Frame order model nesting

The concept of model nesting is used to hugely speed up the optimisation in the automated protocol. The most complex models have 15 independent parameters, and performing a grid search over 15 dimensions of the pseudo-ellipse frame order model is not feasible when using PCS numerical integration. The idea is to use the optimised parameters of a simpler model as the starting point for a more complex model, avoiding the need for a grid search for those copied parameters. This appears to work as the PCS value is dominated by the average domain position, hence the average domain parameters are very similar in all models.

Model categories

The modelling of the σ torsion angle gives a number of categories of related models, those with no torsion, those with restricted torsion, and the free rotors.

No torsion

When σ = 0, the following models are defined:

Restricted torsion

When 0 < σ < π, the following models are defined:

Free rotors

When σ = π, i.e. there is no torsional restriction, the following models are defined:

Multiple torsion angles

This covers a single model - the double rotor.

Parameter categories

There are three major parameter categories - the average domain position, the eigenframe of the motion, and the amplitude of the motion.

Average domain position

Let the translational parameters be

$\displaystyle \mathfrak{T}= \left\{ P_x, P_y, P_z\right\},$ (12.87)

and the rotational or orientational parameters be

$\displaystyle \mathfrak{O}= \left\{ P_\alpha , P_\beta , P_\gamma \right\}.$ (12.88)

Two full average position parameter sets used in the frame order models are

\begin{subequations}\begin{align}&\mathfrak{P}= \mathfrak{T}+ \mathfrak{O}= \lef...
...left\{ P_x, P_y, P_z, P_\beta , P_\gamma \right\}. \end{align}\end{subequations}

The motional eigenframe

This consists of either the full eigenframe or a single axis, combined with the pivot point(s) defining the origin of the frame(s) within the PDB space. The eigenframe parameters themselves are

\begin{subequations}\begin{align}&\mathfrak{E}_{\alpha\beta\gamma}= \left\{ E_\a...
...xtrm{ax}}= \left\{ E_\alpha^{\textrm{ax}}\right\}. \end{align}\end{subequations}

The pivot parameter sets are

\begin{subequations}\begin{align}&\mathfrak{p}_1= \left\{ p_x, p_y, p_z\right\}, \\ &\mathfrak{p}_2= \left\{ p_d\right\}, \end{align}\end{subequations}

The rigid body ordering

The parameters of order are

$\displaystyle \mathfrak{S}= \left\{ \theta , \theta_x , \theta_y , \sigma_{\textrm{max}}, \sigma_{\textrm{max,2}}\right\}.$ (12.92)

Frame order parameter nesting in the automated protocol

Table 12.2: The nesting of frame order model parameters and the resultant grid search dimensionality. The boxes highlight parameter sets which are optimised in the initial grid search. The start of each train of arrows are the optimised parameters which will be copied for the more complex model and excluded from the grid search. The non-nested gird search dimensionality is given in brackets.

The parameter nesting used in the automated protocol is shown in table 12.2. This massively collapses the dimensionality of the initial grid search.

The relax user manual (PDF), created 2016-10-28.