Determination of signal-to-noise ratios and spectral SNRs in cryo-EM low-dose imaging of molecules

A Bayesian view on cryo-EM structure determination

Three-dimensional (3D) structure determination by single-particle analysis of cryo-electron microscopy (cryo-EM) images requires many parameters to be determined from extremely noisy data. This makes the method prone to overfitting, that is, when structures describe noise rather than signal, in particular near their resolution limit where noise levels are highest. Cryo-EM structures are typically filtered using ad hoc procedures to prevent overfitting, but the tuning of arbitrary parameters may lead to subjectivity in the results. I describe a Bayesian interpretation of cryo-EM structure determination, where smoothness in the reconstructed density is imposed through a Gaussian prior in the Fourier domain. The statistical framework dictates how data and prior knowledge should be combined, so that the optimal 3D linear filter is obtained without the need for arbitrariness and objective resolution estimates may be obtained. Application to experimental data indicates that the statistical approach yields more reliable structures than existing methods and is capable of detecting smaller classes in data sets that contain multiple different structures.

Surface-Constrained 3D Reconstruction in Cryo-EM

Random spherically-constrained (RSC) reconstruction is a new form of single particle reconstruction (SPR) using cryo-EM images of membrane proteins embedded in spherical lipid vesicles to generate a 3D protein structure. The method has many advantages over conventional SPR, including a more native environment for protein particles and an initial estimate of the particle's angular orientation. These advances allow us to determine structures of membrane proteins such as ion channels and derive more reliable structure estimates. We present an algorithm that relates conventional SPR to the RSC model, and generally, to projection images of particles embedded with an axis parallel to the local normal of a general 2D manifold. We illustrate the performance of this algorithm in the spherical system using synthetic data.

An adaptation of the Wiener filter suitable for analyzing images of isolated single particles

The Wiener filter is a standard means of optimizing the signal in sums of aligned, noisy images obtained by electron cryo-microscopy (cryo-EM). However, estimation of the resolution-dependent ("spectral") signal-to-noise ratio (SSNR) from the input data has remained problematic, and error reduction due to specific application of the SSNR term within a Wiener filter has not been reported. Here we describe an adjustment to the Wiener filter for optimal summation of images of isolated particles surrounded by large regions of featureless background, as is typically the case in single-particle cryo-EM applications. We show that the density within the particle area can be optimized, in the least-squares sense, by scaling the SSNR term found in the conventional Wiener filter by a factor that reflects the fraction of the image field occupied by the particle. We also give related expressions that allow the SSNR to be computed for application in this new filter, by incorporating a masking step into a Fourier Ring Correlation (FRC), a standard resolution measure. Furthermore, we show that this masked FRC estimation scheme substantially improves on the accuracy of conventional SSNR estimation methods. We demonstrate the validity of our new approach in numeric tests with simulated data corresponding to realistic cryo-EM imaging conditions. This variation of the Wiener filter and accompanying derivation should prove useful for a variety of single-particle cryo-EM applications, including 3D reconstruction.

Fully three-dimensional defocus-gradient corrected backprojection in cryoelectron microscopy

Recognizing that the microscope depth of field is a significant resolution-limiting factor in 3D cryoelectron microscopy, Jensen and Kornberg proposed a concept they called defocus-gradient corrected backprojection (DGCBP) and illustrated by computer simulations that DGCBP can effectively eliminate the depth of field limitation. They did not provide a mathematical justification for their concept. Our paper provides this, by showing (in the idealized case of noiseless data being available for all projection directions) that the reconstructions obtained based on DGCBP from data produced with distance-dependent blurring are essentially the same as what is obtained by a classical method of reconstruction of a 3D object from its line integrals. The approach is general enough to be applicable for correcting for any distance-dependent blurring during projection data collection. We present a new implementation of the DGCBP concept, one that closely follows the mathematics of its justifications, and illustrate it using mathematically described phantoms and their reconstructions from finitely many distance-dependently blurred projections.

A clustering approach to multireference alignment of single-particle projections in electron microscopy

Two-dimensional analysis of projections of single-particles acquired by an electron microscope is a useful tool to help identifying the different kinds of projections present in a dataset and their different projection directions. Such analysis is also useful to distinguish between different kinds of particles or different particle conformations. In this paper we introduce a new algorithm for performing two-dimensional multireference alignment and classification that is based on a Hierarchical clustering approach using correntropy (instead of the more traditional correlation) and a modified criterion for the definition of the clusters specially suited for cases in which the Signal-to-Noise Ratio of the differences between classes is low. We show that our algorithm offers an improved sensitivity over current methods in use for distinguishing between different projection orientations and different particle conformations. This algorithm is publicly available through the software package Xmipp.

An introduction to maximum-likelihood methods in cryo-EM

The maximum-likelihood method provides a powerful approach to many problems in cryo-electron microscopy (cryo-EM) image processing. This contribution aims to provide an accessible introduction to the underlying theory and reviews existing applications in the field. In addition, current developments to reduce computational costs and to improve the statistical description of cryo-EM images are discussed. Combined with the increasing power of modern computers and yet unexplored possibilities provided by theory, these developments are expected to turn the statistical approach into an essential image-processing tool for the electron microscopist.

