Reconstruction of low-resolution molecular structures from simulated atomic force microscopy images

BackgroundAtomic Force Microscopy (AFM) is an experimental technique to study structure-function relationship of biomolecules. AFM provides images of biomolecules at nanometer resolution. High-speed AFM experiments produce a series of images following dynamics of biomolecules. To further understand biomolecular functions, information on three-dimensional (3D) structures is beneficial.MethodWe aim to recover 3D information from an AFM image by computational modeling. The AFM image includes only low-resolution representation of a molecule; therefore we represent the structures by a coarse grained model (Gaussian mixture model). Using Monte-Carlo sampling, candidate models are generated to increase similarity between AFM images simulated from the models and target AFM image.ResultsThe algorithm was tested on two proteins to model their conformational transitions. Using a simulated AFM image as reference, the algorithm can produce a low-resolution 3D model of the target molecule. Effect of molecular orientations captured in AFM images on the 3D modeling performance was also examined and it is shown that similar accuracy can be obtained for many orientations.ConclusionsThe proposed algorithm can generate 3D low-resolution protein models, from which conformational transitions observed in AFM images can be interpreted in more detail.General significanceHigh-speed AFM experiments allow us to directly observe biomolecules in action, which provides insights on biomolecular function through dynamics. However, as only partial structural information can be obtained from AFM data, this new AFM based hybrid modeling method would be useful to retrieve 3D information of the entire biomolecule.     

A Fast Image Alignment Approach for 2D Classification of Cryo-EM Images Using Spectral Clustering

Three-dimensional (3D) reconstruction in single-particle cryo-electron microscopy (cryo-EM) is a significant technique for recovering the 3D structure of proteins or other biological macromolecules from their two-dimensional (2D) noisy projection images taken from unknown random directions. Class averaging in single-particle cryo-EM is an important procedure for producing high-quality initial 3D structures, where image alignment is a fundamental step. In this paper, an efficient image alignment algorithm using 2D interpolation in the frequency domain of images is proposed to improve the estimation accuracy of alignment parameters of rotation angles and translational shifts between the two projection images, which can obtain subpixel and subangle accuracy. The proposed algorithm firstly uses the Fourier transform of two projection images to calculate a discrete cross-correlation matrix and then performs the 2D interpolation around the maximum value in the cross-correlation matrix. The alignment parameters are directly determined according to the position of the maximum value in the cross-correlation matrix after interpolation. Furthermore, the proposed image alignment algorithm and a spectral clustering algorithm are used to compute class averages for single-particle 3D reconstruction. The proposed image alignment algorithm is firstly tested on a Lena image and two cryo-EM datasets. Results show that the proposed image alignment algorithm can estimate the alignment parameters accurately and efficiently. The proposed method is also used to reconstruct preliminary 3D structures from a simulated cryo-EM dataset and a real cryo-EM dataset and to compare them with RELION. Experimental results show that the proposed method can obtain more high-quality class averages than RELION and can obtain higher reconstruction resolution than RELION even without iteration.     

Structure prediction of domain insertion proteins from structures of individual domains

Multidomain proteins continue to be a major challenge in protein structure prediction. Here we present a Monte Carlo (MC) algorithm, implemented within Rosetta, to predict the structure of proteins in which one domain is inserted into another. Three MC moves combine rigid-body and loop movements to search the constrained conformation by structure disruption and subsequent repair of chain breaks. Local searches find that the algorithm samples and recovers near-native structures consistently. Further global searches produced top-ranked structures within 5 A in 31 of 50 cases in low-resolution mode, and refinement of top-ranked low-resolution structures produced models within 2 A in 21 of 50 cases. Rigid-body orientations were often correctly recovered despite errors in linker conformation. The algorithm is broadly applicable to de novo structure prediction of both naturally occurring and engineered domain insertion proteins.     

Iterative stable alignment and clustering of 2D transmission electron microscope images

Identification of homogeneous subsets of images in a macromolecular electron microscopy (EM) image data set is a critical step in single-particle analysis. The task is handled by iterative algorithms, whose performance is compromised by the compounded limitations of image alignment and K-means clustering. Here we describe an approach, iterative stable alignment and clustering (ISAC) that, relying on a new clustering method and on the concepts of stability and reproducibility, can extract validated, homogeneous subsets of images. ISAC requires only a small number of simple parameters and, with minimal human intervention, can eliminate bias from two-dimensional image clustering and maximize the quality of group averages that can be used for ab initio three-dimensional structural determination and analysis of macromolecular conformational variability. Repeated testing of the stability and reproducibility of a solution within ISAC eliminates heterogeneous or incorrect classes and introduces critical validation to the process of EM image clustering.     

