arXiv subjects

Taoyang Wu

Publications and source records attributed to Taoyang Wu.

31 records · Page 2Linked to original sources

arXiv2016

Transforming phylogenetic networks: Moving beyond tree space

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent reticulate evolution. Unrooted phylogenetic networks form a special class of such networks, which naturally generalize unrooted phylogenetic trees. In this paper we define two operations on unrooted phylogenetic networks, one of which is a generalization of the well-known nearest-neighbor interchange (NNI) operation on phylogenetic trees. We show that any unrooted phylogenetic network can be transformed into any other such network using only these operations. This generalizes the well-known fact that any phylogenetic tree can be transformed into any other such tree using only NNI operations. It also allows us to define a generalization of tree space and to define some new metrics on unrooted phylogenetic networks. To prove our main results, we employ some fascinating new connections between phylogenetic networks and cubic graphs that we have recently discovered. Our results should be useful in developing new strategies to search for optimal phylogenetic networks, a topic that has recently generated some interest in the literature, as well as for providing new ways to compare networks.

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arXiv2015

On joint subtree distributions under two evolutionary models

In population and evolutionary biology, hypotheses about micro-evolutionary and macro-evolutionary processes are commonly tested by comparing the shape indices of empirical evolutionary trees with those predicted by neutral models. A key ingredient in this approach is the ability to compute and quantify distributions of various tree shape indices under random models of interest. As a step to meet this challenge, in this paper we investigate the joint distribution of cherries and pitchforks (that is, subtrees with two and three leaves) under two widely used null models: the Yule-Harding-Kingman (YHK) model and the proportional to distinguishable arrangements (PDA) model. Based on two novel recursive formulae, we propose a dynamic approach to numerically compute the exact joint distribution (and hence the marginal distributions) for trees of any size. We also obtained insights into the statistical properties of trees generated under these two models, including a constant correlation between the cherry and the pitchfork distributions under the YHK model, the log-concavity and unimodality of cherry distributions under both models. In particular, we show the existence of a unique change point for cherry distribution between the two models, that is, there exists a critical value $τ_n$ for each $n\geq 4$ such that the probability that a random tree with $n$ leaves generated under the YHK model contains $k$ cherries is lower than that under the PDA model if $1<k< τ_n$, and higher if $τ_n<k\le n/2$.

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arXiv2015

Folding and unfolding phylogenetic trees and networks

Phylogenetic networks are rooted, labelled directed acyclic graphs which are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network $N$ can be 'unfolded' to obtain a MUL-tree $U(N)$ and, conversely, a MUL-tree $T$ can in certain circumstances be 'folded' to obtain a phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study properties of the operations $U$ and $F$ in more detail. In particular, we introduce the class of stable networks, phylogenetic networks $N$ for which $F(U(N))$ is isomorphic to $N$, characterise such networks, and show that that they are related to the well-known class of tree-sibling networks. We also explore how the concept of displaying a tree in a network $N$ can be related to displaying the tree in the MUL-tree $U(N)$. To do this, we develop a phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in $U(N)$ and reconciling phylogenetic trees with networks.

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arXiv2015

Optimal realisations of two-dimensional, totally-decomposable metrics

A realisation of a metric $d$ on a finite set $X$ is a weighted graph $(G,w)$ whose vertex set contains $X$ such that the shortest-path distance between elements of $X$ considered as vertices in $G$ is equal to $d$. Such a realisation $(G,w)$ is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations always exist, although it is NP-hard to compute them in general, and they have applications in areas such as phylogenetics, electrical networks and internet tomography. In [Adv. in Math. 53, 1984, 321-402] A.~Dress showed that the optimal realisations of a metric $d$ are closely related to a certain polytopal complex that can be canonically associated to $d$ called its tight-span. Moreover, he conjectured that the (weighted) graph consisting of the zero- and one-dimensional faces of the tight-span of $d$ must always contain an optimal realisation as a homeomorphic subgraph. In this paper, we prove that this conjecture does indeed hold for a certain class of metrics, namely the class of totally"=decomposable metrics whose tight-span has dimension two. As a corollary, it follows that the minimum Manhattan network problem is a special case of finding optimal realisations of two-dimensional totally-decomposable metrics.

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arXiv2014

Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets

Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level-1 phylogenetic network displaying a given set $\mathcal{T}$ of binary binets or trinets over a set $X$ of taxa, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an $O(3^{|X|} poly(|X|))$ time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted phylogenetic networks.

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arXiv2014

Clades and clans: a comparison study of two evolutionary models

The Yule-Harding-Kingman (YHK) model and the proportional to distinguishable arrangements (PDA) model are two binary tree generating models that are widely used in evolutionary biology. Understanding the distributions of clade sizes under these two models provides valuable insights into macro-evolutionary processes, and is important in hypothesis testing and Bayesian analyses in phylogenetics. Here we show that these distributions are log-convex, which implies that very large clades or very small clades are more likely to occur under these two models. Moreover, we prove that there exists a critical value $κ(n)$ for each $n\geqslant 4$ such that for a given clade with size $k$, the probability that this clade is contained in a random tree with $n$ leaves generated under the YHK model is higher than that under the PDA model if $1<k<κ(n)$, and lower if $κ(n)<k<n$. Finally, we extend our results to binary unrooted trees, and obtain similar results for the distributions of clan sizes.

