arXiv subjects

Kristina Wicke

Publications and source records attributed to Kristina Wicke.

At least 37 records · Page 2Linked to original sources

arXiv2023

Universal rooted phylogenetic tree shapes and universal tanglegrams

We provide an $Ω(n\log n) $ lower bound and an $O(n^2)$ upper bound for the smallest size of rooted binary trees (a.k.a. phylogenetic tree shapes), which are universal for rooted binary trees with $n$ leaves, i.e., contain all of them as induced binary subtrees. We explicitly compute the smallest universal trees for $n\leq 11$. We also provide an $Ω(n^2) $ lower bound and an $O(n^4)$ upper bound for the smallest size of tanglegrams, which are universal for size $n$ tanglegrams, i.e., which contain all of them as induced subtanglegrams. Some of our results generalize to rooted $d$-ary trees and to $d$-ary tanglegrams.

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arXiv2023

Exploring spaces of semi-directed phylogenetic networks

Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While the spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and various rearrangement moves to traverse these spaces have been introduced, the results do not immediately carry over to semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks, called cut edge transfer (CET), and show that the space of semi-directed level-$1$ networks with precisely $k$ reticulations is connected under CET. This level-$1$ space is currently the predominantly used search space for most algorithms that reconstruct semi-directed phylogenetic networks. Hence, every semi-directed level-$1$ network with a fixed number of reticulations and leaf set can be reached from any other such network by a sequence of CETs. By introducing two additional moves, CET$^+$ and CET$^-$, that allow for the addition or deletion of reticulations, we then establish connectedness for the space of all semi-directed phylogenetic networks on a fixed leaf set. As a byproduct of our results for semi-directed phylogenetic networks, we also show that the space of rooted level-$1$ networks with a fixed number of reticulations and leaf set is connected under CET, when translated into the rooted setting.

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arXiv2022

Effects of discordance between species and gene trees on phylogenetic diversity conservation

Phylogenetic diversity indices such as the Fair Proportion (FP) index are frequently discussed as prioritization criteria in biodiversity conservation. They rank species according to their contribution to overall diversity by taking into account the unique and shared evolutionary history of each species as indicated by its placement in an underlying phylogenetic tree. Traditionally, phylogenetic trees were inferred from single genes and the resulting gene trees were assumed to be a valid estimate for the species tree, i.e., the "true" evolutionary history of the species under consideration. However, nowadays it is common to sequence whole genomes of hundreds or thousands of genes, and it is often the case that conflicting genealogical histories exist in different genes throughout the genome, resulting in discordance between individual gene trees and the species tree. Here, we analyze the effects of gene and species tree discordance on prioritization decisions based on the FP index. In particular, we consider the ranking order of taxa induced by (i) the FP index on a species tree, and (ii) the expected FP index across all gene tree histories associated with the species tree. On one hand, we show that for particular tree shapes, the two rankings always coincide. On the other hand, we show that for all leaf numbers greater than or equal to five, there exist species trees for which the two rankings differ. Finally, we illustrate the variability in the rankings obtained from the FP index across different gene tree and species tree estimates for an empirical multilocus mammal data set.

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arXiv2022

How far is my network from being edge-based? Proximity measures for edge-basedness of unrooted phylogenetic networks

Phylogenetic networks which are, as opposed to trees, suitable to describe processes like hybridization and horizontal gene transfer, play a substantial role in evolutionary research. However, while non-treelike events need to be taken into account, they are relatively rare, which implies that biologically relevant networks are often assumed to be similar to trees in the sense that they can be obtained by taking a tree and adding some additional edges. This observation led to the concept of so-called tree-based networks, which recently gained substantial interest in the literature. Unfortunately, though, identifying such networks in the unrooted case is an NP-complete problem. Therefore, classes of networks for which tree-basedness can be guaranteed are of the utmost interest. The most prominent such class is formed by so-called edge-based networks, which have a close relationship to generalized series-parallel graphs known from graph theory. They can be identified in linear time and are in some regards biologically more plausible than general tree-based networks. While concerning the latter proximity measures for general networks have already been introduced, such measures are not yet available for edge-basedness. This means that for an arbitrary unrooted network, the "distance" to the nearest edge-based network could so far not be determined. The present manuscript fills this gap by introducing two classes of proximity measures for edge-basedness, one based on the given network itself and one based on its so-called leaf shrink graph (LS graph). Both classes contain four different proximity measures, whose similarities and differences we study subsequently.