Optimizing cardiac material parameters with a genetic algorithm

Determining the unknown material parameters of intact ventricular myocardium can be challenging due to highly nonlinear material behavior. Previous studies combining a gradient-search optimization procedure with finite element analysis (FEA) were limited to two-dimensional (2D) models or simplified three-dimensional (3D) geometries. Here we present a novel scheme to estimate unknown material parameters for ventricular myocardium by combining a genetic algorithm (GA) with nonlinear finite element analysis. This approach systematically explores the domain of the material parameters. The objective function to minimize was the error between simulated strain data and finite element model strains. The proposed scheme was validated for a 2D problem using a realistic material law for ventricular myocardium. Optimized material parameters were generally within 0.5% of the true values. To demonstrate the robustness of the new scheme, unknown material parameters were also determined for a realistic 3D heart model with an exponential hyperelastic material law. When using strains from two material points, the algorithm converged to parameters within 5% of the true values. We conclude that the proposed scheme is robust when estimating myocardial material parameters in 2D and 3D models.     

Three-dimensional crystals of CaATPase from sarcoplasmic reticulum. Symmetry and molecular packing.

Structural studies of CaATPase from sarcoplasmic reticulum have so far been restricted to low resolution due to the poor order of two-dimensional crystal forms. However, we report that three-dimensional microcrystals of detergent-solubilized CaATPase diffract to 7.2 A in x-ray powder patterns and may therefore provide an opportunity to study CaATPase structure at higher resolutions. In the present study, we have characterized the symmetry and molecular packing of negatively stained crystals by electron microscopy (em). By altering the detergent-to-lipid ratio, different sized crystals were produced, which adhere to an em grid in different orientations. Thus, we obtained micrographs of three different projections and from these determined unit cell dimensions to be 151 X 51 X 158 A and the three-dimensional space group to be C2 with an angle beta very close to 90 degrees; x-ray powder patterns of hydrated, unstained crystals yielded dimensions of 166 X 58 X 164 A. Micrographs from each of two principal projections were averaged to produce two-dimensional density maps. Based on these maps and on the previously determined low-resolution structure of CaATPase, a packing diagram for these three-dimensional crystals is presented and major intermolecular contacts are proposed.     

Automated correlation of single particle tilt pairs for Random Conical Tilt and Orthogonal Tilt Reconstructions

One of the major methodological challenges in single particle electron microscopy is obtaining initial reconstructions which represent the structural heterogeneity of the dataset. Random Conical Tilt and Orthogonal Tilt Reconstruction techniques in combination with 3D alignment and classification can be used to obtain initial low-resolution reconstructions which represent the full range of structural heterogeneity of the dataset. In order to achieve statistical significance, however, a large number of 3D reconstructions, and, in turn, a large number of tilted image pairs are required. The extraction of single particle tilted image pairs from micrographs can be tedious and time-consuming, as it requires intensive user input even for semi-automated approaches. To overcome the bottleneck of manual selection of a large number of tilt pairs, we developed an algorithm for the correlation of single particle images from tilted image pairs in a fully automated and user-independent manner. The algorithm reliably correlates correct pairs even from noisy micrographs. We further demonstrate the applicability of the algorithm by using it to obtain initial references both from negative stain and unstained cryo datasets.     

Symmetry embedding in the reconstruction of macromolecular assemblies via the discrete Radon transform

In this paper we discuss the embedding of symmetry information in an algorithm for three-dimensional reconstruction, which is based on the discrete Radon transform. The original algorithm was designed for randomly oriented and in principal asymmetric particles. The expanded version presented here covers all symmetry point groups which can be exhibited by macromolecular protein assemblies. The orientations of all symmetry equivalent projections, based on the orientation of an experimental projection, are obtained using global group operators. Further, an improved interpolation scheme for the recovery of the three-dimensional discrete Radon transform has been designed for greater computational efficiency. The algorithm has been tested on phantom structures as well as on real data, a virus structure possessing icosahedral symmetry.     

