Correction of Missing-Wedge Artifacts in Filamentous Tomograms by Template-Based Constrained Deconvolution.

Cryo-electron tomography maps often exhibit considerable noise and anisotropic resolution, due to the low-dose requirements and the missing wedge in Fourier space. These spurious features are visually unappealing and, more importantly, prevent an automated segmentation of geometric shapes, requiring a subjective and labor-intensive manual tracing. We developed a novel computational strategy for objectively denoising and correcting missing-wedge artifacts in homogeneous specimen areas of tomograms, where it is assumed that a template repeats itself across the volume under consideration, as happens in the case of filaments. In our deconvolution approach, we use a template and a map of corresponding template locations, allowing us to compensate for the information lost in the missing wedge. We applied the method to tomograms of actin-filament bundles of inner-ear stereocilia, which are critical for the senses of hearing and balance. In addition, we demonstrate that our method can be used for cell membrane detection.

A cryo-tomography-based volumetric model of the actin core of mouse vestibular hair cell stereocilia lacking plastin 1.

Electron cryo-tomography allows for high-resolution imaging of stereocilia in their native state. Because their actin filaments have a higher degree of order, we imaged stereocilia from mice lacking the actin crosslinker plastin 1 (PLS1). We found that while stereocilia actin filaments run 13 nm apart in parallel for long distances, there were gaps of significant size that were stochastically distributed throughout the actin core. Actin crosslinkers were distributed through the stereocilium, but did not occupy all possible binding sites. At stereocilia tips, protein density extended beyond actin filaments, especially on the side of the tip where a tip link is expected to anchor. Along the shaft, repeating density was observed that corresponds to actin-to-membrane connectors. In the taper region, most actin filaments terminated near the plasma membrane. The remaining filaments twisted together to make a tighter bundle than was present in the shaft region; the spacing between them decreased from 13 nm to 9 nm, and the apparent filament diameter decreased from 6.4 to 4.8 nm. Our models illustrate detailed features of distinct structural domains that are present within the stereocilium.

Accurate flexible refinement of atomic models against medium-resolution cryo-EM maps using damped dynamics.

BACKGROUND: Dramatic progress has recently been made in cryo-electron microscopy technologies, which now make possible the reconstruction of a growing number of biomolecular structures to near-atomic resolution. However, the need persists for fitting and refinement approaches that address those cases that require modeling assistance. METHODS: In this paper, we describe algorithms to optimize the performance of such medium-resolution refinement methods. These algorithms aim to automatically optimize the parameters that define the density shape of the flexibly fitted model, as well as the time-dependent damper cutoff distance. Atomic distance constraints can be prescribed for cases where extra containment of parts of the structure is helpful, such as in regions where the density map is poorly defined. Also, we propose a simple stopping criterion that estimates the probable onset of overfitting during the simulation. RESULTS: The new set of algorithms produce more accurate fitting and refinement results, and yield a faster rate of convergence of the trajectory toward the fitted conformation. The latter is also more reliable due to the overfitting warning provided to the user. CONCLUSIONS: The algorithms described here were implemented in the new Damped-Dynamics Flexible Fitting simulation tool "DDforge" in the Situs package.

Tracing Actin Filament Bundles in Three-Dimensional Electron Tomography Density Maps of Hair Cell Stereocilia.

Cryo-electron tomography (cryo-ET) is a powerful method of visualizing the three-dimensional organization of supramolecular complexes, such as the cytoskeleton, in their native cell and tissue contexts. Due to its minimal electron dose and reconstruction artifacts arising from the missing wedge during data collection, cryo-ET typically results in noisy density maps that display anisotropic XY versus Z resolution. Molecular crowding further exacerbates the challenge of automatically detecting supramolecular complexes, such as the actin bundle in hair cell stereocilia. Stereocilia are pivotal to the mechanoelectrical transduction process in inner ear sensory epithelial hair cells. Given the complexity and dense arrangement of actin bundles, traditional approaches to filament detection and tracing have failed in these cases. In this study, we introduce BundleTrac, an effective method to trace hundreds of filaments in a bundle. A comparison between BundleTrac and manually tracing the actin filaments in a stereocilium showed that BundleTrac accurately built 326 of 330 filaments (98.8%), with an overall cross-distance of 1.3 voxels for the 330 filaments. BundleTrac is an effective semi-automatic modeling approach in which a seed point is provided for each filament and the rest of the filament is computationally identified. We also demonstrate the potential of a denoising method that uses a polynomial regression to address the resolution and high-noise anisotropic environment of the density map.

