Detection and separation of heterogeneity in molecular complexes by statistical analysis of their two-dimensional projections

Progress in molecular structure determination by cryo electron microscopy and single particle analysis has led to improvements in the resolution achievable. However, in many cases the limiting factor is structural heterogeneity of the sample. To address this problem, we have developed a method based on statistical analysis of the two-dimensional images to detect and sort localised structural variations caused, for example, by variable occupancy of a ligand. Images are sorted by two consecutive stages of multivariate statistical analysis (MSA) to dissect out the two main sources of variation, namely out of plane orientation and local structural changes. Heterogeneity caused by local changes is detected by MSA that reveals significant peaks in the higher order eigenimages. The eigenimages revealing local peaks are used for automated classification. Evaluation of differences between classes allows discrimination of molecular images with and without ligand. This method is very rapid, independent of any initial three-dimensional model, and can detect even minor subpopulations in an image ensemble. A strategy for using this technique was developed on model data sets. Here, we demonstrate the successful application of this method to both model and real EM data on chaperonin-substrate and ribosome-ligand complexes.     

Combinatorial docking approach for structure prediction of large proteins and multi-molecular assemblies

Protein folding and protein binding are similar processes. In both, structural units combinatorially associate with each other. In the case of folding, we mostly handle relatively small units, building blocks or domains, that are covalently linked. In the case of multi-molecular binding, the subunits are relatively large and are associated only by non-covalent bonds. Experimentally, the difficulty in the determination of the structures of such large assemblies increases with the complex size and the number of components it contains. Computationally, the prediction of the structures of multi-molecular complexes has largely not been addressed, probably owing to the magnitude of the combinatorial complexity of the problem. Current docking algorithms mostly target prediction of pairwise interactions. Here our goal is to predict the structures of multi-unit associations, whether these are chain-connected as in protein folding, or separate disjoint molecules in the assemblies. We assume that the structures of the single units are known, either through experimental determination or modeling. Our aim is to combinatorially assemble these units to predict their structure. To address this problem we have developed CombDock. CombDock is a combinatorial docking algorithm for the structural units assembly problem. Below, we briefly describe the algorithm and present examples of its various applications to folding and to multi-molecular assemblies. To test the robustness of the algorithm, we use inaccurate models of the structural units, derived either from crystal structures of unbound molecules or from modeling of the target sequences. The algorithm has been able to predict near-native arrangements of the input structural units in almost all of the cases, suggesting that a combinatorial approach can overcome the imperfect shape complementarity caused by the inaccuracy of the models. In addition, we further show that through a combinatorial docking strategy it is possible to enhance the predictions of pairwise interactions involved in a multi-molecular assembly.     

Statistical pattern matching facilitates the design of polyvalent inhibitors of anthrax and cholera toxins

Numerous biological processes involve the recognition of a specific pattern of binding sites on a target protein or surface. Although ligands displayed by disordered scaffolds form stochastic rather than specific patterns, theoretical models predict that recognition will occur between patterns that are characterized by similar or "matched" statistics. Endowing synthetic biomimetic structures with statistical pattern matching capabilities may improve the specificity of sensors and resolution of separation processes. We demonstrate that statistical pattern matching enhances the potency of polyvalent therapeutics. We functionalized liposomes with an inhibitory peptide at different densities and observed a transition in potency at an interpeptide separation that matches the distance between ligand-binding sites on the heptameric component of anthrax toxin. Pattern-matched polyvalent liposomes inhibited anthrax toxin in vitro at concentrations four orders of magnitude lower than the corresponding monovalent peptide, and neutralized this toxin in vivo. Statistical pattern matching also enhanced the potency of polyvalent inhibitors of cholera toxin. This facile strategy should be broadly applicable to the detection and neutralization of toxins and pathogens.     

Modeling regionalized volumetric differences in protein-ligand binding cavities

Identifying elements of protein structures that create differences in protein-ligand binding specificity is an essential method for explaining the molecular mechanisms underlying preferential binding. In some cases, influential mechanisms can be visually identified by experts in structural biology, but subtler mechanisms, whose significance may only be apparent from the analysis of many structures, are harder to find. To assist this process, we present a geometric algorithm and two statistical models for identifying significant structural differences in protein-ligand binding cavities. We demonstrate these methods in an analysis of sequentially nonredundant structural representatives of the canonical serine proteases and the enolase superfamily. Here, we observed that statistically significant structural variations identified experimentally established determinants of specificity. We also observed that an analysis of individual regions inside cavities can reveal areas where small differences in shape can correspond to differences in specificity.     

