Model-based statistical inference problems concerning non-linear three-dimensional tomography with applications to the structural biology of asymmetric virus particles

Electron microscopy provides noisy images of viruses that, quantitatively, are projections of the 3-D virus structure. The ensuing tomographic reconstruction problem is nonlinear since the orientation of each particle is unknown and only one image can be measured for each particle and hence, the measured image is related to an unknown 2-D projection of the structure. Statistical models can be formulated for both measurement noise and orientation uncertainty. Inference problems on parameters describing the 3-D virus structure can then be investigated by fusing data from single images of many identical particles. High-order symmetry of the particle, especially icosahedral symmetry, has been helpful in previous computations of such structures. In this work we consider reconstruction methods that are not based on total-particle symmetry. First we consider reconstruction of a spatially localized deviation from symmetry. We describe mathematical models and a two-stage algorithm for 3-D reconstruction of a tailed bacteriophage where the symmetries of the capsid and tail are incompatible. Numerical results from experimental images are presented for bacteriophage P22. In the second phase of this work we present statistical algorithms in which an initial symmetrical reconstruction undergoes relaxation of the symmetry constraint. Both algorithms and Cramer-Rao Bounds on performance of algorithms are presented. We believe such algorithms would be most useful in computing reconstructions of viruses having a symmetrical capsid and an asymmetrical nucleic acid core. A primary focus of this work is development of statistical tools for experimental design