Three-dimensional reconstructions from incomplete data: interpretability of density maps at "atomic" resolution

Three dimensional data collection in electron microscopy is normally limited to a range of tilt angles that is less than +/- 90 degrees. Thus, even under the best conditions, experimental values of the structure factors will be missing within a solid cone in reciprocal space. Previous work has already shown that the missing cone of data can produce serious artifacts in three-dimensional density maps at modest resolution, for example approximately 15A. We now report, however, that a missing cone as large as +/- 30 degrees appears to have no serious effect on the three-dimensional density map of a protein at 3.6 A resolution, and we attribute this result to the fact that the electron density features are quite well separated from one another at that resolution. The map calculated with a +/- 30 degree missing cone is, furthermore, no more sensitive to noise (error) than is the full (isotropic) Fourier map. This result does not seem to be unreasonable in view of the fact that less than 14% of the data (i.e., signal) is lost due to the missing cone. Our numerical simulations therefore indicate that there should be no difficulty in interpreting high-resolution Fourier maps that can be produced with data that fall within realistic estimates of achievable resolution, tilt angles, and experimental error.