Computing Spatiotemporal Heat Maps of Lipid Electropore Formation: A Statistical Approach.

We extend the multiscale spatiotemporal heat map strategies originally developed for interpreting molecular dynamics simulations of well-structured proteins to liquids such as lipid bilayers and solvents. Our analysis informs the experimental and theoretical investigation of electroporation, that is, the externally imposed breaching of the cell membrane under the influence of an electric field of sufficient magnitude. To understand the nanoscale architecture of electroporation, we transform time domain data of the coarse-grained interaction networks of lipids and solvents into spatial heat maps of the most relevant constituent molecules. The application takes advantage of our earlier graph-based activity functions by accounting for the contact-forming and -breaking activity of the lipids in the bilayer. Our novel analysis of lipid interaction networks under periodic boundary conditions shows that the disruption of the bilayer, as measured by the breaking activity, is associated with the externally imposed pore formation. Moreover, the breaking activity can be used for statistically ranking the importance of individual lipids and solvent molecules through a bridging between fast and slow degrees of freedom. The heat map approach highlighted a small number of important lipids and solvent molecules, which allowed us to efficiently search the trajectories for any functionally relevant mechanisms. Our algorithms are freely disseminated with the open-source package TimeScapes.

A Balanced Approach to Adaptive Probability Density Estimation.

Our development of a Fast (Mutual) Information Matching (FIM) of molecular dynamics time series data led us to the general problem of how to accurately estimate the probability density function of a random variable, especially in cases of very uneven samples. Here, we propose a novel Balanced Adaptive Density Estimation (BADE) method that effectively optimizes the amount of smoothing at each point. To do this, BADE relies on an efficient nearest-neighbor search which results in good scaling for large data sizes. Our tests on simulated data show that BADE exhibits equal or better accuracy than existing methods, and visual tests on univariate and bivariate experimental data show that the results are also aesthetically pleasing. This is due in part to the use of a visual criterion for setting the smoothing level of the density estimate. Our results suggest that BADE offers an attractive new take on the fundamental density estimation problem in statistics. We have applied it on molecular dynamics simulations of membrane pore formation. We also expect BADE to be generally useful for low-dimensional applications in other statistical application domains such as bioinformatics, signal processing and econometrics.

Comparing an Atomic Model or Structure to a Corresponding Cryo-electron Microscopy Image at the Central Axis of a Helix.

Three-dimensional density maps of biological specimens from cryo-electron microscopy (cryo-EM) can be interpreted in the form of atomic models that are modeled into the density, or they can be compared to known atomic structures. When the central axis of a helix is detectable in a cryo-EM density map, it is possible to quantify the agreement between this central axis and a central axis calculated from the atomic model or structure. We propose a novel arc-length association method to compare the two axes reliably. This method was applied to 79 helices in simulated density maps and six case studies using cryo-EM maps at 6.4-7.7 Å resolution. The arc-length association method is then compared to three existing measures that evaluate the separation of two helical axes: a two-way distance between point sets, the length difference between two axes, and the individual amino acid detection accuracy. The results show that our proposed method sensitively distinguishes lateral and longitudinal discrepancies between the two axes, which makes the method particularly suitable for the systematic investigation of cryo-EM map-model pairs.

Spatial Heat Maps from Fast Information Matching of Fast and Slow Degrees of Freedom: Application to Molecular Dynamics Simulations.

