Measurement of charges and chemical bonding in a cryo-EM structure

Hydrogen bonding, bond polarity, and charges in protein molecules play critical roles in the stabilization of protein structures, as well as affecting their functions such as enzymatic catalysis, electron transfer, and ligand binding. These effects can potentially be measured in Coulomb potentials using cryogenic electron microscopy (cryo-EM). We here present charges and bond properties of hydrogen in a sub-1.2 Å resolution structure of a protein complex, apoferritin, by single-particle cryo-EM. A weighted difference map reveals positive densities for most hydrogen atoms in the core region of the complex, while negative densities around acidic amino-acid side chains are likely related to negative charges. The former positive densities identify the amino- and oxo-termini of asparagine and glutamine side chains. The latter observations were verified by spatial-resolution selection and a dose-dependent frame series. The average position of the hydrogen densities depends on the parent bonded-atom type, and this is validated by the estimated level of the standard uncertainties in the bond lengths.


Minimization of beam tilts off the axial coma-free axis
Large fluctuations in beam tilt occurred in our previous data collections using a CFE gun equipped in a CRYO ARM 300 microscope (JEOL), even though such images were collected exclusively through the use of mechanical stage shifts 1 -5 . The data gave reconstructions to ~1.5 Å resolution 1,3 . Beam tilts from the axial coma-free axis caused a marked increase in phase errors in proportion to the third power of spatial frequencies. Although it is possible to estimate the beam tilts in acquired images and correct errors 6 -8 , such a post-correction scheme has its limitations particularly for larger beam-tilt errors at higher resolutions (see below).
The fluctuations sometimes range over several mrad [1][2][3][4][5] , and this is likely due to offpivot points of the beam on the specimen plane. Adjustment of the pivot points needs accurate compensation for beam tilts and shifts by means of condenser lens deflector coils. We had used the diffraction mode for compensation of beam shifts, as per instructions in the manual   10 .
We then adjusted compensation of beam shifts in imaging mode as done for the data acquisition condition (see also Methods and Efremov & Stroobants, 2021 11  Averaging and symmetrization are shown to reduce phase errors caused by beam tilts off the axial coma-free axis to some extent even without post-correction of phase shift by beam tilts 12,13 . The post-correction can further improve the resolution 6 . The Rosenthal-Henderson plots 14 for Datasets A and B appear similar up to ~ 1.7 Å ( Supplementary Fig. 3b), with the most prominent difference being the beam tilt fluctuations ( Supplementary Fig. 2). Thus, the present data would suggest that the post-correction is probably effective to this resolution range but beyond this range becomes less so with such large beam tilt fluctuations.
The catalogue value of the spherical aberration coefficient, Cs = 2.7 mm, means that even 0.1 mrad off-estimation of beam tilt introduces significant phase errors in high-resolution ranges ( Supplementary Fig. 2d). Errors are much worse with an accelerating voltage of 200 kV compared with 300 kV, and a new Cs corrector was introduced to remove phase errors by both on-axial and off-axial coma 15 . Nevertheless, our data indicate that the resolution can be extended to sub-1.2 Å without this device. The Cs corrector and the monochromator would be impractical for many users due to instability, additional complexities in electron optics and operation, and high price.

Estimation of the standard uncertainties in bond lengths
In macromolecular X-ray crystallography, diffraction-component precision index (DPI) 16 is a standard metric for estimation of coordinate errors in the atomic model. According to Cruickshank, the root mean square differences (RMSD) in atomic positions <Δr> between two structures solved independently are in good agreement with the estimated standard uncertainties of positions σ(r) 16,17 . For unrestrained refinement of crystal structures of small molecules, the standard uncertainty of a bond length between two atoms with similar coordinate errors is approximated by multiplying the coordinate error, i.e. DPI or σ(r), by the square root of 2 16,17 .
DPI relies on accuracy of diffraction intensity, but again CTF changes amplitude in EM images, yielding less accurate measures for amplitude. Still, REFMAC5 18 can provide DPI during refinement of a model against a single-particle reconstruction, but we think a more suitable criterion is needed here. We calculated RMSD between two models obtained by restrained MD refinement against two half maps reconstructed independently (see Methods) to estimate the standard uncertainty in atomic positions in the model. The half maps are shown to contain information corresponding to Fobs and σ(Fobs) of X-ray crystallography 19 -21 . We then plotted calculated values of DPI for various X-ray structures and those of RMSD1/2 named here for cryo-EM reconstructions in a given resolution range (Supplementary Fig. 8; Supplementary Table 2). The graphs show consistency between them. Thus, RMSD1/2 could be used for estimation of coordinate errors in single-particle cryo-EM models. RMSD1/2 for the apoferritin reconstruction in this study yields ~ 0.05 Å and this becomes ~ 0.05 × √2 = ~ 0.07 Å for the standard uncertainty between two atoms. Considering that RMSD1/2 is calculated based on half maps and might be underestimated, differences between the average lengths of N-H and C-Hn bonds are 1 -2 σ level (Table 1).  23 . ‡ Defined in this study.  All the residues except for Tyr 34 are located on the protein surface exposed to solvent. The density levels of blue and purple nets are 2σ and 7σ, respectively.