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arXiv2022

Classes of Explicit Phylogenetic Networks and their Biological and Mathematical Significance

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. Over the last two decades, several subclasses of rooted phylogenetic networks (characterized by certain structural constraints) have been introduced in the literature, either to model specific biological phenomena or to enable tractable mathematical and computational analyses. In the present manuscript, we provide a thorough review of these network classes, as well as provide a biological interpretation of the structural constraints underlying these networks where possible. In addition, we discuss how imposing structural constraints on the network topology can be used to address the scalability and identifiability challenges faced in the estimation of phylogenetic networks from empirical data.

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arXiv2021

On the Complexity of Optimising Variants of Phylogenetic Diversity on Phylogenetic Networks

Phylogenetic Diversity (PD) is a prominent quantitative measure of the biodiversity of a collection of present-day species (taxa). This measure is based on the evolutionary distance among the species in the collection. Loosely speaking, if $\mathcal{T}$ is a rooted phylogenetic tree whose leaf set $X$ represents a set of species and whose edges have real-valued lengths (weights), then the PD score of a subset $S$ of $X$ is the sum of the weights of the edges of the minimal subtree of $\mathcal{T}$ connecting the species in $S$. In this paper, we define several natural variants of the PD score for a subset of taxa which are related by a known rooted phylogenetic network. Under these variants, we explore, for a positive integer $k$, the computational complexity of determining the maximum PD score over all subsets of taxa of size $k$ when the input is restricted to different classes of rooted phylogenetic networks

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arXiv2021

Phylogenetic Diversity Rankings in the Face of Extinctions: the Robustness of the Fair Proportion Index

Planning for the protection of species often involves difficult choices about which species to prioritize, given constrained resources. One way of prioritizing species is to consider their "evolutionary distinctiveness", i.e. their relative evolutionary isolation on a phylogenetic tree. Several evolutionary isolation metrics or phylogenetic diversity indices have been introduced in the literature, among them the so-called Fair Proportion index (also known as the "evolutionary distinctiveness" score). This index apportions the total diversity of a tree among all leaves, thereby providing a simple prioritization criterion for conservation. Here, we focus on the prioritization order obtained from the Fair Proportion index and analyze the effects of species extinction on this ranking. More precisely, we analyze the extent to which the ranking order may change when some species go extinct and the Fair Proportion index is re-computed for the remaining taxa. We show that for each phylogenetic tree, there are edge lengths such that the extinction of one leaf per cherry completely reverses the ranking. Moreover, we show that even if only the lowest ranked species goes extinct, the ranking order may drastically change. We end by analyzing the effects of these two extinction scenarios (extinction of the lowest ranked species and extinction of one leaf per cherry) for a collection of empirical and simulated trees. In both cases, we can observe significant changes in the prioritization orders, highlighting the empirical relevance of our theoretical findings.

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arXiv2021

Combinatorial perspectives on Dollo-$k$ characters in phylogenetics

Recently, the perfect phylogeny model with persistent characters has attracted great attention in the literature. It is based on the assumption that complex traits or characters can only be gained once and lost once in the course of evolution. Here, we consider a generalization of this model, namely Dollo parsimony, that allows for multiple character losses. More precisely, we take a combinatorial perspective on the notion of Dollo-$k$ characters, i.e. traits that are gained at most once and lost precisely $k$ times throughout evolution. We first introduce an algorithm based on the notion of spanning subtrees for finding a Dollo-$k$ labeling for a given character and a given tree in linear time. We then compare persistent characters (consisting of the union of Dollo-0 and Dollo-1 characters) and general Dollo-$k$ characters. While it is known that there is a strong connection between Fitch parsimony and persistent characters, we show that Dollo parsimony and Fitch parsimony are in general very different. Moreover, while it is known that there is a direct relationship between the number of persistent characters and the Sackin index of a tree, a popular index of tree balance, we show that this relationship does not generalize to Dollo-$k$ characters. In fact, determining the number of Dollo-$k$ characters for a given tree is much more involved than counting persistent characters, and we end this manuscript by introducing a recursive approach for the former. This approach leads to a polynomial time algorithm for counting the number of Dollo-$k$ characters, and both this algorithm as well as the algorithm for computing Dollo-$k$ labelings are publicly available in the Babel package for BEAST 2.