We introduce a fast information matching (FIM) method for transforming time domain data into spatial images through handshaking between fast and slow degrees of freedom. The analytics takes advantage of the detailed time series available from biomolecular computer simulations, and it yields spatial heat maps that can be visualized on 3D molecular structures or in the form of interaction networks. The speed of our efficient mutual information solver is on the order of a basic Pearson cross-correlation calculation. We demonstrate that the FIM method is superior to linear cross-correlation for the detection of nonlinear dependence in challenging situations where measures for the global dynamics (the "activity") diverge. The analytics is applied to the detection of hinge-bending hot spots and to the prediction of pairwise contacts between residues that are relevant for the global activity exhibited by the molecular dynamics (MD) trajectories. Application examples from various MD laboratories include the millisecond bovine pancreatic trypsin inhibitor (BPTI) trajectory using canonical MD, a Gaussian accelerated MD folding trajectory of chignolin, and the heat-induced unfolding of engrailed homeodomain (EnHD). The FIM implementation will be freely disseminated with our open-source package, TimeScapes.

Comparison of an Atomic Model and Its Cryo-EM Image at the Central Axis of a Helix.

Cryo-electron microscopy (cryo-EM) is an important biophysical technique that produces three-dimensional (3D) density maps at different resolutions. Because more and more models are being produced from cryo-EM density maps, validation of the models is becoming important. We propose a method for measuring local agreement between a model and the density map using the central axis of the helix. This method was tested using 19 helices from cryo-EM density maps between 5.5 Å and 7.2 Å resolution and 94 helices from simulated density maps. This method distinguished most of the well-fitting helices, although challenges exist for shorter helices.

Interaction of the mediator head module with RNA polymerase II.

Mediator, a large (21 polypeptides, MW ∼1 MDa) complex conserved throughout eukaryotes, plays an essential role in control of gene expression by conveying regulatory signals that influence the activity of the preinitiation complex. However, the precise mode of interaction between Mediator and RNA polymerase II (RNAPII), and the mechanism of regulation by Mediator remain elusive. We used cryo-electron microscopy and reconstituted in vitro transcription assays to characterize a transcriptionally-active complex including the Mediator Head module and components of a minimum preinitiation complex (RNAPII, TFIIF, TFIIB, TBP, and promoter DNA). Our results reveal how the Head interacts with RNAPII, affecting its conformation and function.

FRODOCK: a new approach for fast rotational protein-protein docking.

MOTIVATION: Prediction of protein-protein complexes from the coordinates of their unbound components usually starts by generating many potential predictions from a rigid-body 6D search followed by a second stage that aims to refine such predictions. Here, we present and evaluate a new method to effectively address the complexity and sampling requirements of the initial exhaustive search. In this approach we combine the projection of the interaction terms into 3D grid-based potentials with the efficiency of spherical harmonics approximations to accelerate the search. The binding energy upon complex formation is approximated as a correlation function composed of van der Waals, electrostatics and desolvation potential terms. The interaction-energy minima are identified by a novel, fast and exhaustive rotational docking search combined with a simple translational scanning. Results obtained on standard protein-protein benchmarks demonstrate its general applicability and robustness. The accuracy is comparable to that of existing state-of-the-art initial exhaustive rigid-body docking tools, but achieving superior efficiency. Moreover, a parallel version of the method performs the docking search in just a few minutes, opening new application opportunities in the current 'omics' world. AVAILABILITY: http://sbg.cib.csic.es/Software/FRODOCK/

Damped-dynamics flexible fitting.

In fitting atomic structures into EM maps, it often happens that the map corresponds to a different conformation of the structure. We have developed a new methodology to handle these situations that preserves the covalent geometry of the structure and allows the modeling of large deformations. The first goal is achieved by working in generalized coordinates (positional and internal coordinates), and the second by avoiding harmonic potentials. Instead, we use dampers (shock absorbers) between every pair of atoms, combined with a force field that attracts the atomic structure toward incompletely occupied regions of the EM map. The trajectory obtained by integrating the resulting equations of motion converges to a conformation that, in our validation cases, was very close to the target atomic structure. Compared to current methods, our approach is more efficient and robust against wrong solutions and to overfitting, and does not require user intervention or subjective decisions. Applications to the computation of transition pathways between known conformers, homology and loop modeling, as well as protein docking, are also discussed.

