Frame order axis permutations

Multiple local minima exist in the optimisation space for the isotropic and pseudo-elliptic cone frame order models. In the case of the pseudo-ellipse, the eigenframes at each minimum are identical, however the θx, θx, and σmax half-angles are permuted. Because of the constraint θxθy in the pseudo-ellipse model, there are exactly three local minima (out of 6 possible permutations). In the isotropic cone, the θxθy condition collapses this to two. The multiple minima correspond to permutations of the motional system - the eigenframe x, y and z-axes as well as the cone opening angles θx, θy, and σmax associated with these axes. But as the mechanics of the cone angles is not identical to that of the torsion angle, only one of the three local minima is the global minimum.

As the minfx library used in the frame order analysis currently only implements local optimisation algorithms, and because a global optimiser cannot be guaranteed to converge to the correct minima, a different approach is required:

These steps have been incorporated into the automated analysis protocol.

The permutation step has been implemented as the frame_order.permute_axes user function. It is complicated by the fact that θx is defined as a rotation about the y-axis and θy is about the x-axis. See table 12.1 on page [*] for the pseudo-ellipse model permutations. These are also illustrated in figure 12.2 on page [*].

For the isotropic cone model, the same permutations exist but with some differences:

The new isotropic cone angle is defined as

θ' = $\displaystyle {\frac{{\theta_x ' + \theta_y '}}{{2}}}$. (12.78)

The isotropic cone axis permutations are shown in figure 12.3 on page [*].


Table 12.1: The pseudo-ellipse motional eigenframe and half-angle permutations implemented in the frame_order.permute_axes user function.
\begin{table}\begin{center}
\begin{threeparttable}\begin{tabular*}{\textwidth}...
... optimised solution.
\end{tablenotes}\end{threeparttable}\end{center}\end{table}


Figure 12.2: Pseudo-ellipse axis permutations. This uses synthetic data for a rotor model applied to CaM, with the rotor axis defined as being between the centre of the two helices between the domains (the centre of all cones in the figure) and the centre of mass of the C-terminal domain, and the rotor half-angle set to 30o. The condition θxθyσmax is shown in A and B. The condition θxσmaxθy is shown in C and D. The condition σmaxθxθy is shown in E and F. A, C, and E are the axis permutations for a set of starting half-angles and B, D, and F are the results after low quality optimisation demonstrating the presence of the multiple local minima.
\includegraphics[
width=0.9\textwidth,
bb=14 14 1036 1384
]{images/perm_pseudo_ellipse}

Figure 12.3: Isotropic cone axis permutations. This uses synthetic data for a rotor model applied to CaM, with the rotor axis defined as being between the centre of the two helices between the domains (the centre of all cones in the figure) and the centre of mass of the C-terminal domain, and the rotor half-angle set to 30o. The condition θσmax is shown in A and B. The condition σmaxθ is shown in C and D. A and C are the axis permutations for a set of starting half-angles and B and D are the results after low quality optimisation demonstrating the presence of the multiple local minima.
\includegraphics[
width=0.75\textwidth,
bb=14 14 835 949
]{images/perm_iso_cone}

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