Many discrete mathematics problems in phylogenetics are defined in terms of the relative labeling of pairs of leaf-labeled trees. These relative labelings are naturally formalized as tanglegrams, which have previously been an object of study in coevolutionary analysis. Although there has been considerable work on planar drawings of tanglegrams, they have not been fully explored as combinatorial objects until recently. In this paper, we describe how many discrete mathematical questions on trees "factor" through a problem on tanglegrams, and how understanding that factoring can simplify analysis. Depending on the problem, it may be useful to consider a unordered version of tanglegrams, and/or their unrooted counterparts. For all of these definitions, we show how the isomorphism types of tanglegrams can be understood in terms of double cosets of the symmetric group, and we investigate their automorphisms. Understanding tanglegrams better will isolate the distinct problems on leaf-labeled pairs of trees and reveal natural symmetries of spaces associated with such problems.
The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees. In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics. …
Enumerating monophyletic characters in mathematical phylogenetics
Institute of Mathematics and Computer Science, Greifswald University（格赖夫斯瓦尔德大学数学与计算机科学研究所）；School of Mathematics and Statistics, University of New South Wales（新南威尔士大学数学与统计学院）；Department of Mathematical Sciences, New Jersey Institute of Technology（新泽西理工学院数学科学系）；National Institute for Theory and Mathematics in Biology, Northwestern University and The University of Chicago（西北大学和芝加哥大学生物理论与数学国家研究所）
In phylogenetics it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction of computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most re…
Measures of phylogenetic balance, such as the Colless and Sackin indices, play an important role in phylogenetics. Unfortunately, these indices are specifically designed for phylogenetic trees, and do not extend naturally to phylogenetic networks (which are increasingly used to describe reticulate evolution). This led us to consider a lesser-known balance index, whose definition is based on a probabilistic interpretation that is equally applicable to trees and to networks. This index, known as the $B_2$ index, was first proposed by Shao and Sokal in 1990. Surprisingly, it does not seem to have…
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. They are also objects of interest in pure mathematics, such as algebraic geometry and combinatorics, due to their discrete geometry. Although they are important data structures, they face the significant challenge that sets of trees form a non-Euclidean phylogenetic tree space, which means that standard computational and statistical methods cannot be directly applied. In this work, we explore the statistical feasibility of a pure mathematical representation of the set of all phylogenetic tr…
The aim of this review is to present and analyze the probabilistic models of mathematical phylogenetics which have been intensively used in recent years in biology as the cornerstone of attempts to infer and reconstruct the ancestral relationships between species. We outline the development of theoretical phylogenetics, from the earliest studies based on morphological characters, through to the use of molecular data in a wide variety of forms. We bring the lens of mathematical physics to bear on the formulation of theoretical models, focussing on the applicability of many methods from the tool…
One of the main aims of phylogenetics is the reconstruction of the correct evolutionary tree when data concerning the underlying species set are given. These data typically come in the form of DNA, RNA or protein alignments, which consist of various characters (also often referred to as sites). Often, however, tree reconstruction methods based on criteria like maximum parsimony may fail to provide a unique tree for a given dataset, or, even worse, reconstruct the `wrong' tree (i.e. a tree that differs from the one that generated the data). On the other hand it has long been known that if the a…
Background. The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set $L$ is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results. A linear-time algorithm, LinCR, for constructing the common refinement $T$ of $k$ input trees with a common leaf set is proposed that explicitly computes the parent function of $T$ in a bottom-up approach. Conclusion. LinCR is…
The Tree of Life is the graphical structure that represents the evolutionary process from single-cell organisms at the origin of life to the vast biodiversity we see today. Reconstructing this tree from genomic sequences is challenging due to the variety of biological forces that shape the signal in the data, and many of those processes like incomplete lineage sorting and hybridization can produce confounding information. Here, we present the mathematical version of the identifiability proofs of phylogenetic networks under the pseudolikelihood model in SNaQ. We establish that the ability to de…
Throughout the last decade, we have seen much progress towards characterising and computing the minimum hybridisation number for a set P of rooted phylogenetic trees. Roughly speaking, this minimum quantifies the number of hybridisation events needed to explain a set of phylogenetic trees by simultaneously embedding them into a phylogenetic network. From a mathematical viewpoint, the notion of agreement forests is the underpinning concept for almost all results that are related to calculating the minimum hybridisation number for when |P|=2. However, despite various attempts, characterising thi…
Phylogenetic networks generalize phylogenetic trees in order to model reticulation events. Although the comparison of phylogenetic trees is well studied, and there are multiple ways to do it in an efficient way, the situation is much different for phylogenetic networks. Some classes of phylogenetic networks, mainly tree-child networks, are known to be classified efficiently by their $μ$-representation, which essentially counts, for every node, the number of paths to each leaf. In this paper, we introduce the extended $μ$-representation of networks, where the number of paths to reticulations is…
Effects like selection in evolution as well as fertility inheritance in the development of populations can lead to a higher degree of asymmetry in evolutionary trees than expected under a null hypothesis. To identify and quantify such influences, various balance indices were proposed in the phylogenetic literature and have been in use for decades. However, so far no balance index was based on the number of \emph{symmetry nodes}, even though symmetry nodes play an important role in other areas of mathematical phylogenetics and despite the fact that symmetry nodes are a quite natural way to meas…
Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality…
TU Delft（代尔夫特理工大学）；Middlesex University（密德萨斯大学）；LIGM, Université Gustave Eiffel（拉格朗日计算与建模实验室，古斯塔夫·埃菲尔大学）；Centre National de Recherche Scientifique（法国国家科学研究中心）
Phylogenetic trees (i.e. evolutionary trees, additive trees or X-trees) play a key role in the processes of modeling and representing species evolution. Genome evolution of a given group of species is usually modeled by a species phylogenetic tree that represents the main patterns of vertical descent. However, the evolution of each gene is unique. It can be represented by its own gene tree which can differ substantially from a general species tree representation. Consensus trees and supertrees have been widely used in evolutionary studies to combine phylogenetic information contained in indivi…
