Accounting for the Ewald Spheres in CryoEM Reconstructions and Their Relationship to 3D Fourier Transforms of Focal Series

In cryoEM, the goal is to achieve as high a resolution as possible. One of the limitations is the widespread use of the projection approximation, which ignores the spherical geometry of electron scattering and focusing. Scattered electrons form a coherent wave front, the Ewald sphere, while focusing by the objective lens constitutes a second, conjugate wave front. These two Ewald spheres are convolved to form the coherent signal in the recorded image and includes the focus gradient through the specimen [1]. Reconstructing according to the projection approximation results in a map with limited detail (Figure 1A), while taking the Ewald spheres into account extends the resolution considerably (Figure 1B). The projection approximation equates the Fourier transform of a micrograph to the central section in 3D frequency space (Figure 1C), so that reconstruction involves the integration of particle transforms as central sections into the 3D frequency space volume. Instead, the correct method is to integrate along the Ewald spheres (Figure 1D). The corresponding Fourier shell correlation (Figure 1E) indicates that the Ewald sphere integration can recover the phases to Nyquist frequency, while central section integration limits the resolution to the first phase reversal. One proposal to better recover the phases from the Ewald sphere for higher resolution reconstructions was to analyze a focal series of micrographs [2]. The 3D Fourier transform of such a stack of micrographs gives curves that resemble the Ewald spheres [3]. We acquired focal series of micrographs of carbon film and graphene oxide, stack each series in a 3D volume and Fourier transform the volume to obtain the 3D power spectrum. The 3D power spectrum from the carbon series shows two curves that represent the coherent wave fronts resulting from the change in focus (Figure 2A). The 3D power spectrum of a focal series of graphene oxide shows the characteristic 2.13 Å reflections (Figure 2B). The pair of reflections are offset from the central section by 0.0027 Å-1 (inset in Figure 2B), consistent with the Ewald sphere offset. However, the reflections are separated with a minimum on the central section, where the 3D power spectrum of the graphene lattice itself has a maximum. This illustrates that the 3D power spectrum of the focal series is different from true Ewald spheres. The reason why it appears the same is that the geometry of electron scattering and defocusing is the same. Unfortunately, this means that the phase information associated with the individual Ewald spheres are already averaged in the images and it cannot be recovered (see [1] for a comprehensive explanation) [4]. (A) Simulated reconstructions of an adeno-associated virus capsid (∼290 Å diameter) assuming the projection approximation and (B) integrating along the Ewald spheres. (C) Illustration of a central section and (D) Ewald spheres within frequency space reconstruction boxes. (E) Fourier shell correlation of simulated reconstructions compared to the original map, showing recovery of phases with integration along the Ewald spheres, while central section integration according to the projection approximation leads to phase inversions [1]. (A) Orthogonal views of the 3D power spectrum of a stacked focal series of micrographs of carbon, showing spheres that mimic Ewald spheres. (B) Orthogonal views of the 3D power spectrum of stacked focal series of micrographs of graphene oxide. The inset shows orthogonal views of a reflection at 2.13 Å with peaks on the apparent Ewald spheres. Micrographs were acquired on a JEOL CryoARM200 at 200 kV with a Gatan K3 camera with a pixel size of 0.16 Å, and a total dose of 160 e/Å2 distributed over 113 frames over 6 s for each of 20 micrograph movies with a focal step size of 37 Å.