TMaCS: A hybrid template matching and classification system for partially-automated particle selection

Selection of particle images from electron micrographs presents a bottleneck in determining the structures of macromolecular assemblies by single particle electron cryomicroscopy (cryo-EM). The problem is particularly important when an experimentalist wants to improve the resolution of a 3D map by increasing by tens or hundreds of thousands of images the size of the dataset used for calculating the map. Although several existing methods for automatic particle image selection work well for large protein complexes that produce high-contrast images, it is well known in the cryo-EM community that small complexes that give low-contrast images are often refractory to existing automated particle image selection schemes. Here we develop a method for partially-automated particle image selection when an initial 3D map of the protein under investigation is already available. Candidate particle images are selected from micrographs by template matching with template images derived from projections of the existing 3D map. The candidate particle images are then used to train a support vector machine, which classifies the candidates as particle images or non-particle images. In a final step in the analysis, the selected particle images are subjected to projection matching against the initial 3D map, with the correlation coefficient between the particle image and the best matching map projection used to assess the reliability of the particle image. We show that this approach is able to rapidly select particle images from micrographs of a rotary ATPase, a type of membrane protein complex involved in many aspects of biology.     

Use of residual dipolar couplings as restraints in ab initio protein structure prediction

NMR residual dipolar couplings (RDCs), in the form of the projection angles between the respective internuclear bond vectors, are used as structural restraints in the ab initio structure prediction of a test set of six proteins. The restraints are applied using a recently developed SICHO (SIde-CHain-Only) lattice protein model that employs a replica exchange Monte Carlo (MC) algorithm to search conformational space. Using a small number of RDC restraints, the quality of the predicted structures is improved as reflected by lower RMSD/dRMSD (root mean square deviation/distance root mean square deviation) values from the corresponding native structures and by the higher correlation of the most cooperative mode of motion of each predicted structure with that of the native structure. The latter, in particular, has possible implications for the structure-based functional analysis of predicted structures.     

Analysis of conformational variation in macromolecular structural models

Experimental conditions or the presence of interacting components can lead to variations in the structural models of macromolecules. However, the role of these factors in conformational selection is often omitted by in silico methods to extract dynamic information from protein structural models. Structures of small peptides, considered building blocks for larger macromolecular structural models, can substantially differ in the context of a larger protein. This limitation is more evident in the case of modeling large multi-subunit macromolecular complexes using structures of the individual protein components. Here we report an analysis of variations in structural models of proteins with high sequence similarity. These models were analyzed for sequence features of the protein, the role of scaffolding segments including interacting proteins or affinity tags and the chemical components in the experimental conditions. Conformational features in these structural models could be rationalized by conformational selection events, perhaps induced by experimental conditions. This analysis was performed on a non-redundant dataset of protein structures from different SCOP classes. The sequence-conformation correlations that we note here suggest additional features that could be incorporated by in silico methods to extract dynamic information from protein structural models.     

Statistical potentials for fold assessment

A protein structure model generally needs to be evaluated to assess whether or not it has the correct fold. To improve fold assessment, four types of a residue-level statistical potential were optimized, including distance-dependent, contact, Phi/Psi dihedral angle, and accessible surface statistical potentials. Approximately 10,000 test models with the correct and incorrect folds were built by automated comparative modeling of protein sequences of known structure. The criterion used to discriminate between the correct and incorrect models was the Z-score of the model energy. The performance of a Z-score was determined as a function of many variables in the derivation and use of the corresponding statistical potential. The performance was measured by the fractions of the correctly and incorrectly assessed test models. The most discriminating combination of any one of the four tested potentials is the sum of the normalized distance-dependent and accessible surface potentials. The distance-dependent potential that is optimal for assessing models of all sizes uses both C(alpha) and C(beta) atoms as interaction centers, distinguishes between all 20 standard residue types, has the distance range of 30 A, and is derived and used by taking into account the sequence separation of the interacting atom pairs. The terms for the sequentially local interactions are significantly less informative than those for the sequentially nonlocal interactions. The accessible surface potential that is optimal for assessing models of all sizes uses C(beta) atoms as interaction centers and distinguishes between all 20 standard residue types. The performance of the tested statistical potentials is not likely to improve significantly with an increase in the number of known protein structures used in their derivation. The parameters of fold assessment whose optimal values vary significantly with model size include the size of the known protein structures used to derive the potential and the distance range of the accessible surface potential. Fold assessment by statistical potentials is most difficult for the very small models. This difficulty presents a challenge to fold assessment in large-scale comparative modeling, which produces many small and incomplete models. The results described in this study provide a basis for an optimal use of statistical potentials in fold assessment.     

