Full contrast transfer function correction in 3D cryo-EM reconstruction

Over the past years electron cryo-microscopy (cryo-EM) has established itself as an important tool in studying the three dimensional structure of biological molecules up to the resolution of 6-9 Å. However, as we pursue even higher resolution (i.e., 3-4 Å), the depth-of-field problem inherent in the contrast transfer function emerges as a limiting factor. This problem has been previously addressed in the research community (Jensen, G.J., 2000; DeRosier, D.J., 2000; Zhou, Z.H. and Chiu, W., 2003; Cohen, H.A. et al., 1984). We develop a full theoretical solution to this problem. We show that the projected image from the electron microscope corresponds to neither a slice, nor an Ewald sphere, in the Fourier space, but a pair of quadratic surfaces in that space. The general solutions to this problem for both single and double defocus exposures are developed. Simulations show the correctness of the theory.