A self-consistent formulation for analysis and generation of non-uniform DNA structures

A fully self-consistent formulation is described here for the analysis and generation of base-pairs in non-uniform DNA structures, in terms of various local parameters. It is shown that the internal "wedge parameters" are mathematically related to the parameters describing the base-pair orientation with respect to an external helix axis. Hence any one set of three translation and three rotation parameters are necessary and sufficient to completely describe the relative orientation of the base-pairs comprising a step (or doublet). A general procedure is outlined for obtaining an average or global helix axis from the local helix axes for each step. A graphical representation of the local helix axes in the form of a polar plot is also shown and its application for estimating the curvature of oligonucleotide structures is illustrated, with examples of both A and B type structures.     

A Self-Consistent Formulation for Analysis and Generation of Non-Uniform DNA Structures

Abstract

A fully self-consistent formulation is described here for the analysis and generation of base-pairs in non-uniform DNA structures, in terms of various local parameters. It is shown that the internal “wedge parameters” are mathematically related to the parameters describing the base-pair orientation with respect to an external helix axis. Hence any one set of three translation and three rotation parameters are necessary and sufficient to completely describe the relative orientation of the base-pairs comprising a step (or doublet). A general procedure is outlined for obtaining an average or global helix axis from the local helix axes for each step. A graphical representation of the local helix axes in the form of a polar plot is also shown and its application for estimating the curvature of oligonucleotide structures is illustrated, with examples of both A and B type structures.     

Topological classification of RNA structures

We present a novel topological classification of RNA secondary structures with pseudoknots. It is based on the topological genus of the circular diagram associated to the RNA base-pair structure. The genus is a positive integer number whose value quantifies the topological complexity of the folded RNA structure. In such a representation, planar diagrams correspond to pure RNA secondary structures and have zero genus, whereas non-planar diagrams correspond to pseudoknotted structures and have higher genus. The topological genus allows for the definition of topological folding motifs, similar in spirit to those introduced and commonly used in protein folding. We analyze real RNA structures from the databases Worldwide Protein Data Bank and Pseudobase and classify them according to their topological genus. For simplicity, we limit our analysis by considering only Watson-Crick complementary base pairs and G-U wobble base pairs. We compare the results of our statistical survey with existing theoretical and numerical models. We also discuss possible applications of this classification and show how it can be used for identifying new RNA structural motifs.     

Surface growth kinematics via local curve evolution

A mathematical framework is developed to model the kinematics of surface growth for objects that can be generated by evolving a curve in space, such as seashells and horns. Growth is dictated by a growth velocity vector field defined at every point on a generating curve. A local orthonormal basis is attached to each point of the generating curve and the velocity field is given in terms of the local coordinate directions, leading to a fully local and elegant mathematical structure. Several examples of increasing complexity are provided, and we demonstrate how biologically relevant structures such as logarithmic shells and horns emerge as analytical solutions of the kinematics equations with a small number of parameters that can be linked to the underlying growth process. Direct access to cell tracks and local orientation enables for connections to be made to the underlying growth process.     

Silver-Stained accessory structures on human sex chromosomes

Summary Using a combination of silver-staining and light microscopic techniques on human male meiotic preparations, it is feasible to study the morphology and behavior of both autosomal synaptonemal complexes and sex chromosome axes. During leptotene and early zygotene, the X and Y chromosomes are separate; their axes appearing as thin, filamentous structures. During late zygotene/early pachytene, the sex chromosomes come close to each other and a distinct sex vesicle is formed. We confirm the existence of a short synaptonemal complex between the terminal ends of the X and Y chromosomes. In our preparations, a number of accessory structures can be seen along the axes of the sex chromosomes. These structures appear to be similar in morphology to those previously observed in several other mammalian species.     