Structural characterization of oligosaccharides by high-performance liquid chromatography, fast-atom bombardment-mass spectrometry, and exoglycosidase digestion

A method for structural characterization of oligosaccharides after preparing uv-absorbing derivatives is described. The derivatives can be rapidly analyzed and purified by high-performance liquid chromatography, with separation of various structures determined primarily by size and sugar composition. Derivatization requires as little as 0.5-1.0 nmol of oligosaccharide, and detection of down to 50 pmol of oligosaccharide is possible by monitoring absorbance at 229 nm. In addition, the carbohydrate portion of the derivative was found to retain its sensitivity to exoglycosidases, allowing sequential enzymatic digestions for determination of sugar sequence and anomerity to be performed. The derivatives also possessed a site of potential positive charge, making them amenable to analysis by fast-atom bombardment-mass spectrometry. Permethylation of the derivatives permitted their separation by capillary gas chromatography, thus allowing investigation of their structures by gas chromatography-mass spectrometry. The combination of these techniques will allow almost the complete structure of small amounts of oligosaccharides to be determined.     

A concerted polymerization-mesophase separation model for formation of trichocyte intermediate filaments and macrofibril templates. 1: relating phase separation to structural development

The concept that macrofibril templates, the fibrillar precursor to complete macrofibrils incorporating matrix proteins in trichokeratins, are formed by intracellular anisotropic phase separation of intermediate filaments (IFs), is here developed in detail. The factors affecting structural development, including IF length dispersion, and presence of other macromolecular solutes, are discussed in terms of the statistical thermodynamic models presented over 30 years ago by P.J. Flory and co-workers. The crucial role of pendant IF head groups in controlling IF separation and stabilizing the mesophase is emphasised. In particular, a concerted process of polymerization of unit-length IF precursors coupled with continuous transfer of longer IFs to the anisotropic phase is invoked. Observed structures in differentiated cell lines in mature fibres are rationalised in terms of different possible nematic or double-twist liquid crystalline precursor structures, with varying degrees of anisotropic phase coalescence. The occurrence of rarely observed macromolecular double-twist structures is made plausible by qualitative analysis of mesophase mechanics and reference to alternative structures in other macromolecular mesogens. The model is consistent with, and explains, certain well-known features of mature fibre structure, such as filament-matrix ratios in different cell lines. A few comments relating to the infill of the template by keratin intermediate filament associated proteins (IFAPs) are presented.     

Quantitative microcomputed tomography analysis of collateral vessel development after ischemic injury

Transgenic mouse models are increasingly being used to investigate the functions of specific growth factors or matrix proteins to design therapeutic strategies for controlling blood vessel growth. However, the available methodologies for evaluating angiogenesis and arteriogenesis in these models are limited by animal size, user subjectivity, the power to visualize the three-dimensional vessel networks, or the capability to employ a vigorous quantitative analysis. In this study, we employed contrast-enhanced microcomputed tomography imaging to assess collateral development after induction of hindlimb ischemia in the mouse. The morphological parameters vessel volume, connectivity, number, thickness, thickness distribution, separation, and degree of anisotropy were evaluated in control and surgery limbs 0, 3, and 14 days postsurgery. Results indicate that the vascular volume of the surgically manipulated limb was reconstituted as early as 3 days after femoral artery excision through development of a series of highly connected, small caliber, closely spaced, and isotropically oriented collateral vessels. Parametric analyses were completed to assess the sensitivity of the calculated morphological parameters to variations in image binarization threshold and voxel size. Images taken at the 36-microm voxel size were found to be optimal for evaluating collateral vessel formation, whereas 8- to 16-microm voxel sizes were needed to resolve smaller vascular structures. This study demonstrates the utility of microcomputed tomography as a robust method for quantitative, three-dimensional analysis of blood vessel networks. Whereas these initial efforts focused on the mouse hindlimb ischemia model, the developed techniques may be applied to a variety of model systems to investigate mechanisms of angiogenesis and arteriogenesis.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Quaternary organization of the Staphylothermus marinus phosphoenolpyruvate synthase: angular reconstitution from cryoelectron micrographs with molecular modeling