Three-dimensional structure of bovine NADH:ubiquinone oxidoreductase of the mitochondrial respiratory chain

We have studied the structure of bovine heart mitochondrial NADH:ubiquinone (Q) oxidoreductase (EC 1.6.99.3) by image analysis of electron micrographs. A three-dimensional reconstruction was calculated from a tilt-series of a two-dimensional crystal of the molecule. Our interpretation of the position of the molecule in the unit cell of the crystal is supported by additional (low-resolution) analysis of images of single molecules. The three-dimensional reconstruction was calculated with the aid of an iterative real-space reconstruction algorithm. The various projections used as input to the algorithm were obtained by averaging the images of the tilted crystal through a Fourier-space peak-filtering procedure. The reconstructed unit cell measures 15.2 X 15.2 nm in the plane of the two-dimensional crystal and has a height of 10-11 nm. The unit cell contains one molecule consisting of four large subunits. At the present resolution of about 1.3 nm in the untilted projection, these four monomers are seen as two dimers related by a two-fold axis. Two views of the single particles have been recognized; they are the top and side view of the building block of the crystal. After computer image alignment and correspondence analysis, clusters of similar particles have been averaged. In the averages an uneven stain distribution is seen around the molecules, which may result from preferential staining of hydrophilic parts of the molecule. The molecular mass of the whole molecule was determined from scanning transmission electron microscopy measurements as (1.6 +/- 0.2) X 10(6) daltons.     

Situs: A package for docking crystal structures into low-resolution maps from electron microscopy

Three-dimensional image reconstructions of large-scale protein aggregates are routinely determined by electron microscopy (EM). We combine low-resolution EM data with high-resolution structures of proteins determined by x-ray crystallography. A set of visualization and analysis procedures, termed the Situs package, has been developed to provide an efficient and robust method for the localization of protein subunits in low-resolution data. Topology-representing neural networks are employed to vector-quantize and to correlate features within the structural data sets. Microtubules decorated with kinesin-related ncd motors are used as model aggregates to demonstrate the utility of this package of routines. The precision of the docking has allowed for the extraction of unique conformations of the macromolecules and is limited only by the reliability of the underlying structural data.     

Unsupervised noise removal algorithms for three-dimensional confocal fluorescence microscopy

Algorithms are presented for effective suppression of the quantum noise artifact that is inherent to three-dimensional confocal fluorescence microscopy images of extended spatial objects such as neurons. The specific advances embodied in these algorithms are as follows: (i) they incorporate an automatic and pattern-constrained three-dimensional segmentation of the image field, and use it to limit any smoothing to the interiors of specified image regions and hence avoid the blurring that is inevitably associated with conventional noise removal algorithms; (ii) they are ‘unsupervised’ in the sense that they automatically estimate and adapt to the unknown spatially and temporally varying noise level in the microscopy data. Fast computation is achieved by parallel computation methods, rather than by algorithmic or modelling compromises.The quantum noise artifact is modelled using a mixture of spatially non-homogeneous Poisson point processes. The intensity of each component process is constrained to lie in specific intervals. A set of segmentation and edge-site variables are used to determine the intensity of the mixture process. Using this model, the noise-removal process is formulated as the joint optimal estimation of the segmentation labels, edge-sites and intensity of the mixture Poisson point process, subject to a combination of stochastic priors and syntactic pattern constraints. The computations proceed iteratively, starting from a set of approximate user-supplied, or default initial guesses of the underlying random process parameters. An Expectation Maximization algorithm is used to obtain a more precise characterization of these parameters, that are then input to a joint estimation algorithm.Stereoscopic renderings of processed three-dimensional images of murine hippocampal neurons are presented to demonstrate the effectiveness of the method. The processed images exhibit increased contrast and significant smoothing and reduction of the background intensity while avoiding any blurring of the foreground neuronal structures.     

Maximizing quantitative accuracy of lung airway lumen and wall measures obtained from X-ray CT imaging.

To objectively quantify airway geometry from three-dimensional computed tomographic (CT) images, an idealized (circular cross section) airway model is parameterized by airway luminal caliber, wall thickness, and tilt angle. Using a two-dimensional CT slice, an initial guess for the airway center, and the full-width-half-maximum principle, we form an estimate of inner and outer airway wall locations. We then fit ellipses to the inner and outer airway walls via a direct least squares fit and use the major and minor axes of the ellipses to estimate the tilt and in-plane rotation angles. Convolving the airway model, initialized with these estimates, with the three-dimensional scanner point-spread function forms the predicted image. The difference between predicted and actual images is minimized by refining the model parameter estimates via a multidimensional, unconstrained, nonlinear minimization routine. When optimization converges, airway model parameters estimate the airway inner and outer radii and tilt angle. Results using a Plexiglas phantom show that tilt angle is estimated to within +/-4 degrees and both inner and outer radii to within one-half pixel when a "standard" CT reconstruction kernel is used. By opening up the ability to measure airways that are not oriented perpendicular to the scanning plane, this method allows evaluation of a greater sampling of airways in a two-dimensional CT slice than previously possible. In addition, by combining the tilt-angle compensation with the deconvolution method, we provide significant improvement over the previous full-width-half-maximum method for assessing location of the luminal edge but not the outer edge of the airway wall.     

