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 θ_x, θ_x, and σ_max half-angles are permuted. Because of the constraint θ_x≤θ_y in the pseudo-ellipse 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 z-axes 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 pseudo-ellipse 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 y-axis and θ_y 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 x and y axes are not defined in the x-y plane, therefore there are only two permutations (the first solution and `permutation A').
• Any axis in the x-y plane can be used for the permutation, however different axes will result in different χ² 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

$${\frac{{\theta_x ' + \theta_y '}}{{2}}}$$ (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.

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 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 half-angles 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 C-terminal domain, and the rotor half-angle 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 half-angles and B and D are the results after low quality optimisation demonstrating the presence of the multiple local minima.