Frame order axis permutations
Multiple local minima exist in the optimisation space for the isotropic and pseudoelliptic cone frame order models.
In the case of the pseudoellipse, the eigenframes at each minimum are identical, however the θ_{x}, θ_{x}, and
σ_{max} halfangles are permuted.
Because of the constraint
θ_{x}≤θ_{y} in the pseudoellipse model, there are exactly three local minima (out of 6 possible permutations).
In the isotropic cone, the
θ_{x}≡θ_{y} condition collapses this to two.
The multiple minima correspond to permutations of the motional system  the eigenframe x, y and zaxes as well as the cone opening angles θ_{x}, θ_{y}, and
σ_{max} 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:
 Optimise to one solution.
 Duplicate the data pipe for the model as `permutation A'.
 Permute the axes and amplitude parameters to jump from one local minimum to the other.
 Optimise the new permuted model, as the permuted parameters will not be exactly at the minimum.
 Repeat for the remaining `permutation B' solution (only for the pseudoellipse models).
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 θ_{x} is defined as a rotation about the yaxis and θ_{y} is about the xaxis.
See table 12.1 on page for the pseudoellipse 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 x and y axes are not defined in the xy plane, therefore there are only two permutations (the first solution and `permutation A').
 Any axis in the xy plane can be used for the permutation, however different axes will result in different χ^{2} values.
 As
θ_{x}≡θ_{y}, the condition
θ_{x}≤σ_{max}≤θ_{y} can only exist if the torsion and cone angles are identical.
 Permutations A and B create identical cones as the x and y axes are equivalent.
The new isotropic cone angle is defined as
θ' = . 
(12.78) 
The isotropic cone axis permutations are shown in figure 12.3 on page .
Table 12.1:
The pseudoellipse motional eigenframe and halfangle permutations implemented in the frame_order.permute_axes user function.

Figure 12.2:
Pseudoellipse 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 Cterminal domain, and the rotor halfangle set to 30^{o}.
The condition
θ_{x}≤θ_{y}≤σ_{max} is shown in A and B.
The condition
θ_{x}≤σ_{max}≤θ_{y} is shown in C and D.
The condition
σ_{max}≤θ_{x}≤θ_{y} is shown in E and F.
A, C, and E are the axis permutations for a set of starting halfangles and B, D, and F are the results after low quality optimisation demonstrating the presence of the multiple local minima.

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 Cterminal domain, and the rotor halfangle set to 30^{o}.
The condition
θ≤σ_{max} is shown in A and B.
The condition
σ_{max}≤θ is shown in C and D.
A and C are the axis permutations for a set of starting halfangles and B and D are the results after low quality optimisation demonstrating the presence of the multiple local minima.

The relax user manual (PDF), created 20240608.