Visualizing molecular machines in action: Single-particle analysis with structural variability

Many of the electron microscopy (EM) samples that are analyzed by single-particle reconstruction are flexible macromolecular assemblies that adopt multiple structural states in their functioning. Consequently, EM samples often contain a mixture of different structural states. This structural variability has long been regarded as a severe hindrance for single-particle analysis because the combination of projections from different structures into a single reconstruction may cause severe artifacts. This chapter reviews recent developments in image processing that may turn structural variability from an obstacle into an advantage. Modern algorithms now allow classifying projection images according to their underlying three-dimensional (3D) structures, so that multiple reconstructions may be obtained from a single data set. This places 3D-EM in a unique position to study the intricate dynamics of functioning molecular assemblies.

Methods for segmentation and interpretation of electron tomographic reconstructions

Electron tomography has become a powerful tool for revealing the molecular architecture of biological cells and tissues. In principle, electron tomography can provide high-resolution mapping of entire proteomes. The achievable resolution (3-8 nm) is capable of bridging the gap between live-cell imaging and atomic resolution structures. However, the relevant information is not readily accessible from the data and needs to be identified, extracted, and processed before it can be used. Because electron tomography imaging and image acquisition technologies have enjoyed major advances in the last few years and continue to increase data throughput, the need for approaches that allow automatic and objective interpretation of electron tomograms becomes more and more urgent. This chapter provides an overview of the state of the art in this field and attempts to identify the major bottlenecks that prevent approaches for interpreting electron tomography data to develop their full potential.

Automation in single-particle electron microscopy connecting the pieces

Throughout the history of single-particle electron microscopy (EM), automated technologies have seen varying degrees of emphasis and development, usually depending upon the contemporary demands of the field. We are currently faced with increasingly sophisticated devices for specimen preparation, vast increases in the size of collected data sets, comprehensive algorithms for image processing, sophisticated tools for quality assessment, and an influx of interested scientists from outside the field who might lack the skills of experienced microscopists. This situation places automated techniques in high demand. In this chapter, we provide a generic definition of and discuss some of the most important advances in automated approaches to specimen preparation, grid handling, robotic screening, microscope calibrations, data acquisition, image processing, and computational infrastructure. Each section describes the general problem and then provides examples of how that problem has been addressed through automation, highlighting available processing packages, and sometimes describing the particular approach at the National Resource for Automated Molecular Microscopy (NRAMM). We contrast the more familiar manual procedures with automated approaches, emphasizing breakthroughs as well as current limitations. Finally, we speculate on future directions and improvements in automated technologies. Our overall goal is to present automation as more than simply a tool to save time. Rather, we aim to illustrate that automation is a comprehensive and versatile strategy that can deliver biological information on an unprecedented scale beyond the scope available with classical manual approaches.

CLASSIFICATION BY BOOTSTRAPPING IN SINGLE PARTICLE METHODS

In single-particle reconstruction methods, projections of macromolecules at random orientations are collected. Often, several classes of conformations or binding states coexist in a biological sample, which requires classification, so that each conformation can be reconstructed separately. In this work, we examine bootstrap techniques for classifying the projection data. When these techniques are applied to variance estimation, the projection images (particles) are randomly sampled with replacement from the data set and a bootstrap volume is reconstructed from each sample. In a recent extension of the bootstrap technique to classification, each particle is assigned to a volume in the space spanned by the bootstrap volumes, such that the projection of the assigned volume best matches the particle. In this work we explain the rationale of these techniques by discussing the nature of the bootstrap volumes and provide some statistical analyses.

A toolbox for ab initio 3-D reconstructions in single-particle electron microscopy

Structure determination of a novel macromolecular complex via single-particle electron microscopy depends upon overcoming the challenge of establishing a reliable 3-D reconstruction using only 2-D images. There are a variety of strategies that deal with this issue, but not all of them are readily accessible and straightforward to use. We have developed a "toolbox" of ab initio reconstruction techniques that provide several options for calculating 3-D volumes in an easily managed and tightly controlled work-flow that adheres to standard conventions and formats. This toolbox is designed to streamline the reconstruction process by removing the necessity for bookkeeping, while facilitating transparent data transfer between different software packages. It currently includes procedures for calculating ab initio reconstructions via random or orthogonal tilt geometry, tomograms, and common lines, all of which have been tested using the 50S ribosomal subunit. Our goal is that the accessibility of multiple independent reconstruction algorithms via this toolbox will improve the ease with which models can be generated, and provide a means of evaluating the confidence and reliability of the final reconstructed map.

Visual proteomics of the human pathogen Leptospira interrogans

Systems biology conceptualizes biological systems as dynamic networks of interacting elements, whereby functionally important properties are thought to emerge from the structure of such networks. Owing to the ubiquitous role of complexes of interacting proteins in biological systems, their subunit composition and temporal and spatial arrangement within the cell are of particular interest. 'Visual proteomics' attempts to localize individual macromolecular complexes inside of intact cells by template matching reference structures into cryo-electron tomograms. Here we combined quantitative mass spectrometry and cryo-electron tomography to detect, count and localize specific protein complexes in the cytoplasm of the human pathogen Leptospira interrogans. We describe a scoring function for visual proteomics and assess its performance and accuracy under realistic conditions. We discuss current and general limitations of the approach, as well as expected improvements in the future.

Single-particle reconstruction of biological macromolecules in electron microscopy--30 years

This essay gives the autho's personal account on the development of concepts underlying single-particle reconstruction, a technique in electron microscopy of macromolecular assemblies with a remarkable record of achievements as of late. The ribosome proved to be an ideal testing ground for the development of specimen preparation methods, cryo-EM techniques, and algorithms, with discoveries along the way as a rich reward. Increasingly, cryo-EM and single-particle reconstruction, in combination with classification techniques, is revealing dynamic information on functional molecular machines uninhibited by molecular contacts.