🍩 Database of Original & Non-Theoretical Uses of Topology

(found 8 matches in 0.001368s)

  1. Topological Data Analysis of Single-Cell Hi-C Contact Maps (2020)

    Mathieu Carrière, Raúl Rabadán
    Abstract Due to recent breakthroughs in high-throughput sequencing, it is now possible to use chromosome conformation capture (CCC) to understand the three dimensional conformation of DNA at the whole genome level, and to characterize it with the so-called contact maps. This is very useful since many biological processes are correlated with DNA folding, such as DNA transcription. However, the methods for the analysis of such conformations are still lacking mathematical guarantees and statistical power. To handle this issue, we propose to use the Mapper, which is a standard tool of Topological Data Analysis (TDA) that allows one to efficiently encode the inherent continuity and topology of underlying biological processes in data, in the form of a graph with various features such as branches and loops. In this article, we show how recent statistical techniques developed in TDA for the Mapper algorithm can be extended and leveraged to formally define and statistically quantify the presence of topological structures coming from biological phenomena, such as the cell cyle, in datasets of CCC contact maps.

  2. Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution (2020)

    Michael Lesnick, Raúl Rabadán, Daniel I. S. Rosenbloom
    Abstract 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.

  3. Signal Enrichment With Strain-Level Resolution in Metagenomes Using Topological Data Analysis (2019)

    Aldo Guzmán-Sáenz, Niina Haiminen, Saugata Basu, Laxmi Parida
    Abstract Background A metagenome is a collection of genomes, usually in a micro-environment, and sequencing a metagenomic sample en masse is a powerful means for investigating the community of the constituent microorganisms. One of the challenges is in distinguishing between similar organisms due to rampant multiple possible assignments of sequencing reads, resulting in false positive identifications. We map the problem to a topological data analysis (TDA) framework that extracts information from the geometric structure of data. Here the structure is defined by multi-way relationships between the sequencing reads using a reference database. Results Based primarily on the patterns of co-mapping of the reads to multiple organisms in the reference database, we use two models: one a subcomplex of a Barycentric subdivision complex and the other a Čech complex. The Barycentric subcomplex allows a natural mapping of the reads along with their coverage of organisms while the Čech complex takes simply the number of reads into account to map the problem to homology computation. Using simulated genome mixtures we show not just enrichment of signal but also microbe identification with strain-level resolution. Conclusions In particular, in the most refractory of cases where alternative algorithms that exploit unique reads (i.e., mapped to unique organisms) fail, we show that the TDA approach continues to show consistent performance. The Čech model that uses less information is equally effective, suggesting that even partial information when augmented with the appropriate structure is quite powerful.

  4. Topological Data Analysis Generates High-Resolution, Genome-Wide Maps of Human Recombination (2016)

    Pablo G. Camara, Daniel I. S. Rosenbloom, Kevin J. Emmett, Arnold J. Levine, Raul Rabadan
    Abstract Meiotic recombination is a fundamental evolutionary process driving diversity in eukaryotes. In mammals, recombination is known to occur preferentially at specific genomic regions. Using topological data analysis (TDA), a branch of applied topology that extracts global features from large data sets, we developed an efficient method for mapping recombination at fine scales. When compared to standard linkage-based methods, TDA can deal with a larger number of SNPs and genomes without incurring prohibitive computational costs. We applied TDA to 1,000 Genomes Project data and constructed high-resolution whole-genome recombination maps of seven human populations. Our analysis shows that recombination is generally under-represented within transcription start sites. However, the binding sites of specific transcription factors are enriched for sites of recombination. These include transcription factors that regulate the expression of meiosis- and gametogenesis-specific genes, cell cycle progression, and differentiation blockage. Additionally, our analysis identifies an enrichment for sites of recombination at repeat-derived loci matched by piwi-interacting RNAs.

  5. Omics-Based Strategies in Precision Medicine: Toward a Paradigm Shift in Inborn Errors of Metabolism Investigations (2016)

    Abdellah Tebani, Carlos Afonso, Stéphane Marret, Soumeya Bekri
    Abstract The rise of technologies that simultaneously measure thousands of data points represents the heart of systems biology. These technologies have had a huge impact on the discovery of next-generation diagnostics, biomarkers, and drugs in the precision medicine era. Systems biology aims to achieve systemic exploration of complex interactions in biological systems. Driven by high-throughput omics technologies and the computational surge, it enables multi-scale and insightful overviews of cells, organisms, and populations. Precision medicine capitalizes on these conceptual and technological advancements and stands on two main pillars: data generation and data modeling. High-throughput omics technologies allow the retrieval of comprehensive and holistic biological information, whereas computational capabilities enable high-dimensional data modeling and, therefore, accessible and user-friendly visualization. Furthermore, bioinformatics has enabled comprehensive multi-omics and clinical data integration for insightful interpretation. Despite their promise, the translation of these technologies into clinically actionable tools has been slow. In this review, we present state-of-the-art multi-omics data analysis strategies in a clinical context. The challenges of omics-based biomarker translation are discussed. Perspectives regarding the use of multi-omics approaches for inborn errors of metabolism (IEM) are presented by introducing a new paradigm shift in addressing IEM investigations in the post-genomic era.

  6. Inference of Ancestral Recombination Graphs Through Topological Data Analysis (2016)

    Pablo G. Cámara, Arnold J. Levine, Raúl Rabadán
    Abstract 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.

  7. Parametric Inference Using Persistence Diagrams: a Case Study in Population Genetics (2014)

    Kevin Emmett, Daniel Rosenbloom, Pablo Camara, Raul Rabadan
    Abstract Persistent homology computes topological invariants from point cloud data. Recent work has focused on developing statistical methods for data analysis in this framework. We show that, in certain models, parametric inference can be performed using statistics defined on the computed invariants. We develop this idea with a model from population genetics, the coalescent with recombination. We apply our model to an influenza dataset, identifying two scales of topological structure which have a distinct biological interpretation.

  8. Topological Data Analysis of Escherichia Coli O157:H7 and Non-O157 Survival in Soils (2014)

    Abasiofiok M. Ibekwe, Jincai Ma, David E. Crowley, Ching-Hong Yang, Alexis M. Johnson, Tanya C. Petrossian, Pek Y. Lum
    Abstract Shiga toxin-producing E. coli O157:H7 and non-O157 have been implicated in many foodborne illnesses caused by the consumption of contaminated fresh produce. However, data on their persistence in soils are limited due to the complexity in datasets generated from different environmental variables and bacterial taxa. There is a continuing need to distinguish the various environmental variables and different bacterial groups to understand the relationships among these factors and the pathogen survival. Using an approach called Topological Data Analysis (TDA); we reconstructed the relationship structure of E. coli O157 and non-O157 survival in 32 soils (16 organic and 16 conventionally managed soils) from California (CA) and Arizona (AZ) with a multi-resolution output. In our study, we took a community approach based on total soil microbiome to study community level survival and examining the network of the community as a whole and the relationship between its topology and biological processes. TDA produces a geometric representation of complex data sets. Network analysis showed that Shiga toxin negative strain E. coli O157:H7 4554 survived significantly longer in comparison to E. coli O157:H7 EDL933, while the survival time of E. coli O157:NM was comparable to that of E. coli O157:H7 strain 933 in all of the tested soils. Two non-O157 strains, E. coli O26:H11 and E. coli O103:H2 survived much longer than E. coli O91:H21 and the three strains of E. coli O157. We show that there are complex interactions between E. coli strain survival, microbial community structures, and soil parameters.