# What is actually occuring when symmetry is imposed?

Hi all
Apologies if this is a simple question to some, but i have had problem figuring out what is actually happening alignment wise when symmetry is imposed.

To my understanding, symmetry allows us to reduce the angles considered during alignment by a factor equivalent to the point group e.g. we for a C6 reconstruction, we only need to align particles to a range of 360/6 of the rotational sphere. This makes intuitive sense to me as correct alignment of one particle in this orientation would correctly orient all subsequent particles in their correct positions. I am however, confused as to how remaining particles contribute to the alignment/density averaging of the reconstruction. The particle number is not duplicated, and yet we are clearly exploiting the the symmetry related particles to imrprove resolution; I just donâ€™t really undestand how each particle image is being manipulated to achieve this.

Further, i wanted to clarify exactly what symmetry expansion is doing. To my understanding here, each particle image is duplicated by its relevant symmetry, and the rotational paratmers for each duplicate image is reset according to the specified symmetry. E.g.if we had a particle image of a C3 particle with itâ€™s Z rotation determined as 120 degrees, then duplicate images would simply have their Z rotational parameters set to 240 and 360? At which point, we donâ€™t impose symmetry, as each duplicate particle image has filled the rotational sphere, allowing C1 reconstruction. Is this correct?

Apologies if Iâ€™m way off! Much of my SPR understanding is unverified and Iâ€™m a biologist first so Iâ€™m never quite sure if I have understood concepts correctly!

James.

1 Like

what is actually happening alignment wise when symmetry is imposed.

In general, nothing is changed during alignment. Symmetry is imposed by inserting each particleâ€™s Fourier transform N times (for a N-fold group) during reconstruction, once in each possible symmetry-related pose.

symmetry allows us to reduce the angles considered during alignment

It does, but cryoSPARC doesnâ€™t implement this optimization, because branch and bound is so fast that other optimizations have been prioritized over reducing the search space. Other refinement packages do limit the search space via symmetry - you can tell because the orientation distributions become restricted to the angles that have actually been searched.

exactly what symmetry expansion is doing

Your description is correct, except that the images arenâ€™t usually literally duplicated. We make duplicate metadata entries with different symmetry-related poses, but they can all point to the same physical image. The particles could be re-extracted, but then the â€śduplicateâ€ť images will actually be slightly different, since they should be centered on different subunits within the particle. (They could also be smaller than the original box used for the whole particle).

A symmetry-expanded reconstruction should indeed be in C1, but should also use a mask focused on one asymmetric unit (but perhaps including interfaces with other subunits).

1 Like

Hi Daniel

In general, nothing is changed during alignment. Symmetry is imposed by inserting each particleâ€™s Fourier transform N times (for a N-fold group) during reconstruction , once in each possible symmetry-related pose.

Ahh i see. A follow up question then is how is this different from symmetry expansion? If symmetry expansion is only re-writing metadata for symmetry related poses that point to the same particle image, that sounds very much like re-inserting each particle images Fourier transform N times according the symmetry?

Cheers,
James.

Thatâ€™s right, if you symmetry expand first, and then do a simple C1 reconstruction of the expanded particle set, you should get the symmetric map.

Frealign has a feature where instead of giving the group as â€śC4,â€ť you can use a lower case â€śc4,â€ť and itâ€™ll actually give 4 poses for each particle optimized around each symmetry related position. This is pretty similar to what you would get from symmetry expansion and local refinement, without focusing on one monomer.

All these features are usually used to study pseudosymmetry by focusing on one asymmetric unit at a time, though.

Itâ€™s also worth noting that we can symmetrize maps post-facto by summing up symmetry transformed copies, perhaps in order to make a symmetric reference. Thatâ€™s what `relion_image_handler --sym` does.

Thankyou again for all your help, i think i finally get the whole thing!!

So symmetry expansion gives us an edge for more mobile/assymetric structures, as each duplicate particle image can have itâ€™s alignments tweaked during masked local alignment. In contrast, imposed symmetry will re-insert each particles Fourier transforms N times, however every insertion will be aligned in steps exactly 360/N (assuming rotational symmetry) degrees relative to the non-expanded particle images angular assignment? contributing more signal but not allowing each unique particle to be individually aligned?

Right!

Just one point of precision:

contributing more signal but not allowing each unique particle to be individually aligned?

The unique whole particle - the assembly of all the asymmetric units - is always aligned, I think you meant the unique subunits / asymmetric units.

The unique whole particle - the assembly of all the asymmetric units - is always aligned, I think you meant the unique subunits / asymmetric units.

I did indeed! Youâ€™ve been a tremendous help thanks again

James.