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arXiv2014

Representing Partitions on Trees

In evolutionary biology, biologists often face the problem of constructing a phylogenetic tree on a set $X$ of species from a multiset $Π$ of partitions corresponding to various attributes of these species. One approach that is used to solve this problem is to try instead to associate a tree (or even a network) to the multiset $Σ_Π$ consisting of all those bipartitions $\{A,X-A\}$ with $A$ a part of some partition in $Π$. The rational behind this approach is that a phylogenetic tree with leaf set $X$ can be uniquely represented by the set of bipartitions of $X$ induced by its edges. Motivated by these considerations, given a multiset $Σ$ of bipartitions corresponding to a phylogenetic tree on $X$, in this paper we introduce and study the set $P(Σ)$ consisting of those multisets of partitions $Π$ of $X$ with $Σ_Π=Σ$. More specifically, we characterize when $P(Σ)$ is non-empty, and also identify some partitions in $P(Σ)$ that are of maximum and minimum size. We also show that it is NP-complete to decide when $P(Σ)$ is non-empty in case $Σ$ is an arbitrary multiset of bipartitions of $X$. Ultimately, we hope that by gaining a better understanding of the mapping that takes an arbitrary partition system $Π$ to the multiset $Σ_Π$, we will obtain new insights into the use of median networks and, more generally, split-networks to visualize sets of partitions.

math.CO↗

arXiv2012

Reconciliation of Gene and Species Trees With Polytomies

Motivation: Millions of genes in the modern species belong to only thousands of `gene families'. A gene family includes instances of the same gene in different species (orthologs) and duplicate genes in the same species (paralogs). Genes are gained and lost during evolution. With advances in sequencing technology, researchers are able to investigate the important roles of gene duplications and losses in adaptive evolution. Because of gene complex evolution, ortholog identification is a basic but difficult task in comparative genomics. A key method for the task is to use an explicit model of the evolutionary history of the genes being studied, called the gene (family) tree. It compares the gene tree with the evolutionary history of the species in which the genes reside, called the species tree, using the procedure known as tree reconciliation. Reconciling binary gene and specific trees is simple. However, both gene and species trees may be non-binary in practice and thus tree reconciliation presents challenging problems. Here, non-binary gene and species tree reconciliation is studied in a binary refinement model. Results: The problem of reconciling arbitrary gene and species trees is proved NP-hard even for the duplication cost. We then present the first efficient method for reconciling a non-binary gene tree and a non-binary species tree. It attempts to find binary refinements of the given gene and species trees that minimize reconciliation cost. Our algorithms have been implemented into a software to support quick automated analysis of large data sets. Availability: The program, together with the source code, is available at its online server http://phylotoo.appspot.com.

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arXiv2012

Reconstruction of Network Evolutionary History from Extant Network Topology and Duplication History

Genome-wide protein-protein interaction (PPI) data are readily available thanks to recent breakthroughs in biotechnology. However, PPI networks of extant organisms are only snapshots of the network evolution. How to infer the whole evolution history becomes a challenging problem in computational biology. In this paper, we present a likelihood-based approach to inferring network evolution history from the topology of PPI networks and the duplication relationship among the paralogs. Simulations show that our approach outperforms the existing ones in terms of the accuracy of reconstruction. Moreover, the growth parameters of several real PPI networks estimated by our method are more consistent with the ones predicted in literature.

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arXiv2012

On Patchworks and Hierarchies

Motivated by questions in biological classification, we discuss some elementary combinatorial and computational properties of certain set systems that generalize hierarchies, namely, 'patchworks', 'weak patchworks', 'ample patchworks' and 'saturated patchworks' and also outline how these concepts relate to an apparently new 'duality theory' for cluster systems that is based on the fundamental concept of 'compatibility' of clusters.

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arXiv2012

On the neighbourhoods of trees

Tree rearrangement operations typically induce a metric on the space of phylogenetic trees. One important property of these metrics is the size of the neighbourhood, that is, the number of trees exactly one operation from a given tree. We present an expression for the size of the TBR (tree bisection and reconnection) neighbourhood, thus answering a question first posed in [Annals of Combinatorics, 5, 2001 1-15].

math.CO↗

arXiv2009

Species, Clusters and the 'Tree of Life': A graph-theoretic perspective

A hierarchical structure describing the inter-relationships of species has long been a fundamental concept in systematic biology, from Linnean classification through to the more recent quest for a 'Tree of Life.' In this paper we use an approach based on discrete mathematics to address a basic question: Could one delineate this hierarchical structure in nature purely by reference to the 'genealogy' of present-day individuals, which describes how they are related with one another by ancestry through a continuous line of descent? We describe several mathematically precise ways by which one can naturally define collections of subsets of present day individuals so that these subsets are nested (and so form a tree) based purely on the directed graph that describes the ancestry of these individuals. We also explore the relationship between these and related clustering constructions.

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arXiv2007

On regular configurations and disjoint cycles in shift graphs

Configurations are necklaces with prescribed numbers of red and black beads. Among all possible configurations, the regular one plays an important role in many applications. In this paper, several aspects of regular configurations are discussed, including construction, uniqueness, symmetry group and the link with balanced words. Another model of configurations is the polygons formed by a given number of sides of two different lengths. In this context, regular configurations are used to obtain a lower bound for the cycles packing number of shift graphs, a subclass of the directed circulant graphs.

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