# C3 symmetry map is not exactly symmetrical

Dear All,

Sorry for the possibly simple question, but I just noticed how the map I produced with C3 symmetry is not perfectly simmetrical, if I observe high resolution features close to the perifery of my complex in one of the 3 monomers their shape varies slightly, maybe I’m missing out on how the symmetry procedure actually works, but I would have expacted the said features to be exactly the same,

Has anybody enocuntered this and could explain to me why is that so? my map is at a fairly high resolution 2.01Å, but I’ve observed the same phenomenon on other maps on EMDB, both generated with cryosparc and relion,

thanks for you time,
Giovanni

Hi @Giovanni! This is a good question, and gets into some tricky problems that come up when working in the discrete world of computers. I think the most likely explanation here is that you’re observing interpolation artifacts.

## What really is a cryoEM map

You are likely already familiar with some of the concepts I describe here, but I will provide detail just to ensure we have some shared terminology for these concepts.

First, let’s talk about what we’re looking at when we view a map in a program like ChimeraX. The map itself is a 3D array of real numbers. Each position in the 3D array is called a voxel, and the number stored in the voxel is often informally called “density” (this is not formally correct, as electrons are scattered by Coulomb potential). There are two important things to note here:

1. This is a 4D object (three spatial dimensions, one dimension for the value in each voxel), which we of course don’t have a great way to display on a 2D screen
2. The voxels are discretely sampled — each voxel represents a cube with side length equal to your pixel size. In real life, proteins are not made of cubes!

To overcome the first problem, we can display a surface which we expect to have the same potential. When you set your threshold in a program like ChimeraX, you’re essentially displaying a surface which encloses voxels which have a value of at least the chosen threshold.

However, when we display this surface, we want it to be visible throughout the entire box, but our data only tells us about the average potential in each voxel. We therefore perform interpolation to model how the Coulomb potential varies as we move between voxels.

## Interpolation and symmetry

Interpolation is necessary, and typically not noticeable in single-particle data. However, in your case of a high-resolution (congratulations!) reconstruction of a C3 symmetric particle, it is not too surprising that the map is not perfectly symmetric. Consider a simple 2D example:

Here I’ve put a C3 symmetric object on a grid of underlying pixels. You can see that two of the three symmetric points are fully contained within a single pixel, but one is split between several. Thus, when we represent this object using the underlying pixel grid, the C3 symmetry is broken:

I’ve obviously created a worst-possible-case example here to emphasize the effect — we generally look at maps with features much greater in size than a single pixel. But hopefully this helps explain why CryoEM maps in general are not perfectly symmetric even when such symmetry is enforced, especially once you consider features at or near Nyquist.

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Hi Rich,

In this 2D example, the C3 object is not aligned with the center of the box. In reality, wouldn’t the C3 object be (at least in theory) perfectly aligned with the box center (here red is box center, blue is C3 axis)? or am I missing something?

Right – oversight on my part, I apologize (centering on bounding box != centering on symmetry). The central point still stands, and here’s an example with a centered triangle showing that the sampling would still be an issue.

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Dear @rposert and @olibclarke,

Thank you for spending your time on a very thorough answer, I really really appreciated it

All the best,
Gio

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Just as an aside, I’ve seen the same thing happen in RELION before now. With an early beta release of RELION 4, I had a C5 symmetry complex which was completely missing one of the five symmetric densities. I presume it was an unintentional bug which got fixed either accidentally or silently, as I never saw it again.