Unified 3-D structure and projection orientation refinement using quasi-Newton algorithm

A joint alignment and reconstruction algorithm for electron tomography to visualize in-depth cell-to-cell interactions

Electron tomography allows one to obtain 3D reconstructions visualizing a tissue's ultrastructure from a series of 2D projection images. An inherent problem with this imaging technique is that its projection images contain unwanted shifts, which must be corrected for to achieve reliable reconstructions. Commonly, the projection images are aligned with each other by means of fiducial markers prior to the reconstruction procedure. In this work, we propose a joint alignment and reconstruction algorithm that iteratively solves for both the unknown reconstruction and the unintentional shift and does not require any fiducial markers. We evaluate the approach first on synthetic phantom data where the focus is not only on the reconstruction quality but more importantly on the shift correction. Subsequently, we apply the algorithm to healthy C57BL/6J mice and then compare it with non-obese diabetic (NOD) mice, with the aim of visualizing the attack of immune cells on pancreatic beta cells within type 1 diabetic mice at a more profound level through 3D analysis. We empirically demonstrate that the proposed algorithm is able to compute the shift with a remaining error at only the sub-pixel level and yields high-quality reconstructions for the limited-angle inverse problem. By decreasing labour and material costs, the algorithm facilitates further research directed towards investigating the immune system's attacks in pancreata of NOD mice for numerous samples at different stages of type 1 diabetes.

Joint iterative reconstruction and 3D rigid alignment for X-ray tomography

X-ray tomography is widely used for three-dimensional structure determination in many areas of science, from the millimeter to the nanometer scale. The resolution and quality of the 3D reconstruction is limited by the availability of alignment parameters that correct for the mechanical shifts of the sample or sample stage for the images that constitute a scan. In this paper we describe an algorithm for marker-free, fully automated and accurately aligned and reconstructed X-ray tomography data. Our approach solves the tomographic reconstruction jointly with projection data alignment based on a rigid-body deformation model. We demonstrate the robustness of our method on both synthetic phantom and experimental data and show that our method is highly efficient in recovering relatively large alignment errors without prior knowledge of a low resolution approximation of the 3D structure or a reasonable estimate of alignment parameters.

Joint Iterative Fast Projection Matching for Fully Automatic Marker-free Alignment of Nano-tomography Reconstructions

Highly accurate, fully automatic marker-free image alignment plays an important role in nano-tomographic reconstruction, particularly in cases where the spatial resolution of the tomographic system is on the nanometer scale. However, highly accurate marker-free methods such as the projection matching method are computationally complex and time-consuming. Achieving alignment accuracy with reduced computational complexity remains a challenging problem. In this study, we propose an efficient method to achieve marker-free fully automatic alignment. Our method implements three main alignment procedures. First, the frequency-domain common line alignment method is used to correct the in-plane rotational errors of each projection. Second, real-space common line alignment method is used to correct the vertical errors of the projections. Finally, a single layer joint-iterative reconstruction and re-projection method is used to correct the horizontal projection errors. This combined alignment approach significantly reduces the computational complexity of the classical projection matching method, and increases the rate of convergence towards determining the accurate alignment. The total processing time can be reduced by up to 4 orders of magnitude as compared to the classical projection matching method. This suggests that the algorithm can be used to process image alignment of nano-tomographic reconstructions on a conventional personal computer in a reasonable time-frame.

Rapid near-atomic resolution single-particle 3D reconstruction with SIMPLE

Cryogenic electron microscopy (cryo-EM) and single-particle analysis enables determination of near-atomic resolution structures of biological molecules. However, large computational requirements limit throughput and rapid testing of new image processing tools. We developed PRIME, an algorithm part of the SIMPLE software suite, for determination of the relative 3D orientations of single-particle projection images. PRIME has primarily found use for generation of an initial ab initio 3D reconstruction. Here we show that the strategy behind PRIME, iterative estimation of per-particle orientation distributions with stochastic hill climbing, provides a competitive approach to near-atomic resolution single-particle 3D reconstruction. A number of mathematical techniques for accelerating the convergence rate are introduced, leading to a speedup of nearly two orders of magnitude. We benchmarked our developments on numerous publicly available data sets and conclude that near-atomic resolution ab initio 3D reconstructions can be obtained with SIMPLE in a matter of hours, using standard over-the-counter CPU workstations.

