Cryo-electron microscopy (cryo-EM), the subject of the 2017 Nobel Prize in Chemistry, is a technology for determining the 3-D structure of macromolecules from many noisy 2-D projections of instances of these macromolecules, whose orientations and positions are unknown. The molecular structures are not rigid objects, but flexible objects involved in dynamical processes. The different conformations are exhibited by different instances of the macromolecule observed in a cryo-EM experiment, each of which is recorded as a particle image. The range of conformations and the conformation of each particle are not known a priori; one of the great promises of cryo-EM is to map this conformation space. Remarkable progress has been made in determining rigid structures from homogeneous samples of molecules in spite of the unknown orientation of each particle image and significant progress has been made in recovering a few distinct states from mixtures of rather distinct conformations, but more complex heterogeneous samples remain a major challenge. We introduce the hyper-molecule'' framework for modeling structures across different states of heterogeneous molecules, including continuums of states. The key idea behind this framework is representing heterogeneous macromolecules as high-dimensional objects, with the additional dimensions representing the conformation space. This idea is then refined to model properties such as localized heterogeneity. In addition, we introduce an algorithmic framework for recovering such maps of heterogeneous objects from experimental data using a Bayesian formulation of the problem and Markov chain Monte Carlo (MCMC) algorithms to address the computational challenges in recovering these high dimensional hyper-molecules. We demonstrate these ideas in a prototype applied to synthetic data.
The covariance estimation approach proposed in @cite_39 does not rely on a particular model for heterogeneity, be it discrete or continuous. Indeed, the authors present a method for characterizing continuous variability in synthetic data. However, the covariance approach is adapted to a linear model of variability and is therefore not well-suited for continuous, and necessarily non-linear, variability. Furthermore, the limited resolution of the reconstruction precludes the study of heterogeneity at higher level of detail. Another approach has been to study the normal modes of perturbation of a macromolecular structure @cite_45 @cite_13 .
@cite_45 @cite_13 @cite_39
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A theory of elastic normal modes is described for the exploration of global distortions of biological structures and their assemblies based upon low-resolution image data. Structural information at low resolution, e.g. from density maps measured by cryogenic electron microscopy (cryo-EM), is used to construct discrete multi-resolution models for the electron density using the techniques of vector quantization. The elastic normal modes computed based on these discretized low-resolution models are found to compare well with the normal modes obtained at atomic resolution. The quality of the normal modes describing global displacements of the molecular system is found to depend on the resolution of the synthetic EM data and the extent of reductionism in the discretized representation. However, models that reproduce the functional rearrangements of our test set of molecules are achieved for realistic values of experimental resolution. Thus large conformational changes as occur during the functioning of biological macromolecules and assemblies can be elucidated directly from low-resolution structural data through the application of elastic normal mode theory and vector quantization. This article presents a method to study large-scale conformational changes by combining electron microscopy (EM) single-particle image analysis and normal mode analysis (NMA). It is referred to as HEMNMA, which stands for hybrid electron microscopy normal mode analysis. NMA of a reference structure (atomic-resolution structure or EM volume) is used to predict possible motions that are then confronted with EM images within an automatic iterative elastic 3D-to-2D alignment procedure to identify actual motions in the imaged samples. HEMNMA can be used to extensively analyze the conformational changes and may be used in combination with classic discrete procedures. The identified conformations allow modeling of deformation pathways compatible with the experimental data. HEMNMA was tested with synthetic and experimental data sets of E. coli 70S ribosome, DNA polymerase Pol a and B subunit complex of the eukaryotic primosome, and tomato bushy stunt virus. In cryo-electron microscopy, the three-dimensional (3D) electric potentials of an ensemble of molecules are projected along arbitrary viewing directions to yield noisy two-dimensional images. The volume maps representing these potentials typically exhibit a great deal of structural variability, which is described by their 3D covariance matrix. Typically, this covariance matrix is approximately low rank and can be used to cluster the volumes or estimate the intrinsic geometry of the conformation space. We formulate the estimation of this covariance matrix as a linear inverse problem, yielding a consistent least-squares estimator. For @math images of size @math -by- @math pixels, we propose an algorithm for calculating this covariance estimator with computational complexity @math , where the condition number @math is empirically in the range 10--200. Its efficiency relies on the observation that the normal equations are equivalent to a deconvolution problem in six dimensions....