# Tutorial: Ewald Sphere Correction

How to work with Ewald Sphere Correction.

Standard cryo-EM 3D reconstruction algorithms are based on modelling particle images as *tomographic projections* of the underlying 3D structure. This assumption is fundamental to the derivation of reconstruction algorithms, has enabled structures to be solved with resolutions well below 2Å, and is an excellent approximation to the underlying physical process governing the formation of the image. However, due to the specifics of electron wave diffraction, the projection approximation breaks down at very high resolutions. The details behind this follow the geometric construction known as the *Ewald sphere.* The nature of the problem was described well by DeRosier in the context of electron microscopy [1], and since then, various algorithms have been proposed for correcting for Ewald sphere curvature in cryo-EM. Relatively recently, structures have been solved by cryo-EM to such high resolutions that the curvature of the Ewald sphere limits their resolutions [2]. As such, work surrounding the correction of Ewald sphere curvature has recently proved useful in the recovery of structures from experimental data [3, 4].

In v3.3+, CryoSPARC supports the correction of Ewald sphere curvature during refinement and CTF refinement. While the standard reconstruction of a 3D density is based upon maximizing the likelihood of the data given the image poses, the algorithm used for Ewald sphere correction is an improvement on the "simple insertion" method developed by [3]. This itself is an approximation to the maximum likelihood method, while accounting for the geometry of the Ewald sphere.

CryoSPARC supports Ewald sphere corrected reconstruction in the `Homogeneous Refinement`

and `Homogeneous Reconstruction Only`

job types. CryoSPARC also supports Ewald sphere corrected (latent variable) inference in both `Global CTF Refinement`

and `Local CTF Refinement`

jobs, as well as in `Homogeneous Refinement`

. In these jobs, Ewald sphere correction must be activated via the corresponding parameter, which is off by default. Typically, the most significant resolution gains come from Ewald sphere corrected reconstruction; we have not observed significant benefits from using Ewald sphere corrected latent inference, although this may change as cryo-EM continues to break resolution limits.

On this page we detail the main considerations with Ewald sphere correction, including typical cases where it may be fruitful to use. We also present an example workflow that would typically be done following the refinement of a high-resolution structure.

### Practical Notes

The curvature of the Ewald sphere may limit the accuracy of the highest frequency Fourier coefficients in a 3D reconstruction. Because of this, Ewald sphere correction should be done only in the final stages of processing a dataset, after an initial high-resolution structure is obtained via refinement. Considering the curvature of the Ewald sphere may help improve resolutions in datasets where any of the following hold:

The dataset refines to a high-resolution, homogeneous structure (e.g. < 3 Å)

The structure is physically large in size (e.g. a large virus capsid)

The microscope voltage is low (e.g. ≤ 200 kV)

The dependence on resolution is due to the fact that the effect of Ewald sphere curvature grows approximately quadratically with frequency, meaning only the highest resolution Fourier coefficients are significantly affected. In 2000, DeRosier presented a rule of thumb that gives an approximate frequency ($R$) at which Ewald curvature will result in significant phase errors [1]. This is given in terms of the diameter of the protein ($t$) and electron wavelength ($\lambda$):

Since the presence of Ewald sphere curvature results in the breakdown of the projection approximation, images are no longer invariant under reflections. Specifically, reflecting a raw 2D image is equivalent to *rotating* it in 3D space by 180º around the image plane, and this corresponds to *inverting* the sign of the Ewald sphere's curvature. Thus, the presence of any additional reflections in the raw data, which may arise during data collection or image processing, introduces an indeterminacy of the *sign* of the Ewald sphere curvature. In all jobs that consider the curvature of the Ewald sphere, an additional parameter controls the sign of the curvature, which can either be `positive`

or `negative`

.

The implications of this are that in practice, reconstructions of particles using both positive and negative curvatures must be done. In cases where Ewald sphere curvature is significant for your dataset, you should expect to see one curvature sign increase the resolution (FSC) of the structure, and the other to decrease the resolution of the structure. In cases where Ewald sphere curvature is insignificant, both reconstructions with differing curvatures should report the same resolution.

## Tutorial: AAV2 Dataset (EMPIAR-10202)

To illustrate a typical workflow, we will use Ewald sphere correction on the EMPIAR-10202 dataset, which is an adeno-associated virus serotype 2 variant (AAV2) [2]. This virus capsid is an ideal candidate for examining the effect of Ewald sphere curvature, due to its large size, high resolution, and icosahedral symmetry. Indeed, in the original publication, it was solved to a resolution of 1.86 Å by correcting the curvature of the Ewald sphere using the simple insertion algorithm. Here, we show that correcting for the Ewald sphere curvature, together with correcting high-order aberrations (HOA), can result in further resolution increases.