Multi-resolution contour-based fitting of macromolecular structures

A novel contour-based matching criterion is presented for the quantitative docking of high-resolution structures of components into low-resolution maps of macromolecular complexes. The proposed Laplacian filter is combined with a six-dimensional search using fast Fourier transforms to rapidly scan the rigid-body degrees of freedom of a probe molecule relative to a fixed target density map. A comparison of the docking performance with the standard cross-correlation criterion demonstrates that contour matching with the Laplacian filter significantly extends the viable resolution range of correlation-based fitting to resolutions as low as 30 A. The gain in docking precision at medium to low resolution (15-30 A) is critical for image reconstructions from electron microscopy (EM). The new algorithm enables for the first time the reliable docking of smaller molecular components into EM densities of large biomolecular assemblies at such low resolutions. As an example of the practical effectiveness of contour-based fitting, a new pseudo-atomic model of a microtubule was constructed from a 20 A resolution EM map and from atomic structures of alpha and beta tubulin subunits.     

Protein structural class identification directly from NMR spectra using averaged chemical shifts

Knowledge of the three-dimensional structure of proteins is integral to understanding their functions, and a necessity in the era of proteomics. A wide range of computational methods is employed to estimate the secondary, tertiary, and quaternary structures of proteins. Comprehensive experimental methods, on the other hand, are limited to nuclear magnetic resonance (NMR) and X-ray crystallography. The full characterization of individual structures, using either of these techniques, is extremely time intensive. The demands of high throughput proteomics necessitate the development of new, faster experimental methods for providing structural information. As a first step toward such a method, we explore the possibility of determining the structural classes of proteins directly from their NMR spectra, prior to resonance assignment, using averaged chemical shifts. This is achieved by correlating NMR-based information with empirical structure-based information available in widely used electronic databases. The results are analyzed statistically for their significance. The robustness of the method as a structure predictor is probed by applying it to a set of proteins of unknown structure. Our results show that this NMR-based method can be used as a low-resolution tool for protein structural class identification.     

Modeling large RNAs and ribonucleoprotein particles using molecular mechanics techniques.

There is a growing body of low-resolution structural data that can be utilized to devise structural models for large RNAs and ribonucleoproteins. These models are routinely built manually. We introduce an automated refinement protocol to utilize such data for building low-resolution three-dimensional models using the tools of molecular mechanics. In addition to specifying the positions of each nucleotide, the protocol provides quantitative estimates of the uncertainties in those positions, i.e., the resolution of the model. In typical applications, the resolution of the models is about 10-20 A. Our method uses reduced representations and allows us to refine three-dimensional structures of systems as big as the 16S and 23S ribosomal RNAs, which are about one to two orders of magnitude larger than nucleic acids that can be examined by traditional all-atom modeling methods. Nonatomic resolution structural data--secondary structure, chemical cross-links, chemical and enzymatic footprinting patterns, protein positions, solvent accessibility, and so on--are combined with known motifs in RNA structure to predict low-resolution models of large RNAs. These structural constraints are imposed on the RNA chain using molecular mechanics-type potential functions with parameters based on the quality of experimental data. Surface potential functions are used to incorporate shape and positional data from electron microscopy image reconstruction experiments into our models. The structures are optimized using techniques of energy refinement to get RNA folding patterns. In addition to providing a consensus model, the method finds the range of models consistent with the data, which allows quantitative evaluation of the resolution of the model. The method also identifies conflicts in the experimental data. Although our protocol is aimed at much larger RNAs, we illustrate these techniques using the tRNA structure as an example and test-bed.     

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

We describe an algorithm for simultaneous refinement of a three-dimensional (3-D) density map and of the orientation parameters of two-dimensional (2-D) projections that are used to reconstruct this map. The application is in electron microscopy, where the 3-D structure of a protein has to be determined from a set of 2-D projections collected at random but initially unknown angles. The design of the algorithm is based on the assumption that initial low resolution approximation of the density map and reasonable guesses for orientation parameters are available. Thus, the algorithm is applicable in final stages of the structure refinement, when the quality of the results is of main concern. We define the objective function to be minimized in real space and solve the resulting nonlinear optimization problem using a Quasi-Newton algorithm. We calculate analytical derivatives with respect to density distribution and the finite difference approximations of derivatives with respect to orientation parameters. We demonstrate that calculation of derivatives is robust with respect to noise in the data. This is due to the fact that noise is annihilated by the back-projection operations. Our algorithm is distinguished from other orientation refinement methods (i) by the simultaneous update of the density map and orientation parameters resulting in a highly efficient computational scheme and (ii) by the high quality of the results produced by a direct minimization of the discrepancy between the 2-D data and the projected views of the reconstructed 3-D structure. We demonstrate the speed and accuracy of our method by using simulated data.     

