Topological Genus Research Articles

On RNA–RNA interaction structures of fixed topological genus

Interacting RNA complexes are studied via bicellular maps using a filtration via their topological genus. Our main result is a new bijection for RNA–RNA interaction structures and a linear time uniform sampling algorithm for RNA complexes of fixed topological genus. The bijection allows to either reduce the topological genus of a bicellular map directly, or to lose connectivity by decomposing the complex into a pair of single stranded RNA structures. Our main result is proved bijectively. It provides an explicit algorithm of how to rewire the corresponding complexes and an unambiguous decomposition grammar. Using the concept of genus induction, we construct bicellular maps of fixed topological genus g uniformly in linear time. We present various statistics on these topological RNA complexes and compare our findings with biological complexes. Furthermore we show how to construct loop-energy based complexes using our decomposition grammar.

ON THE CONNECTEDNESS OF THE BRANCH LOCI OF NON-ORIENTABLE UNBORDERED KLEIN SURFACES OF LOW GENUS

AbstractThis paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

Combinatorics of γ-Structures

In this article we study canonical γ-structures, a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A γ-structure is composed of specific building blocks that have topological genus less than or equal to γ, where composition means concatenation and nesting of such blocks. Our main result is the derivation of the generating function of γ-structures via symbolic enumeration using so called irreducible shadows. We furthermore recursively compute the generating polynomials of irreducible shadows of genus ≤ γ. The γ-structures are constructed via γ-matchings. For 1 ≤ γ ≤ 10, we compute Puiseux expansions at the unique, dominant singularities, allowing us to derive simple asymptotic formulas for the number of γ-structures.

Shapes of interacting RNA complexes.

Shapes of interacting RNA complexes are studied using a filtration via their topological genus. A shape of an RNA complex is obtained by (iteratively) collapsing stacks and eliminating hairpin loops. This shape projection preserves the topological core of the RNA complex, and for fixed topological genus there are only finitely many such shapes. Our main result is a new bijection that relates the shapes of RNA complexes with shapes of RNA structures. This allows for computing the shape polynomial of RNA complexes via the shape polynomial of RNA structures. We furthermore present a linear time uniform sampling algorithm for shapes of RNA complexes of fixed topological genus.

Double Coverings of Non-orientable Riemann Surfaces Ramified over Discrete Sets

We find an upper bound for the number of non-equivalent twofold coverings of closed non-orientable surfaces with conformal structures, ramified over sets with isolated points, by a given surface of topological genus \(g \ge 3\) with such structure and we characterize configurations for which our bounds are attained. We split our study according to whether the covering surface is symmetric or not and whether the corresponding coverings are decomposable or not in a suitable sense.

Topology in Biology: Singularities and Surgery Transformations in Metazoan Development and Evolution

The review presents a topological description and interpretation (analysis) of some events in metazoan development and evolution through the use of well-known mathematical concepts and theorems (using topological approach). It is the topological language that can provide strict and adequate description of various phenomena in developmental and evolutionary transformations. Topological singularities inevitably arising and transforming during early development destroy the preexisting pattern of symmetry. The symmetry breaking of preexisting spatial pattern plays a critical role in biological morphogenesis in development and evolution. Some events of early development are interpreted in terms of symmetry breakdown and related to well-known mathematical theorems. A topological inevitability of some developmental events through the use of classical topological concepts is discussed. The topological approach makes it possible to consider the succession of spherical surgeries, which change the topological genus of an animal body surface. We model the biological shape as a set of smooth, closed, oriented surfaces—membrane or epithelial layers. Membrane and epithelial surfaces are boundary layers, interfaces between a living structure and its environment, ensuring metabolism. Toroid forms as well as fractal structures in metazoans can be considered as functionally optimized biological design and attractors in biological morphogenesis. The epithelial surface is an interface between the internal medium of an organism and the outside environmental medium; topological and fractal transformations during metazoan evolution and development increase this interface, ensuring better adaptation of organism to the environment. Fractal structures as well as toroid forms can be considered as a functionally optimized design in Metazoa. Topological methodology reveals a certain set of topological rules constraining and directing biological morphogenesis during evolution and development.

