scholarly journals On the Loop Homology of a Certain Complex of RNA Structures

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1749
Author(s):  
Thomas J. X. Li ◽  
Christian M. Reidys
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Secondary Structures ◽  
Rna Structures ◽  
Sequence Structure ◽  
Evolutionary Transitions ◽  
Partial Matching ◽  
Rna Sequence ◽  
Homology Groups ◽  
Topological Framework

In this paper, we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence–structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure and that the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

scholarly journals On the Loop Homology of a Certain Complex of RNA Structures

2021 ◽  
Author(s):  
Thomas J.X. Li ◽  
Christian M. Reidys
Keyword(s):  
Euler Characteristic ◽  
Homology Group ◽  
Secondary Structures ◽  
Rna Structures ◽  
Sequence Structure ◽  
Evolutionary Transitions ◽  
Partial Matching ◽  
Rna Sequence ◽  
Homology Groups ◽  
Topological Framework

In this paper we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence-structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure, and the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

scholarly journals Exponentially few RNA structures are designable

2019 ◽  
Author(s):  
Hua-Ting Yao ◽  
Mireille Regnier ◽  
Cedric Chauve ◽  
Yann Ponty
Keyword(s):  
Secondary Structure ◽  
Secondary Structures ◽  
Rna Structures ◽  
Folding Model ◽  
Rna Sequences ◽  
Rna Sequence ◽  
Energy Models ◽  
Rna Design ◽  
Additional Constraints ◽  
Alternative Structure

ABSTRACTThe problem of RNA design attempts to construct RNA sequences that perform a predefined biological function, identified by several additional constraints. One of the foremost objective of RNA design is that the designed RNA sequence should adopt a predefined target secondary structure preferentially to any alternative structure, according to a given metrics and folding model. It was observed in several works that some secondary structures are undesignable, i.e. no RNA sequence can fold into the target structure while satisfying some criterion measuring how preferential this folding is compared to alternative conformations.In this paper, we show that the proportion of designable secondary structures decreases exponentially with the size of the target secondary structure, for various popular combinations of energy models and design objectives. This exponential decay is, at least in part, due to the existence of undesignable motifs, which can be generically constructed, and jointly analyzed to yield asymptotic upper-bounds on the number of designable structures.

Placozoa: at least two

Biologia ◽  
2007 ◽  
Vol 62 (6) ◽  
Author(s):  
Matthias Wolf ◽  
Christian Selig ◽  
Tobias Müller ◽  
Nicole Philippi ◽  
Thomas Dandekar ◽  
...  
Keyword(s):  
Secondary Structure ◽  
Internal Transcribed Spacer ◽  
Distinct Species ◽  
Secondary Structures ◽  
Its2 Sequence ◽  
Structure Alignment ◽  
Sequence Structure ◽  
Rna Sequence ◽  
Internal Transcribed Spacer 2 ◽  
Compensatory Base Changes

AbstractIt was shown that compensatory base changes (CBCs) in internal transcribed spacer 2 (ITS2) sequence-structure alignments can be used for distinguishing species. Using the ITS2 Database in combination with 4SALE — a tool for synchronous RNA sequence and secondary structure alignment and editing — in this study we present an in-depth CBC analysis for placozoan ITS2 sequences and their respective secondary structures. This analysis indicates at least two distinct species in Trichoplax (Placozoa) supporting a recently suggested hypothesis, that Placozoa is “no longer a phylum of one”.

scholarly journals One-relator groups and algebras related to polyhedral products

2021 ◽  
pp. 1-20
Author(s):  
Jelena Grbić ◽  
George Simmons ◽  
Marina Ilyasova ◽  
Taras Panov
Keyword(s):  
Homology Group ◽  
Homotopy Theory ◽  
Commutator Subgroup ◽  
Homology Groups ◽  
Homological Characterization ◽  
One Relator Groups ◽  
Combinatorial Condition ◽  
The Moment ◽  
Necessary And Sufficient

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

scholarly journals Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

scholarly journals Khovanov Homology of Three-Strand Braid Links

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 720
Author(s):  
Young Kwun ◽  
Abdul Nizami ◽  
Mobeen Munir ◽  
Zaffar Iqbal ◽  
Dishya Arshad ◽  
...  
Keyword(s):  
Euler Characteristic ◽  
Jones Polynomial ◽  
Khovanov Homology ◽  
Homology Groups ◽  
Link Invariants ◽  
Chain Complexes ◽  
Chain Homotopy

Khovanov homology is a categorication of the Jones polynomial. It consists of graded chain complexes which, up to chain homotopy, are link invariants, and whose graded Euler characteristic is equal to the Jones polynomial of the link. In this article we give some Khovanov homology groups of 3-strand braid links Δ 2 k + 1 = x 1 2 k + 2 x 2 x 1 2 x 2 2 x 1 2 ⋯ x 2 2 x 1 2 x 1 2 , Δ 2 k + 1 x 2 , and Δ 2 k + 1 x 1 , where Δ is the Garside element x 1 x 2 x 1 , and which are three out of all six classes of the general braid x 1 x 2 x 1 x 2 ⋯ with n factors.

scholarly journals A heuristic approach for detecting RNA H-type pseudoknots

2005 ◽  
Vol 21 (17) ◽  
pp. 3501-3508 ◽  
Author(s):  
Chun-Hsiang Huang ◽  
Chin Lung Lu ◽  
Hsien-Tai Chiu
Keyword(s):  
Web Server ◽  
Heuristic Approach ◽  
Biological Processes ◽  
Rna Structures ◽  
Online Analysis ◽  
Rna Sequence ◽  
Associated Functions ◽  
Almost All ◽  
The Web

Abstract Motivation RNA H-type pseudoknots are ubiquitous pseudoknots that are found in almost all classes of RNA and thought to play very important roles in a variety of biological processes. Detection of these RNA H-type pseudoknots can improve our understanding of RNA structures and their associated functions. However, the currently existing programs for detecting such RNA H-type pseudoknots are still time consuming and sometimes even ineffective. Therefore, efficient and effective tools for detecting the RNA H-type pseudoknots are needed. Results In this paper, we have adopted a heuristic approach to develop a novel tool, called HPknotter, for efficiently and accurately detecting H-type pseudoknots in an RNA sequence. In addition, we have demonstrated the applicability and effectiveness of HPknotter by testing on some sequences with known H-type pseudoknots. Our approach can be easily extended and applied to other classes of more general pseudoknots. Availability The web server of our HPknotter is available for online analysis at http://bioalgorithm.life.nctu.edu.tw/HPKNOTTER/ Contact [email protected], [email protected]

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