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CryoEM data capture the dynamic character associated with biological macromolecular assemblies by preserving the various conformations of the individual specimens at the moment of flash freezing. Regions of high variation in the data set are apparent in the image reconstruction due to the poor density that results from the lack of superposition of these regions. These observations are qualitative and, to date, only preliminary efforts have been made to quantitate the heterogeneity in the ensemble of particles that are individually imaged. We developed and tested a quantitative method for simultaneously computing a reconstruction of the particle and a map of the space-varying heterogeneity of the particle based on an entire data set. The method uses a maximum likelihood algorithm that explicitly takes into account the continuous variability from one instance to another instance of the particle. The result describes the heterogeneity of the particle as a variance to be plotted at every voxel of the reconstructed density. The test, employing time resolved data sets of virus maturation, not only recapitulated local variations obtained with difference map analysis, but revealed a remarkable time dependent reduction in the overall particle dynamics that was unobservable with classical methods of analysis.

Different instances of a biological macromolecular complex need not share the same 3-D electron scattering distribution function due to stoichiometric variability, flexibility and vibrations, etc. Cryo electron microscopy provides 2-D images of each of many such complexes where each image is roughly a projection. A statistical estimation problem is described and demonstrated for determining a statistical description of the complex from such imagery.

A statistical estimation problem for determining 3-D reconstructions from a single 2-D projection image of each of multiple objects when the objects are heterogeneous is described. The method is based on a Gaussian mixture description of the heterogeneity and is motivated by cryo electron microscopy of biological objects.

Cryo electron microscopy is frequently used on biological specimens that show a mixture of different types of object. Because the electron beam rapidly destroys the specimen, the beam current is minimized which leads to noisy images (SNR substantially less than 1) and only one projection image per object (with an unknown projection direction) is collected. For situations where the objects can reasonably be described as coming from a finite set of classes, an approach based on joint maximum likelihood estimation of the reconstruction of each class and then use of the reconstructions to label the class of each image is described and demonstrated on two challenging problems: an assembly mutant of Cowpea Chlorotic Mottle Virus and portals of the bacteriophage P22.

Two algorithms are described for electron microscopy tasks: (1) 3-D reconstruction of an infectious bacteriophage, including the tail, based on a set of single-particle cryo electron microscopy images of identical phage particles with one image per particle and (2) restoration, resolution enhancement, and classification of spherical viruses in a cell based on a tomographic reconstruction of the entire cell. While different tasks, the statistical algorithms, which are maximum likelihood estimators, have extensive commonality. In order to provide solutions for such tasks in a timely fashion, a software toolkit has been created, based on the commonality, for writing parallel software systems to solve these problems.

Statistical algorithms are described for 2- and 3-D signal reconstruction and classification problems based on electron microscopy images where the objects displayed in the images are heterogeneous. Both discrete heterogeneity (there exist classes of objects but all objects within one class are identical) and continuous heterogeneity (there exist classes of objects but, in addition, the objects within a class differ). With continuous heterogeneity, the idea of reconstruction is generalized from reconstruction of a unique 2- or 3-D object to reconstruction of the statistics of the 2- or 3-D object. The basic process is reconstruction by maximum likelihood estimation computed by an expectation-maximization algorithm with classification following as postprocessing given the reconstruction. Two examples are considered, one focusing on classification and one on reconstruction.

Maximum likelihood statistical algorithms are described for estimating the 3-D variation of the electron scattering intensity of biological objects from cryo electron microscopy images of multiple instances of the object. Three virus objects, two spherical and one helical, are considered. Solution of the maximum likelihood problem by expectation maximization algorithms or by direct maximization of the log likelihood requires large scale computing and end-to-end codesign of biological problem formulation, statistical models, algorithms, and software design and implementation have contributed to achieving practical results.

A statistical approach for computing 3-D reconstructions of virus particles from cryo electron microscope images and minimal prior information has been developed which can solve a range of specific problems. The statistical approach causes high computation and storage complexity in the software implementation. A parallel software toolkit is described which allows the construction of software targeted at commodity PC clusters which is modular, reusable, and user-transparent while at the same time delivering nearly linear speedup on practical problems.

A statistical model for cryo electron microscopy image formation, a maximum likelihood 3-D reconstruction algorithm, and a parallel computer implementation of the algorithm are described. The essential character of this reconstruction problem is that it concerns 2-D projections of a 3-D object where the projection orientations are unknown and the 2-D projections have SNR less than 1. A key component of a variety of algorithms is integration over projection orientation. A fast algorithm is described for performing such integrations.

An algorithm for the simultaneous 3-D reconstruction of several types of object, where each type of object may possibly have a rotational symmetry, from 2-D projection images, where for each image the type of object imaged, the projection orientation used to create the image, and the location of the object in the image are unknown, is described. The motivating application is the determination of the 3-D structure of small spherical viruses from cryo electron microscopy images. The algorithm is a maximum likelihood estimator which is computed by expectation maximization (EM). Due to the structure of the statistical model, the maximization step of EM can be easily computed but the expectation step requires 5-D numerical quadrature. The computational burden of the quadratures necessitates parallel computation and three different implementations of two different types of parallelism have been developed using pthreads (for shared memory processors) and MPI (for distributed memory processors). An example applying one of the MPI implementations, running on a 32 node PC cluster, to experimental images of Flock House Virus with comparison to the x-ray crystal diffraction structure of the virus is described.

A statistical method for computing 3-D reconstructions of virus particles from cryo electron microscope images and minimal prior information - particle symmetry and radii - is described. Two different approaches for parallel implementation of the algorithm are described based on parallel processing of different images and based on parallel numerical integration and numerical results showing nearly linear speedup are presented.

A statistical model for cryo electron microscope images is described. 3-D reconstruction and experimental design methods based on the model are demonstrated. Advantages of this approach to 3-D reconstruction include that minimal prior information, only particle symmetry and radii, is required.