RAGS: Rational geometric splines for surfaces of arbitrary topology

A construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity is proposed. The obtained splines are a direct generalization of bivariate polynomial splines on planar partitions. They are defined as composite functions consisting of rational functions and are parametrized by a single parameter domain, which is a piecewise planar surface, such as a triangulation of a cloud of 3D points. The idea of the construction is to utilize linear rational transformations (or transition maps) to endow the piecewise planar surface with a particular C∞-differentiable structure appropriate for defining rational splines.

Isogeometric spline forests

In this paper we present isogeometric spline forests. An isogeometric spline forest is a hierarchical spline representation capable of representing surfaces or volumes of arbitrarily complex geometry and topological genus. Spline forests can accomodate arbitrary degree and smoothness in the underlying hierarchical basis as well as non-uniform knot interval configurations. We describe adaptive h-refinement and coarsening algorithms for isogeometric spline forests and develop a Bézier extraction framework which provides a simple and efficient single level finite element description of the complex multi-level, unstructured hierarchical spline basis. We then demonstrate the potential of spline forests as a basis for analysis in the context of transient advection–diffusion problems where fully integrated adaptivity is demonstrated for the first time in an isogeometric simulation. In all cases, the adaptive process remains local (even in the case of moving fronts) and preserves exact geometry at the coarsest level of the discretization. The accuracy and robustness of the approach is demonstrated in all cases.

Repulsive interactions in quantum Hall systems as a pairing problem

A subtle relation between quantum Hall physics and the phenomenon of pairing is unveiled. By use of second quantization, we establish a connection between (i) a broad class of rotationally symmetric two-body interactions within the lowest Landau level and (ii) integrable hyperbolic Richardson-Gaudin-type Hamiltonians that arise in $({p}_{x}+i{p}_{y})$ superconductivity. Specifically, we show that general Haldane pseudopotentials (and their sums) can be expressed as a sum of repulsive noncommuting $({p}_{x}+i{p}_{y})$-type pairing Hamiltonians. The determination of the spectrum and individual null spaces of each of these noncommuting Richardson-Gaudin-type Hamiltonians is nontrivial yet is Bethe ansatz solvable. For the Laughlin sequence, it is observed that this problem is frustration free and zero-energy ground states lie in the common null space of all of these noncommuting Hamiltonians. This property allows for the use of a new truncated basis of pairing configurations in which to express Laughlin states at general filling factors. We prove separability of arbitrary Haldane pseudopotentials, providing explicit expressions for their second quantized forms, and further show by explicit construction how to exploit the topological equivalence between different geometries (disk, cylinder, and sphere) sharing the same topological genus number, in the second quantized formalism, through similarity transformations. As an application of the second quantized approach, we establish a ``squeezing principle'' that applies to the zero modes of a general class of Hamiltonians, which includes but is not limited to Haldane pseudopotentials. We also show how one may establish (bounds on) ``incompressible filling factors'' for those Hamiltonians. By invoking properties of symmetric polynomials, we provide explicit second quantized quasihole generators; the generators that we find directly relate to bosonic chiral edge modes and further make aspects of dimensional reduction in the quantum Hall systems precise.

Generation of RNA pseudoknot structures with topological genus filtration

In this paper we present a sampling framework for RNA structures of fixed topological genus. We introduce a novel, linear time, uniform sampling algorithm for RNA structures of fixed topological genus g, for arbitrary g>0. Furthermore we develop a linear time sampling algorithm for RNA structures of fixed topological genus g that are weighted by a simplified, loop-based energy functional. For this process the partition function of the energy functional has to be computed once, which has O(n2) time complexity.

On Topological RNA Interaction Structures

Recently a folding algorithm of topological RNA pseudoknot structures was presented in Reidys et al. (2011). This algorithm folds single-stranded γ-structures, that is, RNA structures composed by distinct motifs of bounded topological genus. In this article, we set the theoretical foundations for the folding of the two backbone analogues of γ structures: the RNA γ-interaction structures. These are RNA-RNA interaction structures that are constructed by a finite number of building blocks over two backbones having genus at most γ. Combinatorial properties of γ-interaction structures are of practical interest since they have direct implications for the folding of topological interaction structures. We compute the generating function of γ-interaction structures and show that it is algebraic, which implies that the numbers of interaction structures can be computed recursively. We obtain simple asymptotic formulas for 0- and 1-interaction structures. The simplest class of interaction structures are the 0-interaction structures, which represent the two backbone analogues of secondary structures.

