Electron crystallography on multilayered three-dimensional microcrystals has been limited in application by the need to de®ne precisely the three-dimensional shape of the diffraction density pro®les. A new method is presented here to obtain this pro®le from experimental spot positions which are shifted in a characteristic way from the expected Bragg positions. While the Bragg positions are de®ned by the diffraction geometry, the characteristic shift additionally depends on the density pro®le in Fourier space. In general, these two effects are intermingled. A new correlation approach is presented which uses characteristic shift patterns to separate these effects. This technique also allows the determination of all three crystallographic unit-cell dimensions from a single tilted electron diffraction pattern. It was tested on simulated diffraction patterns and applied to experimental data of frozen hydrated crystals of the protein catalase. Since multilayered catalase crystals with different numbers of crystallographic layers were studied, an inhomogeneous data set had to be evaluated. Processing of such data is now possible using the new correlation approach.

Electron crystallography of biological samples has been used for reconstruction of several proteins [1]. The high scattering cross section of the electrons as opposed to photons in X-ray crystallography allows the use of smaller crystals. However, this high cross section also leads to dynamical scattering effects, which increase with the thickness of the crystal. Due to this limitation and the difficulty associated with interpreting data from thin 3D crystals, electron crystallography has so far been limited to 2D mono-layered crystals. If multi-layered crystals are to be used for structure determination, dynamical scattering must be understood. The effect of dynamical scattering on the usable resolution of the final reconstruction depends on the stacking of the protein in the crystal, the tilt angles selected, and on the electron energy used. It has been shown previously [2] that favourable stacking and carefully selected tilt angles can lead to reconstructions of samples up to 95 nm. It is shown here that symmetrization of the diffraction data serves as a first order correction of dynamical scattering. The multi-slice simulation software YaMS [3] was used to simulate diffraction patterns and images of 12 different proteins utilizing the atomic coordinates found in the PDB data base. Using the amplitude data from the diffraction pattern and the phase data from the images, reconstructions of the proteins were calculated. This was done for a mono-layered and for multi-layered samples consisting of several layers. To obtain a non-complex reconstruction, diffraction pattern amplitudes were symmetrized by averaging each Friedel pair. To illustrate the influence of dynamical scattering, the scattering data and the final reconstruction of one specific protein (bacteriorhodopsin) was analysed in detail. Assuming that data collected from a mono-layer sample is quasi-kinematical, a comparison with the data from 7 layers shows dynamical scattering effects. Figure 2 shows phase data for 1 and 7 layer simulations. Despite an overall similarity of the patterns, a detailed analysis shows considerable deviation (see Table 1). Note also that the magnitude of the deviation depends strongly on resolution. To assess the effect of dynamical scattering on the reconstruction, Fourier ring correlation coefficients between the 1 and 7 layer reconstructions were calculated and averaged for the 12 proteins. Figure 3 (solid lines) shows the Fourier ring correlation of two reconstructions, one at 120keV and 200keV, respectively. Electrons at 200keV have a longer mean free path compared to lower energies and thus are less inclined to scatter dynamically. The correlation is thus slightly better than in the 120keV case. Symmetrization of the diffraction pattern serves as a first order correction of the amplitude data. For the phase data the imaging process itself serves as a correction mechanism. The phase data taken from the image is anti-symmetrical since the simulated image is real. The dashed lines in figure 3 show reconstruction data from...