Automated angular and translational tomographic alignment and application to phase-contrast imaging

X-ray computerized tomography (CT) is a 3D imaging technique that makes use of x-ray illumination and image reconstruction techniques to reproduce the internal cross-sections of a sample. Tomographic projection data usually require an initial relative alignment or knowledge of the exact object position and orientation with respect to the detector. As tomographic imaging reaches increasingly better resolution, thermal drifts, mechanical instabilities, and equipment limitations are becoming the main dominant factors contributing to sample positioning uncertainties that will further introduce reconstruction artifacts and limit the attained resolution in the final tomographic reconstruction. Alignment algorithms that require manual interaction impede data analysis with ever-increasing data acquisition rates, supplied by more brilliant sources. We present in this paper an iterative reconstruction algorithm for wrapped phase projection data and an alignment algorithm that automatically takes 5 degrees of freedom, including the possible linear and angular motion errors, into consideration. The presented concepts are applied to simulated and real measured phase-contrast data, exhibiting a possible improvement in the reconstruction resolution. A MATLAB implementation is made publicly available and will allow robust analysis of large volumes of phase-contrast tomography data.

Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection

As x-ray and electron tomography is pushed further into the nanoscale, the limitations of rotation stages become more apparent, leading to challenges in the alignment of the acquired projection images. Here we present an approach for rapid post-acquisition alignment of these projections to obtain high quality three-dimensional images. Our approach is based on a joint estimation of alignment errors, and the object, using an iterative refinement procedure. With simulated data where we know the alignment error of each projection image, our approach shows a residual alignment error that is a factor of a thousand smaller, and it reaches the same error level in the reconstructed image in less than half the number of iterations. We then show its application to experimental data in x-ray and electron nanotomography.

Fast Projection Matching for X-ray Tomography

X-ray 3D tomographic techniques are powerful tools for investigating the morphology and internal structures of specimens. A common strategy for obtaining 3D tomography is to capture a series of 2D projections from different X-ray illumination angles of specimens mounted on a finely calibrated rotational stage. However, the reconstruction quality of 3D tomography relies on the precision and stability of the rotational stage, i.e. the accurate alignment of the 2D projections in the correct three-dimensional positions. This is a crucial problem for nano-tomographic techniques due to the non-negligible mechanical imperfection of the rotational stages at the nanometer level which significantly degrades the spatial resolution of reconstructed 3-D tomography. Even when using an X-ray micro-CT with a highly stabilized rotational stage, thermal effects caused by the CT system are not negligible and may cause sample drift. Here, we propose a markerless image auto-alignment algorithm based on an iterative method. This algorithm reduces the traditional projection matching method into two simplified matching problems and it is much faster and more reliable than traditional methods. This algorithm can greatly decrease hardware requirements for both nano-tomography and data processing and can be easily applied to other tomographic techniques, such as X-ray micro-CT and electron tomography.

Mesoscale imaging with cryo-light and X-rays: Larger than molecular machines, smaller than a cell

In the context of cell biology, the term mesoscale describes length scales ranging from that of an individual cell, down to the size of the molecular machines. In this spatial regime, small building blocks self-organise to form large, functional structures. A comprehensive set of rules governing mesoscale self-organisation has not been established, making the prediction of many cell behaviours difficult, if not impossible. Our knowledge of mesoscale biology comes from experimental data, in particular, imaging. Here, we explore the application of soft X-ray tomography (SXT) to imaging the mesoscale, and describe the structural insights this technology can generate. We also discuss how SXT imaging is complemented by the addition of correlative fluorescence data measured from the same cell. This combination of two discrete imaging modalities produces a 3D view of the cell that blends high-resolution structural information with precise molecular localisation data.