The full group of automorphisms of non-orientable unbordered Klein surfaces of topological genus 3, 4 and 5

An important problem in the study of Riemann and Klein surfaces is determining their full automorphism groups. Up to now only very partial results are known, concerning surfaces of low genus or families of surfaces with special properties. This paper deals with non-orientable unbordered Klein surfaces. In this case the solution of the problem is known for surfaces of genus 1 and 2, and for hyperelliptic surfaces. Here we explicitly obtain the full automorphism group of all surfaces of genus 3, 4 and 5.

McGenus: a Monte Carlo algorithm to predict RNA secondary structures with pseudoknots

We present McGenus, an algorithm to predict RNA secondary structures with pseudoknots. The method is based on a classification of RNA structures according to their topological genus. McGenus can treat sequences of up to 1000 bases and performs an advanced stochastic search of their minimum free energy structure allowing for non-trivial pseudoknot topologies. Specifically, McGenus uses a Monte Carlo algorithm with replica exchange for minimizing a general scoring function which includes not only free energy contributions for pair stacking, loop penalties, etc. but also a phenomenological penalty for the genus of the pairing graph. The good performance of the stochastic search strategy was successfully validated against TT2NE which uses the same free energy parametrization and performs exhaustive or partially exhaustive structure search, albeit for much shorter sequences (up to 200 bases). Next, the method was applied to other RNA sets, including an extensive tmRNA database, yielding results that are competitive with existing algorithms. Finally, it is shown that McGenus highlights possible limitations in the free energy scoring function. The algorithm is available as a web server at http://ipht.cea.fr/rna/mcgenus.php.

On the combinatorics of sparsification

BackgroundWe study the sparsification of dynamic programming based on folding algorithms of RNA structures. Sparsification is a method that improves significantly the computation of minimum free energy (mfe) RNA structures.ResultsWe provide a quantitative analysis of the sparsification of a particular decomposition rule, Λ∗. This rule splits an interval of RNA secondary and pseudoknot structures of fixed topological genus. Key for quantifying sparsifications is the size of the so called candidate sets. Here we assume mfe-structures to be specifically distributed (see Assumption 1) within arbitrary and irreducible RNA secondary and pseudoknot structures of fixed topological genus. We then present a combinatorial framework which allows by means of probabilities of irreducible sub-structures to obtain the expectation of the Λ∗-candidate set w.r.t. a uniformly random input sequence. We compute these expectations for arc-based energy models via energy-filtered generating functions (GF) in case of RNA secondary structures as well as RNA pseudoknot structures. Furthermore, for RNA secondary structures we also analyze a simplified loop-based energy model. Our combinatorial analysis is then compared to the expected number of Λ∗-candidates obtained from the folding mfe-structures. In case of the mfe-folding of RNA secondary structures with a simplified loop-based energy model our results imply that sparsification provides a significant, constant improvement of 91% (theory) to be compared to an 96% (experimental, simplified arc-based model) reduction. However, we do not observe a linear factor improvement. Finally, in case of the “full” loop-energy model we can report a reduction of 98% (experiment).ConclusionsSparsification was initially attributed a linear factor improvement. This conclusion was based on the so called polymer-zeta property, which stems from interpreting polymer chains as self-avoiding walks. Subsequent findings however reveal that the O(n) improvement is not correct. The combinatorial analysis presented here shows that, assuming a specific distribution (see Assumption 1), of mfe-structures within irreducible and arbitrary structures, the expected number of Λ∗-candidates is Θ(n2). However, the constant reduction is quite significant, being in the range of 96%. We furthermore show an analogous result for the sparsification of the Λ∗-decomposition rule for RNA pseudoknotted structures of genus one. Finally we observe that the effect of sparsification is sensitive to the employed energy model.