Digital electron images of frozen-hydrated preparations of the 2.25-MDa Staphylothermus marinus phosphoenolpyruvate synthase (EC 2.7.9.2) have been analyzed by single-particle classification and averaging and iterative quaternion-based angular reconstitution. Contrast transfer function correction of micrographs obtained at different defocus values was used to improve the informational quality of the projection averages. Three-dimensional reconstructions were obtained to roughly 3-nm spatial resolution, in which the 24 identical subunits were arranged to form an octahedral complex, although the amino-terminal nucleotide-binding domain was not resolved. An atomic model of the subunit was generated by homology modeling using as the reference the known X-ray crystallographic structure of the related enzyme pyruvate orthophosphate dikinase (EC 2.7.9.1) from Clostridium symbiosum (Protein Data Bank entry 1DIK). The S. marinus protein could be arranged into an assembly of 12 homodimers to match the three-dimensional reconstruction in terms of shape and size of the homodimers, as well as overall shape and size of the complex. The quaternary model indicated that active sites of three monomers were localized around cavities (or putative channels) centered at the threefold axes of rotational symmetry and that carboxyl-terminal alpha-helical segments of four monomers were localized at the fourfold axes of rotational symmetry where they could facilitate interdimer interaction. The quaternary arrangement also indicated numerous potential hydrophobic and electrostatic interactions at the interdimer interfaces that could contribute further to structural stability.     

Biophysical characterization of a designed TMV coat protein mutant, R46G, that elicits a moderate hypersensitivity response in Nicotiana sylvestris

The hypersensitivity resistance response directed by the N' gene in Nicotiana sylvestris is elicited by the tobacco mosaic virus (TMV) coat protein R46G, but not by the U1 wild-type TMV coat protein. In this study, the structural and hydrodynamic properties of R46G and wild-type coat proteins were compared for variations that may explain N' gene elicitation. Circular dichroism spectroscopy reveals no significant secondary or tertiary structural differences between the elicitor and nonelicitor coat proteins. Analytical ultracentrifugation studies, however, do show different concentration dependencies of the weight average sedimentation coefficients at 4 degrees C. Viral reconstitution kinetics at 20 degrees C were used to determine viral assembly rates and as an initial assay of the rate of 20S formation, the obligate species for viral reconstitution. These kinetic results reveal a decreased lag time for reconstitution performed with R46G that initially lack the 20S aggregate. However, experiments performed with 20S initially present reveal no detectable differences indicating that the mechanism of viral assembly is similar for the two coat protein species. Therefore, an increased rate of 20S formation from R46G subunits may explain the differences in the viral reconstitution lag times. The inferred increase in the rate of 20S formation is verified by direct measurement of the 20S boundary as a function of time at 20 degrees C using velocity sedimentation analysis. These results are consistent with the interpretation that there may be an altered size distribution and/or lifetime of the small coat protein aggregates in elicitors that allows N. sylvestris to recognize the invading virus.     

Advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixtures

In this paper, we review recent advances in blind source separation (BSS) and independent component analysis (ICA) for nonlinear mixing models. After a general introduction to BSS and ICA, we discuss in more detail uniqueness and separability issues, presenting some new results. A fundamental difficulty in the nonlinear BSS problem and even more so in the nonlinear ICA problem is that they provide non-unique solutions without extra constraints, which are often implemented by using a suitable regularization. In this paper, we explore two possible approaches. The first one is based on structural constraints. Especially, post-nonlinear mixtures are an important special case, where a nonlinearity is applied to linear mixtures. For such mixtures, the ambiguities are essentially the same as for the linear ICA or BSS problems. The second approach uses Bayesian inference methods for estimating the best statistical parameters, under almost unconstrained models in which priors can be easily added. In the later part of this paper, various separation techniques proposed for post-nonlinear mixtures and general nonlinear mixtures are reviewed.     

The effect of overabundant projection directions on 3D reconstruction algorithms

The experimental process of collecting images from macromolecules in an electron microscope is such that it does not allow for prior specification of the angular distribution of the projection images. As a consequence, an uneven distribution of projection directions may occur. Concerns have been raised recently about the behavior of 3D reconstruction algorithms for the case of unevenly distributed projections. It has been illustrated on experimental data that in the case of a heavily uneven distribution of projection directions some algorithms tend to elongate the reconstructed volumes along the overloaded direction so much as to make a quantitative biological analysis impossible. In answer to these concerns we have developed a strategy for quantitative comparison and optimization of 3D reconstruction algorithms. We apply this strategy to quantitatively analyze algebraic reconstruction techniques (ART) with blobs, simultaneous iterative reconstruction techniques (SIRT) with voxels, and weighted backprojection (WBP). We show that the elongation artifacts that had been previously reported can be strongly reduced. With our specific choices for the free parameters of the three algorithms, WBP reconstructions tend to be inferior to those obtained with either SIRT or ART and the results obtained with ART are comparable to those with SIRT, but at a very small fraction of the computational cost of SIRT.