Abstract
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.
Highlights
Complex of RNA Structures.Ribonucleic acid (RNA) is a biomolecule that folds into a helical configuration of its primary sequence by forming hydrogen bonds between pairs of nucleotides
As a secondary structure can be uniquely decomposed into loops, the free energy of a structure is calculated as the sum of the energy of its individual loops [3]
In order to compute the homology of τ-structures, we shall first compute the homology of components separately and integrate this information via the Mayer–Vietoris sequence [18,19]
Summary
The most prominent class of coarse-grained structures are the RNA secondary structures [1,2] These are contact structures that can be represented as diagrams where vertices are nucleotides and arcs are base pairs drawn in the upper half-plane. In this paper we study triples, (S, T, φ), consisting of an ordered pair of RNA secondary structures, (S, T ), [1,2,10,11,12,13] together with a non-crossing partial matching, φ, between the two backbones We shall denote such a triple a transition structure or τ-structure and note that the mapping φ relates homologous bases between two underlying RNA sequences. The loop complex of a bi-structure exhibits only a nontrivial zeroth and second homology, the latter being freely generated by crossing components [7,8]. The first homology H1 ( R) is zero and the second homology H2 ( R) is free and its rank equals the number of crossing components of the bi-structure R [8]
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