Visualization of conformational distribution of short to medium size segments in globular proteins and identification of local structural motifs

Analysis of the conformational distribution of polypeptide segments in a conformational space is the first step for understanding a principle of structural diversity of proteins. Here, we present a statistical analysis of protein local structures based on interatomic C(alpha) distances. Using principal component analysis (PCA) on the intrasegment C(alpha)-C(alpha) atomic distances, the conformational space of protein segments, which we call the protein segment universe, has been visualized, and three essential coordinate axes, suitable for describing the universe, have been identified. Three essential axes specified radius of gyration, structural symmetry, and separation of hairpin structures from other structures. Among the segments of arbitrary length, 6-22 residues long, the conservation of those axes was uncovered. Further application of PCA to the two largest clusters in the universe revealed local structural motifs. Although some of motifs have already been reported, we identified a possibly novel strand motif. We also showed that a capping box, which is one of the helix capping motifs, was separated into independent subclusters based on the C(alpha) geometry. Implications of the strand motif, which may play a role for protein-protein interaction, are discussed. The currently proposed method is useful for not only mapping the immense universe of protein structures but also identification of structural motifs.     

Definitions and analysis of DNA Holliday junction geometry

A number of single-crystal structures have now been solved of the four-stranded antiparallel stacked-X form of the Holliday junction. These structures demonstrate how base sequence, substituents, and drug and ion interactions affect the general conformation of this recombination intermediate. The geometry of junctions had previously been described in terms of a specific set of parameters that include: (i) the angle relating the ends of DNA duplexes arms of the junction (interduplex angle); (ii) the relative rotation of the duplexes about the helix axes of the stacked duplex arms (J_roll); and (iii) the translation of the duplexes along these helix axes (J_slide). Here, we present a consistent set of definitions and methods to accurately calculate each of these parameters based on the helical features of the stacked duplex arms in the single-crystal structures of the stacked-X junction, and demonstrate how each of these parameters contributes to an overall conformational feature of the structure. We show that the values for these parameters derived from global rather than local helical axes through the stacked bases of the duplex arms are the most representative of the stacked-X junction conformation. In addition, a very specific parameter (J_twist) is introduced which relates the relative orientation of the stacked duplex arms across the junction which, unlike the interduplex angle, is length independent. The results from this study provide a general means to relate the geometric features seen in the crystal structures to those determined in solution.     

Automated segmentation of molecular subunits in electron cryomicroscopy density maps

Electron cryomicroscopy (cryoEM) is capable of imaging large macromolecular machines composed of multiple components. However, it is currently only possible to achieve moderate resolution at which it may be possible to computationally extract the individual components in the machine. In this work, we present application details of an automated method for detecting and segmenting the components of a large machine in an experimentally determined density map. This method is applicable to object with and without symmetry and takes advantage of global and local symmetry axes if present. We have applied this segmentation algorithm to several cryoEM data sets already deposited in EMDB with various complexities, symmetries and resolutions and validated the results using manually segmented density and available structures of the components in the PDB. As such, automated segmentation could become a useful tool for the analysis of the ever-increasing number of structures of macromolecular machines derived from cryoEM.     

Topological mirror images in protein structure computation: an underestimated problem.

When calculating three-dimensional structures from NMR data, alternative solutions with very large RMS deviation can be obtained. Sometimes local or global inversions of the protein folding can be observed. We call these different solutions topological mirror images, as they keep the correct amino acid chirality. They are observed when the number of restraints is insufficient and represent different solutions from the same scalar information. Therefore they are common in small peptides where the NMR data are often limited and the secondary structure is not very well defined. They can also be observed in large molecules in regions of higher flexibility. In our experience the observation of topological mirror images is independent of the efficiency of sampling of the algorithm used. We present four examples of proteins with different size and folding. We also discuss ways to distinguish among the different solutions.     