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arXiv2020

On the maximum agreement subtree conjecture for balanced trees

We give a counterexample to the conjecture of Martin and Thatte that two balanced rooted binary leaf-labelled trees on $n$ leaves have a maximum agreement subtree (MAST) of size at least $n^{\frac{1}{2}}$. In particular, we show that for any $c>0$, there exist two balanced rooted binary leaf-labelled trees on $n$ leaves such that any MAST for these two trees has size less than $c n^{\frac{1}{2}}$. We also improve the lower bound of the size of such a MAST to $n^{\frac{1}{6}}$.

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arXiv2020

Unrooted non-binary tree-based phylogenetic networks

Phylogenetic networks are a generalization of phylogenetic trees allowing for the representation of non-treelike evolutionary events such as hybridization. Typically, such networks have been analyzed based on their `level', i.e. based on the complexity of their 2-edge-connected components. However, recently the question of how `treelike' a phylogenetic network is has become the center of attention in various studies. This led to the introduction of \emph{tree-based networks}, i.e. networks that can be constructed from a phylogenetic tree, called the \emph{base tree}, by adding additional edges. While the concept of tree-basedness was originally introduced for rooted phylogenetic networks, it has recently also been considered for unrooted networks. In the present study, we compare and contrast findings obtained for unrooted \emph{binary} tree-based networks to unrooted \emph{non-binary} networks. In particular, while it is known that up to level 4 all unrooted binary networks are tree-based, we show that in the case of non-binary networks, this result only holds up to level 3.

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arXiv2020

Combinatorial views on persistent characters in phylogenetics

The so-called binary perfect phylogeny with persistent characters has recently been thoroughly studied in computational biology as it is less restrictive than the well known binary perfect phylogeny. Here, we focus on the notion of (binary) persistent characters, i.e. characters that can be realized on a phylogenetic tree by at most one $0 \rightarrow 1$ transition followed by at most one $1 \rightarrow 0$ transition in the tree, and analyze these characters under different aspects. First, we illustrate the connection between persistent characters and Maximum Parsimony, where we characterize persistent characters in terms of the first phase of the famous Fitch algorithm. Afterwards we focus on the number of persistent characters for a given phylogenetic tree. We show that this number solely depends on the balance of the tree. To be precise, we develop a formula for counting the number of persistent characters for a given phylogenetic tree based on an index of tree balance, namely the Sackin index. Lastly, we consider the question of how many (carefully chosen) binary characters together with their persistence status are needed to uniquely determine a phylogenetic tree and provide an upper bound for the number of characters needed.

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arXiv2020

On the minimum value of the Colless index and the bifurcating trees that achieve it

Measures of tree balance play an important role in the analysis of phylogenetic trees. One of the oldest and most popular indices in this regard is the Colless index for rooted bifurcating trees, introduced by Colless (1982). While many of its statistical properties under different probabilistic models for phylogenetic trees have already been established, little is known about its minimum value and the trees that achieve it. In this manuscript, we fill this gap in the literature. To begin with, we derive both recursive and closed expressions for the minimum Colless index of a tree with $n$ leaves. Surprisingly, these expressions show a connection between the minimum Colless index and the so-called Blancmange curve, a fractal curve. We then fully characterize the tree shapes that achieve this minimum value and we introduce both an algorithm to generate them and a recurrence to count them. After focusing on two extremal classes of trees with minimum Colless index (the maximally balanced trees and the greedy from the bottom trees), we conclude by showing that all trees with minimum Colless index also have minimum Sackin index, another popular balance index.

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arXiv2020

Non-binary universal tree-based networks

A tree-based network $N$ on $X$ is called universal if every phylogenetic tree on $X$ is a base tree for $N$. Recently, binary universal tree-based networks have attracted great attention in the literature and their existence has been analyzed in various studies. In this note, we extend the analysis to non-binary networks and show that there exist both a rooted and an unrooted non-binary universal tree-based network with $n$ leaves for all positive integers $n$.

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arXiv2019

Classes of treebased networks

Recently, so-called treebased phylogenetic networks have gained considerable interest in the literature, where a treebased network is a network that can be constructed from a phylogenetic tree, called the base tree, by adding additional edges. The main aim of this manuscript is to provide some sufficient criteria for treebasedness by reducing phylogenetic networks to related graph structures. While it is generally known that deciding whether a network is treebased is NP-complete, one of these criteria, namely edgebasedness, can be verified in linear time. Surprisingly, the class of edgebased networks is closely related to a well-known family of graphs, namely the class of generalized series parallel graphs, and we will explore this relationship in full detail. Additionally, we introduce further classes of treebased networks and analyze their relationships.

