Identification of secondary structure elements in intermediate-resolution density maps

An increasing number of structural studies of large macromolecular complexes, both in X-ray crystallography and cryo-electron microscopy, have resulted in intermediate-resolution (5-10 A) density maps. Despite being limited in resolution, significant structural and functional information may be extractable from these maps. To aid in the analysis and annotation of these complexes, we have developed SSEhunter, a tool for the quantitative detection of alpha helices and beta sheets. Based on density skeletonization, local geometry calculations, and a template-based search, SSEhunter has been tested and validated on a variety of simulated and authentic subnanometer-resolution density maps. The result is a robust, user-friendly approach that allows users to quickly visualize, assess, and annotate intermediate-resolution density maps. Beyond secondary structure element identification, the skeletonization algorithm in SSEhunter provides secondary structure topology, which is potentially useful in leading to structural models of individual molecular components directly from the density.     

Detection of secondary and supersecondary structures of proteins from cryo-electron microscopy

Recent advances in three-dimensional electron microscopy (3D EM) have enabled the quantitative visualization of the structural building blocks of proteins at improved resolutions. We provide algorithms to detect the secondary structures (α-helices and β-sheets) from proteins for which the volumetric maps are reconstructed at 6–10 Å resolution. Additionally, we show that when the resolution is coarser than 10 Å, some of the supersecondary structures can be detected from 3D EM maps. For both these algorithms, we employ tools from computational geometry and differential topology, specifically the computation of stable/unstable manifolds of certain critical points of the distance function induced by the molecular surface. Our results connect mathematically well-defined constructions with bio-chemically induced structures observed in proteins.     

EMatch: discovery of high resolution structural homologues of protein domains in intermediate resolution cryo-EM maps

Cryo-EM has become an increasingly powerful technique for elucidating the structure, dynamics, and function of large flexible macromolecule assemblies that cannot be determined at atomic resolution. However, due to the relatively low resolution of cryo-EM data, a major challenge is to identify components of complexes appearing in cryo-EM maps. Here, we describe EMatch, a novel integrated approach for recognizing structural homologues of protein domains present in a 6-10 A resolution cryo-EM map and constructing a quasi-atomic structural model of their assembly. The method is highly efficient and has been successfully validated on various simulated data. The strength of the method is demonstrated by a domain assembly of an experimental cryo-EM map of native GroEL at 6 Aring resolution     

Common structural features of the luxF protein and the subunits of bacterial luciferase: evidence for a (beta alpha)8 fold in luciferase.

The amino acid sequence identity and potential structural similarity between the subunits of bacterial luciferase and the recently determined structure of the luxF molecule are examined. The unique beta/alpha barrel fold found in luxF appears to be conserved in part in the luciferase subunits. From secondary structural predictions of both luciferase subunits, and from structural comparisons between the protein product of the luxF gene, NFP, and glycolate oxidase, we propose that it is feasible for both luciferase subunits to adopt a (beta alpha)8 barrel fold with at least 2 excursions from the (beta alpha)8 topology. Amino acids conserved between NFP and the luciferase subunits cluster together in 3 distinct "pockets" of NFP, which are located at hydrophobic interfaces between the beta-strands and alpha-helices. Several tight turns joining the C-termini of beta-strands and the N-termini of alpha-helices are found as key components of these conserved regions. Helix start and end points are easily demarcated in the luciferase subunit protein sequences; the N-cap residues are the most strongly conserved structural features. A partial model of the luciferase beta subunit from Photobacterium leiognathi has been built based on our crystallographically determined structure of luxF at 1.6 A resolution.     

Determining protein topology from skeletons of secondary structures

We report a novel computational procedure for determining protein native topology, or fold, by defining loop connectivity based on skeletons of secondary structures that can usually be obtained from low to intermediate-resolution density maps. The procedure primarily involves a knowledge-based geometry filter followed by an energetics-based evaluation. It was tested on a large set of skeletons covering a wide range of protein architecture, including one modeled from an experimentally determined 7.6A cryo-electron microscopy (cryo-EM) density map. The results showed that the new procedure could effectively deduce protein folds without high-resolution structural data, a feature that could also be used to recognize native fold in structure prediction and to interpret data in fields like structure genomics. Most importantly, in the energetics-based evaluation, it was revealed that, despite the inevitable errors in the artificially constructed structures and limited accuracy of knowledge-based potential functions, the average energy of an ensemble of structures with slightly different configurations around the native skeleton is a much more robust parameter for marking native topology than the energy of individual structures in the ensemble. This result implies that, among all the possible topology candidates for a given skeleton, evolution has selected the native topology as the one that can accommodate the largest structural variations, not the one rigidly trapped in a deep, but narrow, conformational energy well.     

