# Symmetry expansion query

Dear users,

Apologies this is more of a general EM question rather than cryoSPARC related. I have a question regarding symmetry expansion. I am using `relion_particle_symmetry_expand` to expand particles with a given point group symmetry before performing signal subtraction and focused refinement.

When Relion incrementally adds 360/point group symmetry to a given Euler angle in the star file (`AngleRot`), does it matter why this angle is selected rather than the other two?

Additionally, I am typically performing `relion_reconstruct` to check signal subtractions etc. Should this still work for a set of symmetry expanded particles (ie should the reconstruction look the same as for the original set of particles)?

RJ

The Euler angles certainly are all different in their effect on the particle pose. You should pick up a pen from your desk and rotate it around the long axis, then about an orthogonal axis, then about the long axis again. Not only is each different, but the order matters too. 3D rotations do not commute.

The example you gave only applies to cyclic point groups (Cn). In general Relion, cryoSPARC, pyem and localrec all convert the Euler angles to a useful rotation representation like matrices and generate the symmetry related copies by applying each unique symmetry operator by e.g. matrix multiplication. The resulting new orientations are converted back to Euler angles afterwards. For cyclic groups, the sole symmetry axis is by convention the Z-axis of the map, which also happens to be the axis for AngleRot (same as AnglePhi). For the other groups, it’s not so simple what happens to the Euler angles.

If you do a reconstruction after a symmetry-imposed refinement in Relion, then the angles are only one symmetry equivalent sector of the rotation sphere. You must therefore also impose symmetry during reconstruction. After expansion, there is a (copy of) a particle in every symmetry equivalent sector; the space is full. The reconstruction can then be asymmetric. Since you’re just checking subtraction quality and so forth, I would randomly pick a small set of particles (~10,000), use symmetry, and set the maximum resolution to ~6A. The reconstruction will take just a few minutes, and you can analyze the subtraction quality without worrying about the complexities of the duplicated particles and so forth.

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Dear Daniel,
Thankyou for your reply- I think I realise now where I was going wrong. I was trying to symmetry expand particles from a reconstruction that I had done in C1 rather than with the correct point group symmetry (I was worried that by imposing symmetry, I may be averaging out differences in the region I am now checking for differences in). However, I was wondering how it selects the angle of rotation to coincide with the rotational symmetry axis- as you say I had done the exercise with the pen and realised they are not the same! But, I think symmetry expansion does not make sense unless you are starting from a reconstruction that has been performed with the specified symmetry, otherwise the particles are not aligned to a particular symmetry axis?
This would explain why when I reconstruct from a set of symmetry expanded particles where the original refinement was performed in c1, I end up with nonsense?

Correct, symmetry expansion only means anything if the particles/reference are aligned to the (conventional) axis (axes) of symmetry, such as after a symmetry-imposed refinement.

For cyclic groups, the symmetry axis is the map Z-axis. For other groups, the first axis is the map Z. For example, for Dn groups, the C sub-group’s axis the map Z, and the 2nd one is the map Y.

ah thank you so much!!