Local selection rules that can determine specific pathways of DNA unknotting by type II DNA topoisomerases

We performed numerical simulations of DNA chains to understand how local geometry of juxtaposed segments in knotted DNA molecules can guide type II DNA topoisomerases to perform very efficient relaxation of DNA knots. We investigated how the various parameters defining the geometry of inter-segmental juxtapositions at sites of inter-segmental passage reactions mediated by type II DNA topoisomerases can affect the topological consequences of these reactions. We confirmed the hypothesis that by recognizing specific geometry of juxtaposed DNA segments in knotted DNA molecules, type II DNA topoisomerases can maintain the steady-state knotting level below the topological equilibrium. In addition, we revealed that a preference for a particular geometry of juxtaposed segments as sites of strand-passage reaction enables type II DNA topoisomerases to select the most efficient pathway of relaxation of complex DNA knots. The analysis of the best selection criteria for efficient relaxation of complex knots revealed that local structures in random configurations of a given knot type statistically behave as analogous local structures in ideal geometric configurations of the corresponding knot type.     

Structural complexity of species assemblages and spatial scale of community organization: A case study of marine benthos

In this paper, definitions and measures of complexity with regard to biological communities are briefly considered. A new topological approach that considers the community's complexity in terms of groups of species coherently varied in space or time is proposed. For a given set of samples, the number of such groups is related to the minimal number M of axes necessary to represent the original configuration of the data set. I interpret this “minimal dimensionality of structure” as the number of independent factors of structural variability and its normalized value M/M_MAX as a measure of the organizational complexity of a community. The M value can be estimated as the number of significant axes obtained by ordination procedures. The percentage of total variance explained by these axes, T, is used as measure of structural rigidity. This approach is applied to data on the multi-scaled spatial distribution of marine benthic ciliates and macrofauna. Both the M and the T values obtained by principal component analysis show significant scale-dependence with an evident threshold at some critical area, with values of zero below this threshold, then increasing sharply as the area extends beyond the threshold. The critical scale of community organization ranges from hundreds of meters to kilometers for macroorganisms, whereas several meters are sufficient when considering ciliates.     

Superhelical DNA with local substructures. A generalization of the topological constraint in terms of the intersection number and the ladder-like correspondence surface

The presence of certain local structural elements in superhelical DNA, such as cruciforms and denatured loops, complicates the topological and geometric analysis of these molecules. In particular, the duplex axis is often difficult to define. In consequence, the usual conservation condition, Lk = Tw + Wr, is often inapplicable as formulated in terms of the winding of either strand of the DNA about the duplex axis. We present here a more general formulation of the topological conservation condition in terms of a model in which the two strands of DNA are regarded as twisting about one another, and in which one of the two strands is considered to writhe. We define a ladder-like correspondence surface, which connects the two strands nd is independent of whether or not a unique duplex axis is locally available. These considerations lead to the definition of a new topological property of superhelical DNA, the intersection number, In. This quantity describes the complexity of a local structural element; in the case of a cruciform, for example, the intersection number is a measure of the number of duplex turns removed from the major segment of the DNA by the cruciform formation. In more general terms, the topological constraint applicable to closed circular DNA is given by Lk(W,C) + In(S,C) = Tw(W,C) + Wr (C), where W and C represent the two strands of the DNA and S is the ladder-like correspondence surface that connects the two strands.     

A rapid method for exploring the protein structure universe

We have developed an automatic protein fingerprinting method for the evaluation of protein structural similarities based on secondary structure element compositions, spatial arrangements, lengths, and topologies. This method can rapidly identify proteins sharing structural homologies as we demonstrate with five test cases: the globins, the mammalian trypsinlike serine proteases, the immunoglobulins, the cupredoxins, and the actinlike ATPase domain-containing proteins. Principal components analysis of the similarity distance matrix calculated from an all-by-all comparison of 1,031 unique chains in the Protein Data Bank has produced a distribution of structures within a high-dimensional structural space. Fifty percent of the variance observed for this distribution is bounded by six axes, two of which encode structural variability within two large families, the immunoglobulins and the trypsinlike serine proteases. Many aspects of the spatial distribution remain stable upon reduction of the database to 140 proteins with minimal family overlap. The axes correlated with specific structural families are no longer observed. A clear hierarchy of organization is seen in the arrangement of protein structures in the universe. At the highest level, protein structures populate regions corresponding to the all-alpha, all-beta, and alpha/beta superfamilies. Large protein families are arranged along family-specific axes, forming local densely populated regions within the space. The lowest level of organization is intrafamilial; homologous structures are ordered by variations in peripheral secondary structure elements or by conformational shifts in the tertiary structure.     

