  Advances in imagery at atomic and near-atomic resolution, such as cryogenic
electron microscopy (cryo-EM), have led to an influx of high resolution images
of proteins and other macromolecular structures to data banks worldwide.
Producing a protein structure from the discrete voxel grid data of cryo-EM maps
involves interpolation into the continuous spatial domain. We present a novel
data format called the neural cryo-EM map, which is formed from a set of neural
networks that accurately parameterize cryo-EM maps and provide native,
spatially continuous data for density and gradient. As a case study of this
data format, we create graph-based interpretations of high resolution
experimental cryo-EM maps. Normalized cryo-EM map values interpolated using the
non-linear neural cryo-EM format are more accurate, consistently scoring less
than 0.01 mean absolute error, than a conventional tri-linear interpolation,
which scores up to 0.12 mean absolute error. Our graph-based interpretations of
115 experimental cryo-EM maps from 1.15 to 4.0 Angstrom resolution provide high
coverage of the underlying amino acid residue locations, while accuracy of
nodes is correlated with resolution. The nodes of graphs created from atomic
resolution maps (higher than 1.6 Angstroms) provide greater than 99% residue
coverage as well as 85% full atomic coverage with a mean of than 0.19 Angstrom
root mean squared deviation (RMSD). Other graphs have a mean 84% residue
coverage with less specificity of the nodes due to experimental noise and
differences of density context at lower resolutions. This work may be
generalized for transforming any 3D grid-based data format into non-linear,
continuous, and differentiable format for the downstream geometric deep
learning applications.