Assembly of macromolecular complexes by satisfaction of spatial restraints from electron microscopy images

To obtain a structural model of a macromolecular assembly by single-particle EM, a large number of particle images need to be collected, aligned, clustered, averaged, and finally assembled via reconstruction into a 3D density map. This process is limited by the number and quality of the particle images, the accuracy of the initial model, and the compositional and conformational heterogeneity. Here, we describe a structure determination method that avoids the reconstruction procedure. The atomic structures of the individual complex components are assembled by optimizing a match against 2D EM class-average images, an excluded volume criterion, geometric complementarity, and optional restraints from proteomics and chemical cross-linking experiments. The optimization relies on a simulated annealing Monte Carlo search and a divide-and-conquer message-passing algorithm. Using simulated and experimentally determined EM class averages for 12 and 4 protein assemblies, respectively, we show that a few class averages can indeed result in accurate models for complexes of as many as five subunits. Thus, integrative structural biology can now benefit from the relative ease with which the EM class averages are determined.

Real-space processing of helical filaments in SPARX

We present a major revision of the iterative helical real-space refinement (IHRSR) procedure and its implementation in the SPARX single particle image processing environment. We built on over a decade of experience with IHRSR helical structure determination and we took advantage of the flexible SPARX infrastructure to arrive at an implementation that offers ease of use, flexibility in designing helical structure determination strategy, and high computational efficiency. We introduced the 3D projection matching code which now is able to work with non-cubic volumes, the geometry better suited for long helical filaments, we enhanced procedures for establishing helical symmetry parameters, and we parallelized the code using distributed memory paradigm. Additional features include a graphical user interface that facilitates entering and editing of parameters controlling the structure determination strategy of the program. In addition, we present a novel approach to detect and evaluate structural heterogeneity due to conformer mixtures that takes advantage of helical structure redundancy.

Automatic alignment and reconstruction of images for soft X-ray tomography

Soft X-ray tomography (SXT) is a powerful imaging technique that generates quantitative, 3D images of the structural organization of whole cells in a near-native state. SXT is also a high-throughput imaging technique. At the National Center for X-ray Tomography (NCXT), specimen preparation and image collection for tomographic reconstruction of a whole cell require only minutes. Aligning and reconstructing the data, however, take significantly longer. Here we describe a new component of the high throughput computational pipeline used for processing data at the NCXT. We have developed a new method for automatic alignment of projection images that does not require fiducial markers or manual interaction with the software. This method has been optimized for SXT data sets, which routinely involve full rotation of the specimen. This software gives users of the NCXT SXT instrument a new capability - virtually real-time initial 3D results during an imaging experiment, which can later be further refined. The new code, Automatic Reconstruction 3D (AREC3D), is also fast, reliable, and robust. The fundamental architecture of the code is also adaptable to high performance GPU processing, which enables significant improvements in speed and fidelity.

Single particle analysis at high resolution

Electron cryomicroscopy (cryo-EM) and single particle analysis is emerging as a powerful technique for determining the 3D structure of large biomolecules and biomolecular assemblies in close to their native solution environment. Over the last decade, this technology has improved, first to sub-nanometer resolution, and more recently beyond 0.5 nm resolution. Achieving sub-nanometer resolution is now readily approachable on mid-range microscopes with straightforward data processing, so long as the target specimen meets some basic requirements. Achieving resolutions beyond 0.5 nm currently requires a high-end microscope and careful data acquisition and processing, with much more stringent specimen requirements. This chapter will review and discuss the methodologies for determining high-resolution cryo-EM structures of nonvirus particles to sub-nanometer resolution and beyond, with a particular focus on the reconstruction strategy implemented in the EMAN software suite.