From EMPIAR-10202, we will download image set 6 "Final particle stack of AAV2-L336C using frames 5 to 19", which consists of 30,515 particles. We can use the `Import Particles`

job together with the `.star`

file that accompanies the image set, to import the particles with their CTF values and alignments.

Once the particles have finished importing, we will connect them to a `Homogeneous Reconstruction Only`

job, which will reconstruct a 3D density from the aligned particle stack. This job will allow us to isolate the effects of Ewald sphere correction during reconstruction, while keeping the particles' poses constant. First, we will reconstruct the density with all parameters as default, except using icosahedral symmetry (and without Ewald sphere correction). This initial reconstruction reached an unmasked resolution of 2.39 Å, and a masked resolution of 1.94 Å.

Taking a capsid diameter of ~250 Å and an electron wavelength of 0.0197 Å (from a microscope voltage of 300 kV), DeRosier's rule of thumb gives $1/R \approx 1.9 Å$. This indicates that beyond ~1.9 Å, curvature of the Ewald sphere will result in significant phase errors. Since our reconstruction of this dataset is approaching ~1.9 Å, it will likely be beneficial to continue on with Ewald sphere correction.

Next, we will run two more reconstructions, one with positive curvature, and one with negative curvature. We can do this by launching two more reconstruction jobs, activating the "Do EWS Correction" parameter, and changing the curvature parameter on the jobs to be either positive or negative. Note that the curvature sign depends on various factors during data collection and image processing, and it is frequently unknown what the true curvature sign is. Thus, it is always necessary to perform this step to deduce the true curvature. If Ewald sphere curvature is in fact contributing to the resolution limit, we would expect one reconstruction to produce a worse resolution than the standard reconstruction, and the other to produce a better resolution.

We will also connect the generated mask from the initial reconstruction to all subsequent reconstructions, so that all reported FSCs are calculated using the same mask. After the reconstructions are complete, we found that the reconstruction with positive curvature reached a masked resolution of 2.07 Å, and negative curvature reached a resolution of 1.89 Å. Since the 0.143 FSC threshold value and the FSC curves both appear more favourable when reconstructing with negative curvature, this indicates that the curvature sign in this dataset is negative.

Finally, in order to push the resolution as much as possible, we can repeat the above reconstructions after fitting beam tilt and other high order CTF aberrations. CryoSPARC's implementation of Ewald sphere correction will also simultaneously correct for high order CTF aberrations and anisotropic magnification, if they have been previously estimated for a given particle stack. To do this, we will launch a `Global CTF Refinement`

job and connect the particles to it, as well as the reference map generated by the negative-curvature reconstruction.

The left hand plot shows that there is quite significant beam tilt, which suggests we may be able to further improve resolution by accounting for it. To complete our experiment, we will do two final reconstructions, one with the CTF-refined particles and without Ewald curvature, and one with negative Ewald curvature. Doing both reconstructions will allow us to see the individual effects of accounting for aberrations and Ewald curvature. Below is a tree view for our full set of experiments on this dataset.

Using the five reconstruction jobs and the generated FSC files, we have summarized in the FSC comparison plot shown below.

We can see that accounting for high-order aberrations alone (red curve), or Ewald sphere curvature alone (orange curve) produces similar resolutions between 1.87 - 1.89 Å, whereas accounting for both effects bumps the resolution to 1.74 Å. Below are images taken from two of the maps, after sharpening to a B-factor of -50 and zeropadding in fourier space to a box size of 1600. One can note the slight increase in definition of the aromatic ring.

One may also be interested in re-doing a `Homogeneous Refinement`

that corrects Ewald sphere curvature, now that the sign of the curvature is known. This can be done by building the job and activating the Ewald sphere correction parameter, along with inputting the true curvature sign. On this dataset, we did not find significant resolution improvements after a second refinement.

### References

[1] D. DeRosier, "Correction of high-resolution data for curvature of the Ewald sphere", Ultramicroscopy, vol. 81, no. 2, pp. 83-98, 2000. Available: 10.1016/s0304-3991(99)00120-5

[2] Y. Tan et al., "Sub-2 Å Ewald curvature corrected structure of an AAV2 capsid variant", Nature Communications, vol. 9, no. 1, 2018. Available: 10.1038/s41467-018-06076-6

[3] M. Wolf, D. DeRosier and N. Grigorieff, "Ewald sphere correction for single-particle electron microscopy", Ultramicroscopy, vol. 106, no. 4-5, pp. 376-382, 2006. Available: 10.1016/j.ultramic.2005.11.001

[4] C. Russo and R. Henderson, "Ewald sphere correction using a single side-band image processing algorithm", Ultramicroscopy, vol. 187, pp. 26-33, 2018. Available: 10.1016/j.ultramic.2017.11.001

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