Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution

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

doi:10.1137/18m118150x

Cited by 16 publications

(11 citation statements)

References 46 publications

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“…homology is suitable for any metric space and thus is a useful tool for summarizing genomic data [Camara et al (2016), Blumberg and Rabadan (2017), Lesnick et al (2018)].…”

Section: Discussionmentioning

confidence: 99%

“…For a detailed introduction to the field, see Wakeley (2009). We note that our approach differs from the recent considerations of Lesnick et al (2018) in that we consider a coalescent model with branch lengths and model H 0 behavior. Furthermore, we assume a more restrictive sampling regime where only sequences at contemporaneous terminals of the graph are known, as opposed to sequences all along the genealogy.…”

Section: Coalescent Intuition For Topological Statisticsmentioning

confidence: 95%

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Fast Estimation of Recombination Rates Using Topological Data Analysis

et al. 2019

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Accurate estimation of recombination rates is critical for studying the origins and maintenance of genetic diversity. Because the inference of recombination rates under a full evolutionary model is computationally expensive, we developed an alternative approach using topological data analysis (TDA) on genome sequences. We find that this method can analyze datasets larger than what can be handled by any existing recombination inference software, and has accuracy comparable to commonly used model-based methods with significantly less processing time. Previous TDA methods used information contained solely in the first Betti number (β1) of a set of genomes, which aims to capture the number of loops that can be detected within a genealogy. These explorations have proven difficult to connect to the theory of the underlying biological process of recombination, and, consequently, have unpredictable behavior under perturbations of the data. We introduce a new topological feature, which we call ψ, with a natural connection to coalescent models, and present novel arguments relating β1 to population genetic models. Using simulations, we show that ψ and β1 are differentially affected by missing data, and package our approach as TREE (Topological Recombination Estimator). TREE’s efficiency and accuracy make it well suited as a first-pass estimator of recombination rate heterogeneity or hotspots throughout the genome. Our work empirically and theoretically justifies the use of topological statistics as summaries of genome sequences and describes a new, unintuitive relationship between topological features of the distribution of sequence data and the footprint of recombination on genomes.

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“…homology is suitable for any metric space and thus is a useful tool for summarizing genomic data [Camara et al (2016), Blumberg and Rabadan (2017), Lesnick et al (2018)].…”

Section: Discussionmentioning

confidence: 99%

Section: Coalescent Intuition For Topological Statisticsmentioning

confidence: 95%

Fast Estimation of Recombination Rates Using Topological Data Analysis

et al. 2019

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“…For a detailed introduction to the field, see (29). We note that our approach differs from the recent considerations of Lesnick, Rabadán, and Rosenbloom (30) in that we consider a coalescent model with branch lengths and model H 0 behavior. Furthermore, we assume a more restrictive sampling regime where only sequences at contemporaneous terminals of the graph are known, as opposed to sequences all along the genealogy.…”

Section: Coalescent Intuition For Topological Statisticsmentioning

confidence: 93%

Fast Estimation of Recombination Rates Using Topological Data Analysis

Humphreys

McGuirl

Miyagi

et al. 2018

Preprint

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Accurate estimation of recombination rates is critical for studying the origins and maintenance of genetic diversity. Because the inference of recombination rates under a full evolutionary model is computationally expensive, an alternative approach using topological data analysis (TDA) has been proposed. Previous TDA methods used information contained solely in the first Betti number (β 1 ) of the cloud of genomes, which relates to the number of loops that can be detected within a genealogy. While these methods are considerably less computationally intensive than current biological model-based methods, these explorations have proven difficult to connect to the theory of the underlying biological process of recombination, and consequently have unpredictable behavior under different perturbations of the data. We introduce a new topological feature with a natural connection to coalescent models, which we call ψ. We show that ψ and β 1 are differentially affected by changes to the structure of the data and use them in conjunction to provide a robust, efficient, and accurate estimator of recombination rates, TREE. Compared to previous TDA methods, TREE more closely approximates of the results of commonly used model-based methods. These characteristics make TREE well suited as a first-pass estimator of recombination rate heterogeneity or hotspots throughout the genome. In addition, we present novel arguments relating β 1 to population genetic models; our work justifies the use of topological statistics as summaries of distributions of genome sequences and describes a new, unintuitive relationship between topological summaries of distance and the footprint of recombination on genome sequences.

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“…Persistent homology of sampled data is used to obtain information about a novelty profile. The authors of [LRR18] provide mathematical foundation for several works that have used persistent homology to study recombination. Some other articles showing the use of persistent homology for studying recombination events are [ER14, CRE + 16, ER16].…”

Section: Further Applications To Biologymentioning

confidence: 99%

A Primer on Persistent Homology of Finite Metric Spaces

Mémoli

Singhal

2019

Bull Math Biol

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One of the methods for extracting information from a data set is clustering the data set according to some rule. In this paper, datasets are represented as finite metric spaces. A finite metric space is a pair (X, d X ), where X is a finite set and d X : X × X → R + is a distance function. We denote by M the collection of all finite metric spaces.We start by providing a definition of a clustering method with some examples. For any n ∈ N, we denote the set {1, 2, . . . , n} by [1 : n]. Given (X, d X ) ∈ M, we denote by P (X), the collection of all partitions of X. Precisely, every P ∈ P (X) is a family of setsas a block of P . We denote by P, the collection of all pairs (X, P X ), where X ∈ M and P X ∈ P (X). Formally,Example 2.2 An example of a clustering method is the discrete clustering that partitions every metric space into singletons. Precisely, we have C disc : M → P with C disc ((X, d X )) = (X, S X ), where S X ∈ P (X) is the partition of X into singletons.Example 2.3 Another example of a clustering method is the full clustering that partitions every metric space into a single block. Precisely, we have C full : M → P with C full ((X, d X )) = (X, {X}).There are various other examples of clustering methods such as partitioning into clusters whose diameter is bounded above by a constant, or partitioning into clusters whose diameter is bounded below by a constant, and so on [JS72]. Since we are working with finite metric spaces, the metric structure is the only information we have for determining a partition. Thus, 3 it seems natural that for (X, d X ), (Y, d Y ) ∈ M and a structure preserving map f : X → Y , a partition of Y induced by a clustering method C can be determined, at least partially, using the map f and a partition of X induced by the same clustering method C. Precisely, we want a clustering method C to be a functor, see [CM13].In order to view a clustering method C as a functor, we need to view M and P as categories. We refer the readers to [Jac12, Spi14] for an account on category theory. We define the categorical structure on M and P as follows:Definition 2.4 (Category of Finite Metric Spaces). Let M, by abuse of notation, denote the category of finite metric spaces. The objects of M are finite metric spaces (X, d X

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Quantifying Genetic Innovation: Mathematical Foundations for the Topological Study of Reticulate Evolution

Cited by 16 publications

References 46 publications

Fast Estimation of Recombination Rates Using Topological Data Analysis

Fast Estimation of Recombination Rates Using Topological Data Analysis

Fast Estimation of Recombination Rates Using Topological Data Analysis

A Primer on Persistent Homology of Finite Metric Spaces

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