Image restoration in cryo-electron microscopy

Image restoration techniques are used to obtain, given experimental measurements, the best possible approximation of the original object within the limits imposed by instrumental conditions and noise level in the data. In molecular electron microscopy (EM), we are mainly interested in linear methods that preserve the respective relationships between mass densities within the restored map. Here, we describe the methodology of image restoration in structural EM, and more specifically, we will focus on the problem of the optimum recovery of Fourier amplitudes given electron microscope data collected under various defocus settings. We discuss in detail two classes of commonly used linear methods, the first of which consists of methods based on pseudoinverse restoration, and which is further subdivided into mean-square error, chi-square error, and constrained based restorations, where the methods in the latter two subclasses explicitly incorporates non-white distribution of noise in the data. The second class of methods is based on the Wiener filtration approach. We show that the Wiener filter-based methodology can be used to obtain a solution to the problem of amplitude correction (or "sharpening") of the EM map that makes it visually comparable to maps determined by X-ray crystallography, and thus amenable to comparative interpretation. Finally, we present a semiheuristic Wiener filter-based solution to the problem of image restoration given sets of heterogeneous solutions. We conclude the chapter with a discussion of image restoration protocols implemented in commonly used single particle software packages.

Cryo-EM image alignment based on nonuniform fast Fourier transform

In single particle analysis, two-dimensional (2-D) alignment is a fundamental step intended to put into register various particle projections of biological macromolecules collected at the electron microscope. The efficiency and quality of three-dimensional (3-D) structure reconstruction largely depends on the computational speed and alignment accuracy of this crucial step. In order to improve the performance of alignment, we introduce a new method that takes advantage of the highly accurate interpolation scheme based on the gridding method, a version of the nonuniform fast Fourier transform, and utilizes a multi-dimensional optimization algorithm for the refinement of the orientation parameters. Using simulated data, we demonstrate that by using less than half of the sample points and taking twice the runtime, our new 2-D alignment method achieves dramatically better alignment accuracy than that based on quadratic interpolation. We also apply our method to image to volume registration, the key step in the single particle EM structure refinement protocol. We find that in this case the accuracy of the method not only surpasses the accuracy of the commonly used real-space implementation, but results are achieved in much shorter time, making gridding-based alignment a perfect candidate for efficient structure determination in single particle analysis.

Exploration of parameters in cryo-EM leading to an improved density map of the E. coli ribosome

A number of image processing parameters in the 3D reconstruction of a ribosome complex from a cryo-EM data set were varied to test their effects on the final resolution. The parameters examined were pixel size, window size, and mode of Fourier amplitude enhancement at high spatial frequencies. In addition, the strategy of switching from large to small pixel size during angular refinement was explored. The relationship between resolution (in Fourier space) and the number of particles was observed to follow a lin-log dependence, a relationship that appears to hold for other data, as well. By optimizing the above parameters, and using a lin-log extrapolation to the full data set in the estimation of resolution from half-sets, we obtained a 3D map from 131,599 ribosome particles at 6.7A resolution (FSC=0.5).

3D electron microscopy of biological nanomachines: principles and applications

Transmission electron microscopy is a powerful technique for studying the three-dimensional (3D) structure of a wide range of biological specimens. Knowledge of this structure is crucial for fully understanding complex relationships among macromolecular complexes and organelles in living cells. In this paper, we present the principles and main application domains of 3D transmission electron microscopy in structural biology. Moreover, we survey current developments needed in this field, and discuss the close relationship of 3D transmission electron microscopy with other experimental techniques aimed at obtaining structural and dynamical information from the scale of whole living cells to atomic structure of macromolecular complexes.

SPARX, a new environment for Cryo-EM image processing

SPARX (single particle analysis for resolution extension) is a new image processing environment with a particular emphasis on transmission electron microscopy (TEM) structure determination. It includes a graphical user interface that provides a complete graphical programming environment with a novel data/process-flow infrastructure, an extensive library of Python scripts that perform specific TEM-related computational tasks, and a core library of fundamental C++ image processing functions. In addition, SPARX relies on the EMAN2 library and cctbx, the open-source computational crystallography library from PHENIX. The design of the system is such that future inclusion of other image processing libraries is a straightforward task. The SPARX infrastructure intelligently handles retention of intermediate values, even those inside programming structures such as loops and function calls. SPARX and all dependencies are free for academic use and available with complete source.