Tetrahedral Symmetry Orientation

When applying tetrahedral symmetry during refinement/reconstruction, CryoSPARC aligns the map so that a 3-fold axes is along Z:

My understanding is that with tetrahedral, typically a 2-fold is along each axis (X, Y, and Z) and the 3-fold axes are across the cube diagonals. Similar to this image:


This convention allows for one to easily address the D2/C2 symmetry that is inherently present in T symmetry. While it still requires a rotation to center the C3 symmetry on an axis, CryoSPARC’s current method still only allows for a single C3 axis on X, Y, or Z, and requires rotations to reach the other three.

I’m by no means a symmetry expert here, so I could be wrong, but I was wondering if this placement was intentional, and if so, what the rationale was for it?

Dear @ccgauvin94 ,

Thanks for the post! I believe our convention for tetrahedral symmetry (3 fold axis on Z) is consistent with RELION’s conventions. (Note that this isn’t true for all symmetry groups — in particular, our convention for Dn symmetry places the two fold axis along Y, rather than X). We’d like to document our symmetry conventions better, and we’ve made a note to describe this on our guide.

Could you describe your rationale for preferring the indicated convention? Do you mean that, if T symmetry is implemented with the 2fold axis along Y, then the inherent D1 group inside the T group is already aligned?


Another thought — Since the D1 (equivalent to C2) group, comprising a single 2-fold rotation axis, is effectively contained within the T group, you could use Volume Alignment Tools with the tetrahedral volume but specify a symmetry group of D1 to obtain a rotated output volume with the two fold axis along Y.

Makes sense! RELION even notes:

3-fold axis on Z (deviating from Heymann et al!)

I wonder why they did that.

I’m trying to examine symmetry breaking features along a 3-fold axis, and most of the software I’m used to using places it as described in my original post. By placing the 2-folds along X, Y, and Z, it lets me easily address the two different major subgroups (C3 and D2). With the 3-fold on Z, at most I can address a single 3-fold, but still need rotations to get to the other 3. With 2-folds on X, Y, and Z, I can address all the 2-fold symmetry in the particle with no rotations. That said, my PI was a crystallographer and so many of our tools are coming from that domain.

This is ultimately what I wound up doing.

I think if RELION chose that convention, while the convention doesn’t make a lot of sense to me, CryoSPARC sticking to it does.