Computational prediction of atomic structures of helical membrane proteins aided by EM maps.

Integral membrane proteins pose a major challenge for protein-structure prediction because only approximately 100 high-resolution structures are available currently, thereby impeding the development of rules or empirical potentials to predict the packing of transmembrane alpha-helices. However, when an intermediate-resolution electron microscopy (EM) map is available, it can be used to provide restraints which, in combination with a suitable computational protocol, make structure prediction feasible. In this work we present such a protocol, which proceeds in three stages: 1), generation of an ensemble of alpha-helices by flexible fitting into each of the density rods in the low-resolution EM map, spanning a range of rotational angles around the main helical axes and translational shifts along the density rods; 2), fast optimization of side chains and scoring of the resulting conformations; and 3), refinement of the lowest-scoring conformations with internal coordinate mechanics, by optimizing the van der Waals, electrostatics, hydrogen bonding, torsional, and solvation energy contributions. In addition, our method implements a penalty term through a so-called tethering map, derived from the EM map, which restrains the positions of the alpha-helices. The protocol was validated on three test cases: GpA, KcsA, and MscL.

Molecular modeling and mutagenesis of gap junction channels.

Gap junction channels connect the cytoplasms of adjacent cells through the end-to-end docking of hexameric hemichannels called connexons. Each connexon is formed by a ring of 24 alpha-helices that are staggered by 30 degrees with respect to those in the apposed connexon. Current evidence suggests that the two connexons are docked by interdigitated, anti-parallel beta strands across the extracellular gap. The second extracellular loop, E2, guides selectivity in docking between connexons formed by different isoforms. There is considerably more sequence variability of the N-terminal portion of E2, suggesting that this region dictates connexon coupling. Mutagenesis, biochemical, dye-transfer and electrophysiological data, combined with computational studies, have suggested possible assignments for the four transmembrane alpha-helices within each subunit. Most current models assign M3 as the major pore-lining helix. Mapping of human mutations onto a C(alpha) model suggested that native helix packing is important for the formation of fully functional channels. Nevertheless, a mutant in which the M4 helix has been replaced with polyalanine is functional, suggesting that M4 is located on the perimeter of the channel. In spite of this substantial progress in understanding the structural biology of gap junction channels, an experimentally determined structure at atomic resolution will be essential to confirm these concepts.

DFprot: a webtool for predicting local chain deformability.

UNLABELLED: DFprot is a web-based server for predicting main-chain deformability from a single protein conformation. The server automatically performs a normal-mode analysis (NMA) of the uploaded structure and calculates its capability to deform at each of its residues. Non-specialists can easily and rapidly obtain a quantitative first approximation of the flexibility of their structures with a simple and efficient interface. AVAILABILITY: http://sbg.cib.csic.es/Software/DFprot.

ADP_EM: fast exhaustive multi-resolution docking for high-throughput coverage.

MOTIVATION: Efficient fitting tools are needed to take advantage of a fast growth of atomic models of protein domains from crystallography or comparative modeling, and low-resolution density maps of larger molecular assemblies. Here, we report a novel fitting algorithm for the exhaustive and fast overlay of partial high-resolution models into a low-resolution density map. The method incorporates a fast rotational search based on spherical harmonics (SH) combined with a simple translational scanning. RESULTS: This novel combination makes it possible to accurately dock atomic structures into low-resolution electron-density maps in times ranging from seconds to a few minutes. The high-efficiency achieved with simulated and experimental test cases preserves the exhaustiveness needed in these heterogeneous-resolution merging tools. The results demonstrate its efficiency, robustness and high-throughput coverage. AVAILABILITY: http://sbg.cib.csic.es/Software/ADP_EM. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Representing receptor flexibility in ligand docking through relevant normal modes.

