations}\begin{align} \mathfrak{M}&= \mathfrak{P}+ \mathfrak{E}^\alp... ..._z\right\}+ \left\{ \sigma_{\textrm{max}}\right\}, \end{align}\end{subequations}$

where P_i are the average domain position translations and rotations, E_α^ax is the single angle defining the rotation axis, p_i are the coordinates of the pivot point, and σ_max is the torsion half-angle of the rotor motion.

The rotation axis vector in the system is calculated as

$\displaystyle \hat{{r}}_{{ax}}^{}$ = $\displaystyle \left\vert\vphantom{ R_\alpha \cdot (\hat{r}_{\textrm{CoM}-\mathfrak{p}_1}\times \hat{z}) }\right.$ R_α⋅( $\displaystyle \hat{{r}}_{{\textrm{CoM}-\mathfrak{p}_1}}^{}$ × $\displaystyle \hat{{z}}$ ) $\displaystyle \left.\vphantom{ R_\alpha \cdot (\hat{r}_{\textrm{CoM}-\mathfrak{p}_1}\times \hat{z}) }\right\vert$ .

(16.8)

The angle E_α^ax is obtained from $\hat{{r}}_{{ax}}^{}$ as

$\begin{subequations}\begin{align} E_\alpha^{\textrm{ax}}&= \frac{\pi}{2} - \arcc... ...at{z}), \\ &= \arcsin(\hat{r}_{ax}\cdot \hat{z}), \end{align}\end{subequations}$

The relax user manual (PDF), created 2020-08-26.