Various modes of transmission electron microscopy can provide unique structural information on nanometre‐scale biological machines such as viruses and ribosomes. A maximum‐likelihood statistical reconstruction algorithm using information from two such modalities, single‐particle cryo‐electron microscopy and computed electron tomography, is described for the problem where the machine is nearly symmetrical, but the localised regions of asymmetry are important for the functioning of the machine. The algorithm is demonstrated on experimental bacteriophage Lambda procapsid data. Key features of the algorithm are the use of probability density functions (pdfs) derived from normalised correlation rather than Gaussian pdfs and the solution of a constrained classification problem in which 11 out of 12 sites of asymmetry belong to one class while the 12th site belongs to a second class.

Many micro- to nano-scale 3-D biological objects have a helical symmetry. Cryo electron microscopy provides 2-D projection images where, however, the images have low SNR and unknown projection directions. The object is described as a helical array of identical motifs, where both the parameters of the helical symmetry and the motif are unknown. Using a detailed image formation model, a maximum-likelihood estimator for the parameters of the symmetry and the 3-D motif based on images of many objects and algorithms for computing the estimate are described. The possibility that the objects are not identical but rather come from a small set of homogeneous classes is included. The first example is based on 316 128 × 100 pixel experimental images of Tobacco Mosaic Virus, has one class, and achieves 12.40-Å spatial resolution in the reconstruction. The second example is based on 400 128 × 128 pixel synthetic images of helical objects constructed from NaK ion channel pore macromolecular complexes, has two classes differing in helical symmetry, and achieves 7.84- and 7.90-Å spatial resolution in the reconstructions for the two classes.

In cryo-electron microscopy, the data is comprised of noisy 2-D projection images of the 3-D electron scattering intensity of the object where the orientation of the projections is unknown. Often, the images show randomly selected objects from a mixture of different types of objects. Objects of different type may be unrelated, e.g., different species of virus, or related, e.g., different conformations of the same species of virus. Due to the low SNR and the 2-D nature of the data, it is challenging to determine the type of the object shown in an individual image. A statistical model and maximum likelihood estimator that computes simultaneous 3-D reconstruction and labels using an expectation maximization algorithm exists but requires extensive computation due to the numerical evaluation of 3-D or 5-D integrations of a square matrix of dimension equal to the number of degrees of freedom in the 3-D reconstruction. By exploiting the geometry of rotations in 3-D, the estimation problem can be transformed so that the inner-most numerical integral has a scalar rather than a matrix integrand. This leads to a dramatic reduction in computation, especially as the number of degrees of freedom in the 3-D reconstruction increases. Numerical examples of the 3-D reconstructions are provided based on synthetic and experimental images where the objects are small spherical viruses.

A 3-D reconstruction problem for virus particles based on cryo electron microscope images is described in which a spatially localized deviation from total-particle symmetry is present. In particular, we describe a statistical algorithm for the reconstruction of tailed bacteriophages where the symmetries of the capsid and tail are incompatible and therefore the tail is a deviation from symmetry. For reasons of computational complexity and reduced storage, we propose a two-step approach to reconstruction. Mathematical models and numerical results computed from synthetic data are presented.

The 3D reconstruction of the electron scattering intensity of a virus from cryo electron microscopy is essentially a 3D tomography problem in which the orientation of the 2D projections is unknown. Many biological problems concern mixtures of different types of virus particles or mixtures of different maturation states of the same type of virus particle. For a variety of reasons, especially low SNR, it can be very challenging to label the type or state shown in an individual image. Algorithms capable of computing multiple reconstructions, one for each type or state, based on images which are not labeled according to type or state, are described and demonstrated on experimental images.

A statistical model for the object and the complete image formation process in the cryo electron microscopy of viruses is presented. Using this model, maximum-likelihood reconstructions of the three-dimensional (3-D) structure of viruses are computed using the expectation maximization algorithm, and alternative experimental designs are evaluated based on Cramer-Rao bounds. Numerical examples of the reconstructions and experimental design evaluations are provided based on Cowpea mosaic virus.

An interpolation problem that is important in viral X-ray crystallography is considered. The problem requires new methods because (1) the function is known to have icosahedral symmetry, (2) the data is corrupted by experimental errors and therefore lacks the symmetry, (3) the problem is 3D, (4) the measurements are irregularly spaced, and (5) the number of measurements is large (10**4). A least-squares approach is taken using two sets of basis functions: the functions implied by a minimum-energy band-limited exact interpolation problem and a complete orthonormal set of band-limited functions. A numerical example on Cowpea Mosaic virus is described.