Inspired by the current representation of the ligand-receptor binding process, a normal-mode-based methodology is presented to incorporate receptor flexibility in ligand docking and virtual screening. However, the systematic representation of the deformation space grows geometrically with the number of modes, and furthermore, midscale loop rearrangements like those found in protein kinase binding pockets cannot be accounted for with the first lowest-frequency modes. We thus introduced a measure of relevance of normal modes on a given region of interest and showed that only very few modes in the low-frequency range are necessary and sufficient to describe loop flexibility in cAMP-dependent protein kinase. We used this approach to generate an ensemble of representative receptor backbone conformations by perturbing the structure along a combination of relevant modes. Each ensemble conformation is complexed with known non-native binders to optimize the position of the binding-pocket side chains through a full flexible docking procedure. The multiple receptor conformations thus obtained are used in a small-scale virtual screening using receptor ensemble docking. We evaluated this algorithm on holo and apo structures of cAMP-dependent protein kinase that exhibit backbone rearrangements on two independent loop regions close to the binding pocket. Docking accuracy is improved, since the ligands considered in the virtual screening docked within 1.5 A to at least one of the structures. The discrimination between binders and nonbinders is also enhanced, as shown by the improvement of the enrichment factor. This constitutes a new step toward the systematic integration of flexible ligand-flexible receptor docking tools in structure-based drug discovery.

Predictions of protein flexibility: first-order measures.

The normal modes of a molecule are utilized, in conjunction with classical conformal vector field theory, to define a function that measures the capability of the molecule to deform at each of its residues. An efficient algorithm is presented to calculate the local chain deformability from the set of normal modes of vibration. This is done by considering each mode as an off-grid sample of a deformation vector field. Predictions of deformability are compared with experimental data in the form of dihedral angle differences between two conformations of ten kinases by using a modified correlation function. Deformability calculations correlate well with experimental results and validate the applicability of this method to protein flexibility predictions.

2D fast rotational matching for image processing of biophysical data.

In 3D single particle reconstruction, which involves the translational and rotational matching of a large number of electron microscopy (EM) images, the algorithmic performance is largely dependent on the efficiency and accuracy of the underlying 2D image alignment kernel. We present a novel fast rotational matching kernel for 2D images (FRM2D) that significantly reduces the cost of this alignment. The alignment problem is formulated using one translational and two rotational degrees of freedom. This allows us to take advantage of fast Fourier transforms (FFTs) in rotational space to accelerate the search of the two angular parameters, while the remaining translational parameter is explored, within a limited range, by exhaustive search. Since there are no boundary effects in FFTs of cyclic angular variables, we avoid the expensive zero padding associated with Fourier transforms in linear space. To verify the robustness of our method, efficiency and accuracy tests were carried out over a range of noise levels in realistic simulations of EM images. Performance tests against two standard alignment methods, resampling to polar coordinates and self-correlation, demonstrate that FRM2D compares very favorably to the traditional methods. FRM2D exhibits a comparable or higher robustness against noise and a significant gain in efficiency that depends on the fineness of the angular sampling and linear search range.

Fast rotational matching of rigid bodies by fast Fourier transform acceleration of five degrees of freedom.

The 'fast rotational matching' method (an approach to find the three rotational degrees of freedom in matching problems using just one three-dimensional FFT) is extended to the full six-dimensional (rotation and translation) matching scenario between two three-dimensional objects. By recasting this problem into a formulation involving five angles and just one translational parameter, it was possible to accelerate, by means of fast Fourier transforms, five of the six degrees of freedom of the problem. This method was successfully applied to the docking of atomic structures of components into three-dimensional low-resolution density maps. Timing comparisons performed with our method and with 'fast translational matching' (the standard way to accelerate the translational parameters utilizing fast Fourier transforms) demonstrates that the performance gain can reach several orders of magnitude, especially for large map sizes. This gain can be particularly advantageous for spherical- and toroidal-shaped maps, since the scanning range of the translational parameter would be significantly constrained in these cases. The method can also be harnessed to the complementary surface (or 'exterior docking') problem and to pattern recognition in image processing.