Topological constraints strongly affect chromatin reconstitution in silico

The fundamental building block of chromatin, and of chromosomes, is the nucleosome, a composite material made up from DNA wrapped around a histone octamer. In this study we provide the first computer simulations of chromatin self-assembly, starting from DNA and histone proteins, and use these to understand the constraints which are imposed by the topology of DNA molecules on the creation of a polynucleosome chain. We take inspiration from the in vitro chromatin reconstitution protocols which are used in many experimental studies. Our simulations indicate that during self-assembly, nucleosomes can fall into a number of topological traps (or local folding defects), and this may eventually lead to the formation of disordered structures, characterised by nucleosome clustering. Remarkably though, by introducing the action of topological enzymes such as type I and II topoisomerase, most of these defects can be avoided and the result is an ordered 10-nm chromatin fibre. These findings provide new insight into the biophysics of chromatin formation, both in the context of reconstitution in vitro and in terms of the topological constraints which must be overcome during de novo nucleosome formation in vivo, e.g. following DNA replication or repair.     

Exploring local structural organization of metabolic networks using subgraph patterns

Metabolic networks of many cellular organisms share global statistical features. Their connectivity distributions follow the long-tailed power law and show the small-world property. In addition, their modular structures are organized in a hierarchical manner. Although the global topological organization of metabolic networks is well understood, their local structural organization is still not clear. Investigating local properties of metabolic networks is necessary to understand the nature of metabolism in living organisms. To identify the local structural organization of metabolic networks, we analysed the subgraphs of metabolic networks of 43 organisms from three domains of life. We first identified the network motifs of metabolic networks and identified the statistically significant subgraph patterns. We then compared metabolic networks from different domains and found that they have similar local structures and that the local structure of each metabolic network has its own taxonomical meaning. Organisms closer in taxonomy showed similar local structures. In addition, the common substrates of 43 metabolic networks were not randomly distributed, but were more likely to be constituents of cohesive subgraph patterns.     

Re-orienting in space: do animals use global or local geometry strategies?

Here we compare whether birds encode surface geometry using principal axes, medial axes or local geometry. Birds were trained to locate hidden food in two geometrically identical corners of a rectangular arena and subsequently tested in an L-shaped arena. The chicks showed a primary local geometry strategy, and a secondary medial axes strategy, whereas the pigeons showed a medial axes strategy. Neither species showed behaviour supportive of the use of principal axes. This is, to our knowledge, the first study to directly examine these three current theories of geometric encoding.     

Reorienting in Virtual 3D Environments: Do Adult Humans Use Principal Axes,Medial Axes or Local Geometry?

Studies have shown that animals, including humans, use the geometric properties of environments to orient. It has been proposed that orientation is accomplished primarily by encoding the principal axes (i.e., global geometry) of an environment. However, recent research has shown that animals use local information such as wall length and corner angles as well as local shape parameters (i.e., medial axes) to orient. The goal of the current study was to determine whether adult humans reorient according to global geometry based on principal axes or whether reliance is on local geometry such as wall length and sense information or medial axes. Using a virtual environment task, participants were trained to select a response box located at one of two geometrically identical corners within a featureless rectangular-shaped environment. Participants were subsequently tested in a transformed L-shaped environment that allowed for a dissociation of strategies based on principal axes, medial axes and local geometry. Results showed that participants relied primarily on a medial axes strategy to reorient in the L-shaped test environment. Importantly, the search behaviour of participants could not be explained by a principal axes-based strategy.