Quantitative Comparison of 3d Reconstruction Algorithms under Conditions of Uneven Angular Distribution in Electron Microscopy

The full understanding of the way proteins and other macromolecules work in living beings is highly demanded in the design of selective drugs and development of illness treatments. In this way, the macromolecular structure determination is a key problem in biology in order to understand deeper the functionality of a given complex and the way it interacts with other particles. There are several methods of collecting such structural information. One among them is the 3D reconstruction from projections obtained by means of an electron microscope. Recently, several concerns in the field have been raised about the 3D reconstruction algorithms performance when the angular distribution of the projections is highly uneven, which can be the case due to the differential macromolecular interactions with the supporting film that results in having more projections in some "preferred" directions. In this work a quantitative comparison among Weighted Back Projection (the standard reconstruction method in the field), SIRT and ART is done. At the end we will show that under this uneven distribution ART outperforms by far SIRT, and behaves slightly better than WBP. Introduction to Transmission Electron Microscopy (TEM)[1] Electron microscopy is one of the most versatile and direct techniques used to obtain information about three-dimensional structures. It can provide structural information on biological molecules at atomic resolution. X-ray crystallography and nuclear magnetic resonance (NMR) can also be used for this purpose but are restricted either to crystals with sufficiently large dimensions and good crystal order or to molecules with a maximum weight of 30k/da. The main advantage of Cryo-TEM is to preserve the biological samples in their native environment and to provide images closely related to true 2D projections of the observed particles In a transmission electron microscope the magnitude and phase of the electron beam emitted by the filament are modified as they go through the specimen being visualized. The resolution of the obtained images is limited (in an ideal microscope) by the electron wavelength and the angular aperture of the objective lens which is, under the standard working conditions) about 0.2 nm. Unfortunately, this resolution is never reached due to substantial instrumental aberrations, intrinsic limits in the specimen preparative steps, low contrast, and radiation damage of the sample. Other points to have into account are the Contrast Transfer Function of the microscope (strongly attenuating some frequencies, and even forcing a phase change in the image); the extremely low Signal to Noise Ratio of the images (in the order of 0.55); and the lack of proper control in the data collection strategy, that may lead to problems such as uneven distribution of projection directions, including substantial gaps among the experimentally achieved direction of projection. 3D reconstruction algorithms in Electron Microscopy Among all the possible 3D reconstruction from projections algorithms, the one used the most in the field of electron tomography is Weighted Back Projection [2] (WBP). Another family of algorithms try to iteratively reconstruct the specimen working all time in real space. A volume can be regarded as the sum of a set of basis functions placed on a given 3D grid and multiplied by the right coefficients. The problem of 3D reconstruction, then, can be expressed as the iterative solution of a set of equations in which the unknowns are the coefficients multiplying the basis. SIRT (Simultaneous Iterative Reconstruction Technique) and ART (Algebraic Reconstruction Technique) by blocks belong to this family. The difference between both is that ART updates the volume each time a projection is presented to the algorithm, while SIRT updates the volume each time the whole set of projections is presented. The SIRT[3] and ART[4] compared in this work are implemented using blobs[5] distributed in a Body Centered Cubic grid [4], while WBP uses voxels in a simple Cubic grid (the implementation of SIRT with blobs on a BCC grid is new and has been specially developed for allowing more proper comparisons between ART and SIRT).