The topological filtration of [formula omitted]-structures

In this paper we study γ-structures filtered by topological genus. γ-structures are a class of RNA pseudoknot structures that plays a key role in the context of polynomial time folding of RNA pseudoknot structures. A γ-structure is composed by specific building blocks, that have topological genus less than or equal to γ, where composition means concatenation and nesting of such blocks. Our main results are the derivation of a new bivariate generating function for γ-structures via symbolic methods, the singularity analysis of the solutions and a central limit theorem for the distribution of topological genus in γ-structures of given length. In our derivation specific bivariate polynomials play a central role. Their coefficients count particular motifs of fixed topological genus and they are of relevance in the context of genus recursion and novel folding algorithms.

Geometric characterization of nanoporous metals

Nanoporous metals made by dealloying possess significant geometric complexity—they are random, bicontinuous structures that also possess bubbles within ligaments, regions of very high negative, positive, and saddlepoint curvature, and significant polyfaceting. Here we introduce methods to geometrically quantify the structure of nanoporous metals employing simulated model nanoporous metals generated via large-scale kinetic Monte Carlo simulations as the basis of discussion. A method is introduced to transform these simulated structures into smooth triangulated meshes using new mesh-smoothing algorithms that hybridize mean curvature flow and signal processing approaches to mesh fairing. The technique is assessed by comparing the exact genus of high-genus meshes with the genus calculated via the Gauss–Bonnet formula, and works well to find the local curvature at all points of simulated surfaces of high topological genus. Specific geometric quantification of nanoporous metals is discussed for two quantities: (i) the relative surface area fraction of different crystal facets, which is important for catalysis; and (ii) the curvature distribution on the surface of porous metals, important for applications in which high curvature features are active (e.g. optical sensing).

On electromagnetic waves in complex linear media in nonsmooth domains

Electromagnetic wave propagation in more complex linear materials such as bi‐anisotropic media have come to a considerable attention within the last 15–20 years. We shall propose a general framework to approach a class of highly complex materials. Such problems have been extensively studied mostly in the time‐harmonic case. In this paper, we focus on the time‐dependent case. A well‐posedness result for a large class of media is obtained. We also analyze Drude–Born–Fedorov type media in nonsmooth domains of arbitrary topological genus within this framework. Copyright © 2012 John Wiley & Sons, Ltd.

On the page number of RNA secondary structures with pseudoknots

Let S denote the set of (possibly noncanonical) base pairs {i, j } of an RNA tertiary structure; i.e. {i, j} ∈ S if there is a hydrogen bond between the ith and jth nucleotide. The page number of S, denoted π(S), is the minimum number k such that Scan be decomposed into a disjoint union of k secondary structures. Here, we show that computing the page number is NP-complete; we describe an exact computation of page number, using constraint programming, and determine the page number of a collection of RNA tertiary structures, for which the topological genus is known. We describe an approximation algorithm from which it follows that ω(S) ≤ π(S) ≤ ω(S) ・log n,where the clique number of S, ω(S), denotes the maximum number of base pairs that pairwise cross each other.

Mechanism of Coarsening and Bubble Formation in High-Genus Nanoporous Metals

Coarsening of crystalline nanoporous metals involves complex changes in topology associated with the reduction of genus via both ligament pinch-off and void bubble formation. Although void bubbles in metals are often associated with vacancy agglomeration, we use large-scale kinetic Monte Carlo simulations to show that both bubble formation and ligament pinch-off are natural results of a surface-diffusion-controlled solid-state Rayleigh instability that controls changes in the topology of the porous material during coarsening. This result is used to find an effective activation energy for coarsening in nanoporous metals that is associated with the reduction of topological genus, and not the reduction of local surface roughness.

TT2NE: a novel algorithm to predict RNA secondary structures with pseudoknots

We present TT2NE, a new algorithm to predict RNA secondary structures with pseudoknots. The method is based on a classification of RNA structures according to their topological genus. TT2NE is guaranteed to find the minimum free energy structure regardless of pseudoknot topology. This unique proficiency is obtained at the expense of the maximum length of sequences that can be treated, but comparison with state-of-the-art algorithms shows that TT2NE significantly improves the quality of predictions. Analysis of TT2NE's incorrect predictions sheds light on the need to study how sterical constraints limit the range of pseudoknotted structures that can be formed from a given sequence. An implementation of TT2NE on a public server can be found at http://ipht.cea.fr/rna/tt2ne.php.