[3dem] Symmetry Matrix Differences Between Cryo-EM Software

Thu Feb 4 11:41:01 PST 2021

Hi,

    Just to add a bit of historical information, for anyone who cares.  The first icosahedral virus structures were solved using Fourier Bessel methods (https://urldefense.com/v3/__https://www.jstor.org/stable/2417109?seq=1*metadata_info_tab_contents__;Iw!!Mih3wA!SS7Oh5yTP6zsOJ4J0OuAu2OdNgmq1DfYkZJg5Xg4bhbr7mDtvZDJtjuCW6JVB7xpag$ ).  

    In order to use those methods, the reconstructed object needed to be oriented in a particular orientation:  the 5-fold symmetry axis needed to be along the Cartesian Z-axis and the 2-fold needed to be along the Y-axis (at least I think I remember that it needed to be along Y;  it's possible that it needed to be along X).  The important thing is that by orienting this 2-fold symmetry axis along a particular Cartesian axis, the real and imaginary parts of the Fourier Bessel terms separated into independent equations, which were far easier to solve.  This story is even a bit more complicated than I am describing, but I'll end this bit of history here.

    For non-Fourier Bessel reconstruction methods, this particular orientation of the 5- and 2-fold axes along particular Cartesian axes didn't matter, and the X-ray community adopted another orientation for icosahedral objects where there were 2-fold axes along the Cartesian X, Y and Z axes.  I have never talked with an X-ray person about this, but I have always assumed that since crystals can't exhibit 5-fold symmetry, they didn't want their "standard icosahedral orientation" to assign anything special to the orientation of the 5-fold axis.

    This EM/X-ray difference is also related to and highlights that icosahedral symmetry (point group symmetry 532) includes but is higher symmetry than point group 222.  The X-ray orientation of an icosahedral object puts it into an orientation where the 222 symmetry is easy to enforce, and the 5- and 3-fold symmetry "falls out" (and can be enforced using non-crystallographic symmetry).

    When new EM reconstruction methods became available, additional and different orientations were used by different software suites and so there needed to be ways to convert from one orientation to the other.  In those early days, it was more critical to know what orientation your data had (i.e., which icosahedral convention your data had), and what orientation you wanted to go to (and I think people understood some of these details better).  

    And just to end being a bit pedantic, remember that a symmetric object is defined by the presence and orientation of internal symmetry axes relative to each other, but that the symmetric object itself can adopt any Cartesian orientation.  That's why single particle EM works so well with symmetric objects :-) and also why different software can produce "identical" reconstructions that are not in the same orientation.

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            David Gene Morgan
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________________________________________
From: 3dem <3dem-bounces at ncmir.ucsd.edu> on behalf of Mike Strauss <mike.strauss at mcgill.ca>
Sent: Thursday, February 4, 2021 8:29 AM
To: Rodarte, Justas V
Cc: 3dem at ncmir.ucsd.edu
Subject: [External] Re: [3dem] Symmetry Matrix Differences Between Cryo-EM Software

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Hi Justas,

just to add to what Matthias said: the different conventions have different historical backgrounds.  My default in Relion - I1, puts the 2-fold axes centered on the faces of the cube, and the 5-fold closest to the z-axis on the x=0 plane (assuming the origin is in the center). I like it because many of the virus structures in my field adopt this convention.  I2 puts this 5-fold on the y-0 plane.  I think these two were used because symmetrisation was easier for memory-limited calculations in the early days - just reorder the pixels and add, no interpolation necessary for the 2-folds.

Another convention uses the 5-fold axis pointing along the z-axis, which follows the convention of the highest symmetry along this axis (like D7 symmetry).

To be clear - they all produce the same structure - just rotated versions of one another - and they are all “correct” conventions, albeit not necessarily common.  If you’re getting crazy structures, it may be because the convention of your reference doesn’t match the convention of your output.

Regards,
mike

--------
Prof. Mike Strauss
Anatomy and Cell Biology / FEMR
McGill University
(514) 398-8084
mike.strauss at mcgill.ca

> On 4. Feb 2021, at 01:17, Matthias Wolf <matthias.wolf at oist.jp> wrote:
>
> Hi Justas,
>
> As far as I know, there is no absolute convention – the two possible icososahedral symmetries are called “I” and “I2”.
> One of them describes an icosahedron where you can “walk” between 5-fold axes thru 3- and 2- fold (5-3-2-5), the other thru 2- then 3-fold (5-2-3-5). Which is referred to as “I” and which as “I2” is, in principle, irrelevant. The two icosahedra are different structures, they are not mirrors of one hand.
>
> So just use one of the two option that works in cryosparc. I guess it would be I2…
>
> Best,
>    Matthias
>
> From: 3dem On Behalf Of Rodarte, Justas V
> Sent: Thursday, February 4, 2021 7:53 AM
> To: 3dem at ncmir.ucsd.edu
> Subject: [3dem] Symmetry Matrix Differences Between Cryo-EM Software
>
> Hi all,
>
> I’m a relative newcomer to the field of Cryo-EM, so my apologies if this question is easy/misplaced/previously answered. I’m working on a VLP with I symmetry that displays 80 spikes. Some programs (Relion, cisTEM) apply symmetry correctly when moving onto 3D ab-initio, while others (Cryosparc), don’t. I was able to trace this to the fact that these programs use different matrices to apply symmetry, an example is included below. Does anyone know why this is though? To me it appears to be an oversight that two popular programs apply the same symmetry differently?
>
> Thank you all for any help or advice you can provide!
>
> Best,
> Justas Rodarte
>
>
> I1 Sym Matrix Relion
> rot_axis 2, 1 0 0
> rot_axis 5, 0.85065080702670 0 -0.5257311142635
> rot_axis 3, 0.9341723640 0.3568220765 0
>
> I1 Sym Matrix Cryosparc
> rot_axis 2, 0 1 0
> rot_axis 5, -0.85065080702670 0 0.5257311142635
> rot_axis 3, -0.9341723640 0.3568220765
>
>
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