Statistical analysis of sizes and shapes of virus capsids and their resulting elastic properties

From the analysis of sizes of approximately 130 small icosahedral viruses we find that there is a typical structural capsid protein, having a mean diameter of 5 nm and a mean thickness of 3 nm, with more than two thirds of the analyzed capsid proteins having thicknesses between 2 nm and 4 nm. To investigate whether, in addition to the fairly conserved geometry, capsid proteins show similarities in the way they interact with one another, we examined the shapes of the capsids in detail. We classified them numerically according to their similarity to sphere and icosahedron and an interpolating set of shapes in between, all of them obtained from the theory of elasticity of shells. In order to make a unique and straightforward connection between an idealized, numerically calculated shape of an elastic shell and a capsid, we devised a special shape fitting procedure, the outcome of which is the idealized elastic shape fitting the capsid best. Using such a procedure we performed statistical analysis of a series of virus shapes and we found similarities between the capsid elastic properties of even very different viruses. As we explain in the paper, there are both structural and functional reasons for the convergence of protein sizes and capsid elastic properties. Our work presents a specific quantitative scheme to estimate relatedness between different proteins based on the details of the (quaternary) shape they form (capsid). As such, it may provide an information complementary to the one obtained from the studies of other types of protein similarity, such as the overall composition of structural elements, topology of the folded protein backbone, and sequence similarity.     

Protein titration in the crystal state.

Proteins are complex structures whose overall stability critically depends on a delicate balance of numerous interactions of similar strength, which are markedly influenced by their environment. Here, we present an analysis of the effect of pH on a protein structure in the crystalline state using RNase A as a model system. By altering only one physico-chemical parameter in a controlled manner, we are able to quantify the structural changes induced in the protein. Atomic resolution X-ray diffraction data were collected for crystals at six pH* values ranging from 5.2 to 8.8, and the six independently refined structures reveal subtle, albeit well-defined variations directly related to the pH titration of the protein. The deprotonation of the catalytic His12 residue is clearly evident in the electron density maps, confirming the reaction mechanism proposed by earlier enzymatic and structural studies. The concerted structural changes observed in the regions remote from the active-site point to an adaptation of the protein structure to the changes in the physico-chemical environment. Analysis of the stereochemistry of the six structures provided accurate estimates of p Kavalues of most of the histidine residues. This study gives further evidence for the advantage of atomic resolution X-ray crystallographic analyses for revealing small but significant structural changes which provide clues to the function of a biological macromolecule.     

Automated reconstruction of three-dimensional neuronal morphology from laser scanning microscopy images

Experimental and theoretical studies demonstrate that both global dendritic branching topology and fine spine geometry are crucial determinants of neuronal function, its plasticity and pathology. Importantly, simulation studies indicate that the interaction between local and global morphologic properties is pivotal in determining dendritic information processing and the induction of synapse-specific plasticity. The ability to reconstruct and quantify dendritic processes at high resolution is therefore an essential prerequisite to understanding the structural determinants of neuronal function. Existing methods of digitizing 3D neuronal structure use interactive manual computer tracing from 2D microscopy images. This method is time-consuming, subjective and lacks precision. In particular, fine details of dendritic varicosities, continuous dendritic taper, and spine morphology cannot be captured by these systems. We describe a technique for automated reconstruction of 3D neuronal morphology from multiple stacks of tiled confocal and multiphoton laser scanning microscopy (CLSM and MPLSM) images. The system is capable of representing both global and local structural variations, including gross dendritic branching topology, dendritic varicosities, and fine spine morphology with sufficient resolution for accurate 3D morphometric analyses and realistic biophysical compartment modeling. Our system provides a much needed tool for automated digitization and reconstruction of 3D neuronal morphology that reliably captures detail on spatial scales spanning several orders of magnitude, that avoids the subjective errors that arise during manual tracing with existing digitization systems, and that runs on a standard desktop workstation.     