Optimized 3D coordinate reconstruction from paired stereographs using a calibrated phantom

A refinement to a previously described three-dimensional reconstruction algorithm based on point identification in calibrated non-orthogonal radiograms (stereo-pairs) is described. The modification involves a computation of the focal point magnitude of the point in three dimensions, analogous to focusing in two dimensions, as well as the most likely location of the target point in 3-space; the focal point magnitude may be thought of as the precision of the point identification. Multiple observer studies of the same stereopair can be used to estimate three-dimensional reconstruction accuracy by providing an average location and a mean distance from average. Both measures are useful parameters for initial selection of bone landmark references and for error propagation studies.     

Low-Resolution Density Maps from Atomic Models: How Stepping “Back” Can Be a Step “Forward”

Atomic-resolution structures have had a tremendous impact on modern biological science. Much useful information also has been gleaned by merging and correlating atomic-resolution structural details with lower-resolution (15–40 Å), three-dimensional (3D) reconstructions computed from images recorded with cryo-transmission electron microscopy (cryoTEM) procedures. One way to merge these structures involves reducing the resolution of an atomic model to a level comparable to a cryoTEM reconstruction. A low-resolution density map can be derived from an atomic-resolution structure by retrieving a set of atomic coordinates editing the coordinate file, computing structure factors from the model coordinates, and computing the inverse Fourier transform of the structure factors. This method is a useful tool for structural studies primarily in combination with 3D cryoTEM reconstructions. It has been used to assess the quality of 3D reconstructions, to determine corrections for the phase-contrast transfer function of the transmission electron microscope, to calibrate the dimensions and handedness of 3D reconstructions, to produce difference maps, to model features in macromolecules or macromolecular complexes, and to generate models to initiate model-based determination of particle orientation and origin parameters for 3D reconstruction.     

Visual computation of egomotion using an image interpolation technique

A novel technique is presented for the computation of the parameters of egomotion of a mobile device, such as a robot or a mechanical arm, equipped with two visual sensors. Each sensor captures a panoramic view of the environment. We show that the parameters of egomotion can be computed by interpolating the position of the image captured by one of the sensors at the robot's present location, with respect to the images captured by the two sensors at the robot's previous location. The algorithm delivers the distance travelled and angle rotated, without the explicit measurement or integration of velocity fields. The result is obtained in a single step, without any iteration or successive approximation. Tests of the algorithm on real and synthetic images reveal an accuracy to within 5% of the actual motion. Implementation of the algorithm on a mobile robot reveals that stepwise rotation and translation can be measured to within 10% accuracy in a three-dimensional world of unknown structure. The position and orientation of the robot at the end of a 30-step trajectory can be estimated with accuracies of 5% and 5°, respectively.     

Parameter Identifiability and Estimation of HIV/AIDS Dynamic Models

We use a technique from engineering (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005) to investigate the algebraic identifiability of a popular three-dimensional HIV/AIDS dynamic model containing six unknown parameters. We find that not all six parameters in the model can be identified if only the viral load is measured, instead only four parameters and the product of two parameters (N and λ) are identifiable. We introduce the concepts of an identification function and an identification equation and propose the multiple time point (MTP) method to form the identification function which is an alternative to the previously developed higher-order derivative (HOD) method (Xia and Moog, in IEEE Trans. Autom. Contr. 48(2):330–336, 2003; Jeffrey and Xia, in Tan, W.Y., Wu, H. (Eds.), Deterministic and Stochastic Models of AIDS Epidemics and HIV Infections with Intervention, 2005). We show that the newly proposed MTP method has advantages over the HOD method in the practical implementation. We also discuss the effect of the initial values of state variables on the identifiability of unknown parameters. We conclude that the initial values of output (observable) variables are part of the data that can be used to estimate the unknown parameters, but the identifiability of unknown parameters is not affected by these initial values if the exact initial values are measured with error. These noisy initial values only increase the estimation error of the unknown parameters. However, having the initial values of the latent (unobservable) state variables exactly known may help to identify more parameters. In order to validate the identifiability results, simulation studies are performed to estimate the unknown parameters and initial values from simulated noisy data. We also apply the proposed methods to a clinical data set to estimate HIV dynamic parameters. Although we have developed the identifiability methods based on an HIV dynamic model, the proposed methodologies are generally applicable to any ordinary differential equation systems.