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arXiv2019

Combinatorial properties of phylogenetic diversity indices

Phylogenetic diversity indices provide a formal way to apportion 'evolutionary heritage' across species. Two natural diversity indices are Fair Proportion (FP) and Equal Splits (ES). FP is also called 'evolutionary distinctiveness' and, for rooted trees, is identical to the Shapley Value (SV), which arises from cooperative game theory. In this paper, we investigate the extent to which FP and ES can differ, characterise tree shapes on which the indices are identical, and study the equivalence of FP and SV and its implications in more detail. We also define and investigate analogues of these indices on unrooted trees (where SV was originally defined), including an index that is closely related to the Pauplin representation of phylogenetic diversity.

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arXiv2019

Extremal properties of the Colless balance index for rooted binary trees

Measures of tree balance play an important role in various research areas, for example in phylogenetics. There they are for instance used to test whether an observed phylogenetic tree differs significantly from a tree generated by the Yule model of speciation. One of the most popular indices in this regard is the Colless index, which measures the degree of balance for rooted binary trees. While many statistical properties of the Colless index (e.g. asymptotic results for its mean and variance under different models of speciation) have already been discussed in different contexts, we focus on its extremal properties. While it is relatively straightforward to characterize trees with maximal Colless index, the analysis of the minimal value of the Colless index and the characterization of trees that achieve it, are much more involved. In this note, we therefore focus on the minimal value of the Colless index for any given number of leaves. We derive both a recursive formula for this minimal value, as well as an explicit expression, which shows a surprising connection between the Colless index and the so-called Blancmange curve, a fractal curve that is also known as the Takagi curve. Moreover, we characterize two classes of trees that have minimal Colless index, consisting of the set of so-called \emph{maximally balanced trees} and a class of trees that we call \emph{greedy from the bottom trees}. Furthermore, we derive an upper bound for the number of trees with minimal Colless index by relating these trees with trees with minimal Sackin index (another well-studied index of tree balance).

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arXiv2018

On the statistical inconsistency of Maximum Parsimony for $k$-tuple-site data

One of the main aims of phylogenetics is to reconstruct the \enquote{Tree of Life}. In this respect, different methods and criteria are used to analyze DNA sequences of different species and to compare them in order to derive the evolutionary relationships of these species. Maximum Parsimony is one such criterion for tree reconstruction and, it is the one which we will use in this paper. However, it is well-known that tree reconstruction methods can lead to wrong relationship estimates. One typical problem of Maximum Parsimony is long branch attraction, which can lead to statistical inconsistency. In this work, we will consider a blockwise approach to alignment analysis, namely so-called $k$-tuple analyses. For four taxa it has already been shown that $k$-tuple-based analyses are statistically inconsistent if and only if the standard character-based (site-based) analyses are statistically inconsistent. So, in the four-taxon case, going from individual sites to $k$-tuples does not lead to any improvement. However, real biological analyses often consider more than only four taxa. Therefore, we analyze the case of five taxa for $2$- and $3$-tuple-site data and consider alphabets with two and four elements. We show that the equivalence of single-site data and $k$-tuple-site data then no longer holds. Even so, we can show that Maximum Parsimony is statistically inconsistent for $k$-tuple site data and five taxa.

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arXiv2018

Phylogenetic diversity and biodiversity indices on phylogenetic networks

In biodiversity conservation it is often necessary to prioritize the species to conserve. Existing approaches to prioritization, e.g. the Fair Proportion Index and the Shapley Value, are based on phylogenetic trees and rank species according to their contribution to overall phylogenetic diversity. However, in many cases evolution is not treelike and thus, phylogenetic networks have come to the fore as a generalization of phylogenetic trees, allowing for the representation of non-treelike evolutionary events, such as horizontal gene transfer or hybridization. Here, we extend the concepts of phylogenetic diversity and phylogenetic diversity indices from phylogenetic trees to phylogenetic networks. On the one hand, we consider the treelike content of a phylogenetic network, e.g. the (multi)set of phylogenetic trees displayed by a network and the LSA tree associated with it. On the other hand, we derive the phylogenetic diversity of subsets of taxa and biodiversity indices directly from the internal structure of the network. Furthermore, we introduce our software package NetDiversity, which was implemented in Perl and allows for the calculation of all generalized measures of phylogenetic diversity and generalized phylogenetic diversity indices established in this note that are independent of inheritance probabilities. We apply our methods to a phylogentic network representing the evolutionary relationships among swordtails and platyfishes (Xiphophorus: Poeciliidae), a group of species characterized by widespread hybridization.

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