Related papers: Shapes of topological RNA structures
In this paper we present a novel framework for sequence to shape maps. These combinatorial maps realize exponentially many shapes, and have preimages which contain extended connected subgraphs of diameter n (neutral networks). We prove that…
In this paper we derive polynomial time algorithms that generate random $k$-noncrossing matchings and $k$-noncrossing RNA structures with uniform probability. Our approach employs the bijection between $k$-noncrossing matchings and…
We enumerate possible topologies of pseudoknots in single-stranded RNA molecules. We use a steepest-descent approximation in the large N matrix field theory, and a Feynman diagram formalism to describe the resulting pseudoknot structure.
We review a derivation of the numbers of RNA complexes of an arbitrary topology. These numbers are encoded in the free energy of the hermitian matrix model with potential V(x)=x^2/2-stx/(1-tx), where s and t are respective generating…
Accurate RNA structure modeling remains difficult because RNA backbones are highly flexible, non-canonical interactions are prevalent, and experimentally determined 3D structures are comparatively scarce. We introduce \emph{RiboSphere}, a…
We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed…
Shape graphs are complex geometrical structures commonly found in biological and anatomical systems. A shape graph is a collection of nodes, some connected by curvilinear edges with arbitrary shapes. Their high complexity stems from the…
Sequence data, such as DNA, RNA, and protein sequences, exhibit intricate, multi-scale structures that pose significant challenges for conventional analysis methods, particularly those relying on alignment or purely statistical…
The primary structure of a ribonucleic acid (RNA) molecule can be represented as a sequence of nucleotides (bases) over the alphabet {A, C, G, U}. The secondary or tertiary structure of an RNA is a set of base pairs which form bonds between…
The RNA inverse folding problem, a key challenge in RNA design, involves identifying nucleotide sequences that can fold into desired secondary structures, which are critical for ensuring molecular stability and function. The inherent…
This article proposes a topological method that extracts hierarchical structures of various amorphous solids. The method is based on the persistence diagram (PD), a mathematical tool for capturing shapes of multiscale data. The input to the…
The topological (or graph) structures of real-world networks are known to be predictive of multiple dynamic properties of the networks. Conventionally, a graph structure is represented using an adjacency matrix or a set of hand-crafted…
RNA molecules are known to form complex secondary structures including pseudoknots. A systematic framework for the enumeration, classification and prediction of secondary structures is critical to determine the biological significance of…
This paper is first-line research expanding GANs into graph topology analysis. By leveraging the hierarchical connectivity structure of a graph, we have demonstrated that generative adversarial networks (GANs) can successfully capture…
The relationship between sequences and secondary structures or shapes in RNA exhibits robust statistical properties summarized by three notions: (1) the notion of a typical shape (that among all sequences of fixed length certain shapes are…
An RNA molecule is structured on several layers. The primary and most obvious structure is its sequence of bases, i.e. a word over the alphabet {A,C,G,U}. The higher structure is a set of one-to-one base-pairings resulting in a…
Formation of base pairs between the nucleotides of an RNA sequence gives rise to a complex and often highly branched RNA structure. While numerous studies have demonstrated the functional importance of the high degree of RNA branching --…
In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud…
Networks are important representations in computer science to communicate structural aspects of a given system of interacting components. The evolution of a network has several topological properties that can provide us information on the…
This paper introduces and demonstrates a computational pipeline for the statistical analysis of shape graph datasets, namely geometric networks embedded in 2D or 3D spaces. Unlike traditional abstract graphs, our purpose is not only to…