A high-order graph generating self-organizing structure

A large class of neural network models have their units organized in a lattice with fixed topology or generate their topology during the learning process. These network models can be used as neighborhood preserving map of the input manifold, but such a structure is difficult to manage since these maps are graphs with a number of nodes that is just one or two orders of magnitude less than the number of input points (i.e., the complexity of the map is comparable with the complexity of the manifold) and some hierarchical algorithms were proposed in order to obtain a high-level abstraction of these structures. In this paper a general structure capable to extract high order information from the graph generated by a large class of self-organizing networks is presented. This algorithm will allow to build a two layers hierarchical structure starting from the results obtained by using the suitable neural network for the distribution of the input data. Moreover the proposed algorithm is also capable to build a topology preserving map if it is trained using a graph that is also a topology preserving map.     

Elastic model of highly supercoiled DNA

The equilibrium shapes of supercoiled DNA are investigated by employing an elastic model. First, a set of Euler equations is derived to determine the equilibrium shapes under ring-closure conditions. Two exact solutions that describe circular and figure-8 shapes are obtained. Using these and their topological properties, the configuration change from the circular to the figure-8 form is discussed. Second, more intricate structures of supercoiling DNA are studied by a numerical analysis. Among a class of configurations, the shape that has the minimum elastic energy is explicitly determined. Poisson's ratio, the ratio of the self-avoiding radius to the total length, and the deficit (or excess) of the linking number ΔLk are found to be the important parameters. We conclude that the topology and the elastic theory of looped DNA explain the essential features of the supercoiling phenomena.     

Sub-Angstrom resolution enzyme X-ray structures: is seeing believing?

Recent technical advances in crystallographic analysis, particularly highly focused and high brilliance synchrotron beam lines, have significantly improved the resolutions that are attainable for many macromolecular crystal structures. The Protein Data Bank (http://www.rcsb.org/pdb/) contains an increasing number of atomic resolution structures, which are providing a wealth of structural information that was not previously visible in lower resolution electron density maps. Here, we review the importance of visualizing hydrogen atoms and multiple sidechain conformations or anisotropy, as well as substrate strain, at sub-Angstrom resolution. The additional structural features that are visible in the electron density maps as a result of atomic resolution data provide a better understanding of the catalytic mechanisms of cholesterol oxidase, ribonuclease A, beta-lactamase, serine proteases, triosephosphate isomerase and endoglucanase.     

Topology-preserving smoothing of retinotopic maps

Retinotopic mapping, i.e., the mapping between visual inputs on the retina and neuronal activations in cortical visual areas, is one of the central topics in visual neuroscience. For human observers, the mapping is obtained by analyzing functional magnetic resonance imaging (fMRI) signals of cortical responses to slowly moving visual stimuli on the retina. Although it is well known from neurophysiology that the mapping is topological (i.e., the topology of neighborhood connectivity is preserved) within each visual area, retinotopic maps derived from the state-of-the-art methods are often not topological because of the low signal-to-noise ratio and spatial resolution of fMRI. The violation of topological condition is most severe in cortical regions corresponding to the neighborhood of the fovea (e.g., < 1 degree eccentricity in the Human Connectome Project (HCP) dataset), significantly impeding accurate analysis of retinotopic maps. This study aims to directly model the topological condition and generate topology-preserving and smooth retinotopic maps. Specifically, we adopted the Beltrami coefficient, a metric of quasiconformal mapping, to define the topological condition, developed a mathematical model to quantify topological smoothing as a constrained optimization problem, and elaborated an efficient numerical method to solve the problem. The method was then applied to V1, V2, and V3 simultaneously in the HCP dataset. Experiments with both simulated and real retinotopy data demonstrated that the proposed method could generate topological and smooth retinotopic maps.     

