🍩 Database of Original & Non-Theoretical Uses of Topology
(found 7 matches in 0.001915s)
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Construction of Symbolic Dynamics From Experimental Time Series (1999)
K. Mischaikow, M. Mrozek, J. Reiss, A. SzymczakAbstract
Symbolic dynamics play a central role in the description of the evolution of nonlinear systems. Yet there are few methods for determining symbolic dynamics of chaotic data. One difficulty is that the data contains random fluctuations associated with the experimental process. Using data obtained from a magnetoelastic ribbon experiment we show how a topological approach that allows for experimental error and bounded noise can be used to obtain a description of the dynamics in terms of subshift dynamics on a finite set of symbols. -
Characterizing Scales of Genetic Recombination and Antibiotic Resistance in Pathogenic Bacteria Using Topological Data Analysis (2014)
Kevin J. Emmett, Raul RabadanAbstract
Pathogenic bacteria present a large disease burden on human health. Control of these pathogens is hampered by rampant lateral gene transfer, whereby pathogenic strains may acquire genes conferring resistance to common antibiotics. Here we introduce tools from topological data analysis to characterize the frequency and scale of lateral gene transfer in bacteria, focusing on a set of pathogens of significant public health relevance. As a case study, we examine the spread of antibiotic resistance in Staphylococcus aureus. Finally, we consider the possible role of the human microbiome as a reservoir for antibiotic resistance genes. -
Persistent Homology Analysis of Protein Structure, Flexibility, and Folding (2014)
Kelin Xia, Guo-Wei WeiAbstract
SUMMARYProteins are the most important biomolecules for living organisms. The understanding of protein structure, function, dynamics, and transport is one of the most challenging tasks in biological science. In the present work, persistent homology is, for the first time, introduced for extracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. The method of slicing is proposed to track the geometric origin of protein topological invariants. Both all-atom and coarse-grained representations of MTFs are constructed. A new cutoff-like filtration is proposed to shed light on the optimal cutoff distance in elastic network models. On the basis of the correlation between protein compactness, rigidity, and connectivity, we propose an accumulated bar length generated from persistent topological invariants for the quantitative modeling of protein flexibility. To this end, a correlation matrix-based filtration is developed. This approach gives rise to an accurate prediction of the optimal characteristic distance used in protein B-factor analysis. Finally, MTFs are employed to characterize protein topological evolution during protein folding and quantitatively predict the protein folding stability. An excellent consistence between our persistent homology prediction and molecular dynamics simulation is found. This work reveals the topology–function relationship of proteins. Copyright © 2014 John Wiley & Sons, Ltd. -
Topology Identifies Emerging Adaptive Mutations in SARS-CoV-2 (2021)
Michael Bleher, Lukas Hahn, Juan Angel Patino-Galindo, Mathieu Carriere, Ulrich Bauer, Raul Rabadan, Andreas OttAbstract
The COVID-19 pandemic has lead to a worldwide effort to characterize its evolution through the mapping of mutations in the genome of the coronavirus SARS-CoV-2. Ideally, one would like to quickly identify new mutations that could confer adaptive advantages (e.g. higher infectivity or immune evasion) by leveraging the large number of genomes. One way of identifying adaptive mutations is by looking at convergent mutations, mutations in the same genomic position that occur independently. However, the large number of currently available genomes precludes the efficient use of phylogeny-based techniques. Here, we establish a fast and scalable Topological Data Analysis approach for the early warning and surveillance of emerging adaptive mutations based on persistent homology. It identifies convergent events merely by their topological footprint and thus overcomes limitations of current phylogenetic inference techniques. This allows for an unbiased and rapid analysis of large viral datasets. We introduce a new topological measure for convergent evolution and apply it to the GISAID dataset as of February 2021, comprising 303,651 high-quality SARS-CoV-2 isolates collected since the beginning of the pandemic. We find that topologically salient mutations on the receptor-binding domain appear in several variants of concern and are linked with an increase in infectivity and immune escape, and for many adaptive mutations the topological signal precedes an increase in prevalence. We show that our method effectively identifies emerging adaptive mutations at an early stage. By localizing topological signals in the dataset, we extract geo-temporal information about the early occurrence of emerging adaptive mutations. The identification of these mutations can help to develop an alert system to monitor mutations of concern and guide experimentalists to focus the study of specific circulating variants. -
Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution (2020)
Michael Lesnick, Raúl Rabadán, Daniel I. S. RosenbloomAbstract
A topological approach to the study of genetic recombination, based on persistent homology, was introduced by Chan, Carlsson, and Rabadán in 2013. This associates a sequence of signatures called barcodes to genomic data sampled from an evolutionary history. In this paper, we develop theoretical foundations for this approach. First, we present a novel formulation of the underlying inference problem. Specifically, we introduce and study the novelty profile, a simple, stable statistic of an evolutionary history which not only counts recombination events but also quantifies how recombination creates genetic diversity. We propose that the (hitherto implicit) goal of the topological approach to recombination is the estimation of novelty profiles. We then study the problem of obtaining a lower bound on the novelty profile using barcodes. We focus on a low-recombination regime, where the evolutionary history can be described by a directed acyclic graph called a galled tree, which differs from a tree only by isolated topological defects. We show that in this regime, under a complete sampling assumption, the \$1\textasciicircum\mathrm\st\\$ barcode yields a lower bound on the novelty profile, and hence on the number of recombination events. For \$i\textgreater1\$, the \$i\textasciicircum\\mathrm\th\\\$ barcode is empty. In addition, we use a stability principle to strengthen these results to ones which hold for any subsample of an arbitrary evolutionary history. To establish these results, we describe the topology of the Vietoris--Rips filtrations arising from evolutionary histories indexed by galled trees. As a step towards a probabilistic theory, we also show that for a random history indexed by a fixed galled tree and satisfying biologically reasonable conditions, the intervals of the \$1\textasciicircum\\mathrm\st\\\$ barcode are independent random variables. Using simulations, we explore the sensitivity of these intervals to recombination. -
Inference of Ancestral Recombination Graphs Through Topological Data Analysis (2016)
Pablo G. Cámara, Arnold J. Levine, Raúl RabadánAbstract
The recent explosion of genomic data has underscored the need for interpretable and comprehensive analyses that can capture complex phylogenetic relationships within and across species. Recombination, reassortment and horizontal gene transfer constitute examples of pervasive biological phenomena that cannot be captured by tree-like representations. Starting from hundreds of genomes, we are interested in the reconstruction of potential evolutionary histories leading to the observed data. Ancestral recombination graphs represent potential histories that explicitly accommodate recombination and mutation events across orthologous genomes. However, they are computationally costly to reconstruct, usually being infeasible for more than few tens of genomes. Recently, Topological Data Analysis (TDA) methods have been proposed as robust and scalable methods that can capture the genetic scale and frequency of recombination. We build upon previous TDA developments for detecting and quantifying recombination, and present a novel framework that can be applied to hundreds of genomes and can be interpreted in terms of minimal histories of mutation and recombination events, quantifying the scales and identifying the genomic locations of recombinations. We implement this framework in a software package, called TARGet, and apply it to several examples, including small migration between different populations, human recombination, and horizontal evolution in finches inhabiting the Galápagos Islands., Evolution occurs through different mechanisms, including point mutations, gene duplication, horizontal gene transfer, and recombinations. Some of these mechanisms cannot be captured by tree graphs. We present a framework, based on the mathematical tools of computational topology, that can explicitly accommodate both recombination and mutation events across the evolutionary history of a sample of genomic sequences. This approach generates a new type of summary graph and algebraic structures that provide quantitative information on the evolutionary scale and frequency of recombination events. The accompanying software, TARGet, is applied to several examples, including migration between sexually-reproducing populations, human recombination, and recombination in Darwin’s finches.