Effects of aligned α‐helix peptide dipoles on experimental electrostatic potentials

Aligned protein α‐helix dipoles have been implicated in protein function and structure. The recent breakthroughs in high‐resolution electron microscopy (EM) of macromolecules makes it possible to explore fundamental aspects of structural biology at the detailed molecular level. The electrostatic potential (ESP) generated by aligned protein α‐helix dipole should be observable in high‐resolution EM maps despite the fact that the effect may be partially screened by induced electric fields. Here, we show that aligned backbone dipoles in protein α‐helices account for long‐range features in the protein ESP functions. Our results are consistent with experimental EM maps and density functional theory calculations, including direct Fourier summation for proper calculation of the ESP due to the nonlocal nature of the ESP function from aligned dipoles and other partial atomic charges.     

3D genital shape complexity in female marine mammals

Comparisons of 3D shapes have recently been applied to diverse anatomical structures using landmarking techniques. However, discerning evolutionary patterns can be challenging for structures lacking homologous landmarks. We used alpha shape analyses to quantify vaginal shape complexity in 40 marine mammal specimens including cetaceans, pinnipeds, and sirenians. We explored phylogenetic signal and the potential roles of natural and sexual selection on vaginal shape evolution. Complexity scores were consistent with qualitative observations. Cetaceans had a broad range of alpha complexities, while pinnipeds were comparatively simple and sirenians were complex. Intraspecific variation was found. Three‐dimensional surface heat maps revealed that shape complexity was driven by invaginations and protrusions of the vaginal wall. Phylogenetic signal was weak and metrics of natural selection (relative neonate size) and sexual selection (relative testes size, sexual size dimorphism, and penis morphology) did not explain vaginal complexity patterns. Additional metrics, such as penile shape complexity, may yield interesting insights into marine mammal genital coevolution. We advocate for the use of alpha shapes to discern patterns of evolution that would otherwise not be possible in 3D anatomical structures lacking homologous landmarks.     

204 Polymer translocation

The translocation of a single macromolecule through a protein pore or a solid-state nanopore involves three major stages: (1) approach of the macromolecule towards the pore, (2) capture/recognition of the macromolecule at the pore entrance, and (3) threading through the pore (see the Figure) (Muthukumar, 2011). All of these stages are controlled by conformational entropy of the macromolecule, charge decoration, and the geometry of the pore, hydrodynamics, and electrostatic interactions. Chief among the contributing factors are the entropic barrier presented by the pore to the penetration of the macromolecule, pore–polymer interactions, electro-osmotic flow, and the drift-diffusion of the macromolecule in electrolyte solutions. A unifying theory of these contributing factors will be described in the context of a few illustrative experimental data on DNA translocation and protein translocation through protein pores and solid-state nanopores. Future challenges to specific biological systems will be briefly discussed.     

Symmetry-restrained flexible fitting for symmetric EM maps

Many large biological macromolecules have inherent structural symmetry, being composed of a few distinct subunits, repeated in a symmetric array. These complexes are often not amenable to traditional high-resolution structural determination methods, but can be imaged in functionally relevant states using cryo-electron microscopy (cryo-EM). A number of methods for fitting atomic-scale structures into cryo-EM maps have been developed, including the molecular dynamics flexible fitting (MDFF) method. However, quality and resolution of the cryo-EM map are the major determinants of a method's success. In order to incorporate knowledge of structural symmetry into the fitting procedure, we developed the symmetry-restrained MDFF method. The new method adds to the cryo-EM map-derived potential further restraints on the allowed conformations of a complex during fitting, thereby improving the quality of the resultant structure. The benefit of using symmetry-based restraints during fitting, particularly for medium to low-resolution data, is demonstrated for three different systems.     

EM-fold: de novo atomic-detail protein structure determination from medium-resolution density maps

Electron density maps of membrane proteins or large macromolecular complexes are frequently only determined at medium resolution between 4?? and 10??, either by cryo-electron microscopy or X-ray crystallography. In these density maps, the general arrangement of secondary structure elements (SSEs) is revealed, whereas their directionality and connectivity remain elusive. We demonstrate that the topology of proteins with up to 250 amino acids can be determined from such density maps when combined with a computational protein folding protocol. Furthermore, we accurately reconstruct atomic detail in loop regions and amino acid side chains not visible in the experimental data. The EM-Fold algorithm assembles the SSEs de novo before atomic detail is added using Rosetta. In a benchmark of 27 proteins, the protocol consistently and reproducibly achieves models with root mean square deviation values <3??.