Computing all hybridization networks for multiple binary phylogenetic input trees.

Background The computation of phylogenetic trees on the same set of species that are based on different orthologous genes can lead to incongruent trees. One possible explanation for this behavior are interspecific hybridization events recombining genes of different species. An important approach to analyze such events is the computation of hybridization networks. Results This work presents the first algorithm computing the hybridization number as well as a set of representative hybridization networks for multiple binary phylogenetic input trees on the same set of taxa. To improve its practical runtime, we show how this algorithm can be parallelized. Moreover, we demonstrate the efficiency of the software Hybroscale, containing an implementation of our algorithm, by comparing it to PIRNv2.0, which is so far the best available software computing the exact hybridization number for multiple binary phylogenetic trees on the same set of taxa. The algorithm is part of the software Hybroscale, which was developed specifically for the investigation of hybridization networks including their computation and visualization. Hybroscale is freely available 1 and runs on all three major operating systems. Conclusion Our simulation study indicates that our approach is on average 100 times faster than PIRNv2.0. Moreover, we show how Hybroscale improves the interpretation of the reported hybridization networks by adding certain features to its graphical representation

Computing all hybridization networks for multiple binary phylogenetic input trees.

Background The computation of phylogenetic trees on the same set of species that are based on different orthologous genes can lead to incongruent trees. One possible explanation for this behavior are interspecific hybridization events recombining genes of different species. An important approach to analyze such events is the computation of hybridization networks. Results This work presents the first algorithm computing the hybridization number as well as a set of representative hybridization networks for multiple binary phylogenetic input trees on the same set of taxa. To improve its practical runtime, we show how this algorithm can be parallelized. Moreover, we demonstrate the efficiency of the software Hybroscale, containing an implementation of our algorithm, by comparing it to PIRNv2.0, which is so far the best available software computing the exact hybridization number for multiple binary phylogenetic trees on the same set of taxa. The algorithm is part of the software Hybroscale, which was developed specifically for the investigation of hybridization networks including their computation and visualization. Hybroscale is freely available 1 and runs on all three major operating systems. Conclusion Our simulation study indicates that our approach is on average 100 times faster than PIRNv2.0. Moreover, we show how Hybroscale improves the interpretation of the reported hybridization networks by adding certain features to its graphical representation

Computing hybridization networks using agreement forests

Rooted phylogenetic trees are widely used in biology to represent the evolutionary history of certain species. Usually, such a tree is a simple binary tree only containing internal nodes of in-degree one and out-degree two representing specific speciation events. In applied phylogenetics, however, trees can contain nodes of out-degree larger than two because, often, in order to resolve some orderings of speciation events, there is only insufficient information available and the common way to model this uncertainty is to use nonbinary nodes (i.e., nodes of out-degree of at least three), also denoted as polytomies. Moreover, in addition to such speciation events, there exist certain biological events that cannot be modeled by a tree and, thus, require the more general concept of rooted phylogenetic networks or, more specifically, of hybridization networks. Examples for such reticulate events are horizontal gene transfer, hybridization, and recombination. Nevertheless, in order to construct hybridization networks, the less general concept of a phylogenetic tree can still be used as building block. More precisely, often, in a first step, phylogenetic trees for a set of species, each based on a distinctive orthologous gene, are constructed. In a second step, specific sets containing common subtrees of those trees, known as maximum acyclic agreement forests, are calculated, which are then glued together to a single hybridization network. In such a network, hybridization nodes (i.e., nodes of in-degree larger than or equal to two) can exist representing potential reticulate events of the underlying evolutionary history. As such events are considered as rare phenomena, from a biological point of view, especially those networks representing a minimum number of reticulate events, which is denoted as hybridization number, are of high interest. Consequently, in a mathematical aspect, the problem of calculating hybridization networks can be briefly described as follows. Given a set T of rooted phylogenetic trees sharing the same set of taxa, compute a hybridization network N displaying T with minimum hybridization number. In this context, we say that such a network N displays a phylogenetic tree T, if we can obtain T from N by removing as well as contracting some of its nodes and edges. Unfortunately, this is a computational hard problem (i.e., it is NP-hard), even for the simplest case given just two binary input trees. In this thesis, we present several methods tackling this NP-hard problem. Our first approach describes how to compute a representative set of minimum hybridization networks for two binary input trees. For that purpose, our approach implements the first non-naive algorithm - called allMAAFs - calculating all maximum acyclic agreement forests for two rooted binary phylogenetic trees on the same set of taxa. In a subsequent step, in order to maximize the efficiency of the algorithm allMAAFs, we have developed additionally several modifications each reducing the number of computational steps and, thus, significantly improving its practical runtime. Our second approach is an extension of our first approach making the underlying algorithm accessible to more than two binary input trees. For this purpose, our approach implements the algorithm allHNetworks being the first algorithm calculating all relevant hybridization networks displaying a set of rooted binary phylogenetic trees on the same set of taxa, which is a preferable feature when studying hybridization events. Lastly, we have developed a generalization of our second approach that can now deal with multiple nonbinary input trees. For that purpose, our approach implements the first non-naive algorithm - called allMulMAAFs - calculating a relevant set of nonbinary maximum acyclic agreement forests for two rooted (nonbinary) phylogenetic trees on the same set of taxa. Each of the algorithms above is integrated into our user friendly Java-based software package Hybroscale, which is freely available and platform independent, so that it runs on all major operating systems. Our program provides a graphical user interface for visualizing trees and networks. Moreover, it facilitates the interpretation of computed hybridization networks by adding specific features to its graphical representation and, thus, supports biologists in investigating reticulate evolution. In addition, we have implemented a method using a user friendly SQL-style modeling language for filtering the usually large amount of reported networks

Kernelizations for the hybridization number problem on multiple nonbinary trees

Given a finite set $X$, a collection $\mathcal{T}$ of rooted phylogenetic trees on $X$ and an integer $k$, the Hybridization Number problem asks if there exists a phylogenetic network on $X$ that displays all trees from $\mathcal{T}$ and has reticulation number at most $k$. We show two kernelization algorithms for Hybridization Number, with kernel sizes $4k(5k)^t$ and $20k^2(\Delta^+-1)$ respectively, with $t$ the number of input trees and $\Delta^+$ their maximum outdegree. Experiments on simulated data demonstrate the practical relevance of these kernelization algorithms. In addition, we present an $n^{f(k)}t$-time algorithm, with $n=|X|$ and $f$ some computable function of $k$

A practical approximation algorithm for solving massive instances of hybridization number for binary and nonbinary trees

Reticulate events play an important role in determining evolutionary relationships. The problem of computing the minimum number of such events to explain discordance between two phylogenetic trees is a hard computational problem. Even for binary trees, exact solvers struggle to solve instances with reticulation number larger than 40-50. Here we present CycleKiller and NonbinaryCycleKiller, the first methods to produce solutions verifiably close to optimality for instances with hundreds or even thousands of reticulations. Using simulations, we demonstrate that these algorithms run quickly for large and difficult instances, producing solutions that are very close to optimality. As a spin-off from our simulations we also present TerminusEst, which is the fastest exact method currently available that can handle nonbinary trees: this is used to measure the accuracy of the NonbinaryCycleKiller algorithm. All three methods are based on extensions of previous theoretical work and are publicly available. We also apply our methods to real data

Inferring phylogenetic networks with maximum pseudolikelihood under incomplete lineage sorting

Phylogenetic networks are necessary to represent the tree of life expanded by edges to represent events such as horizontal gene transfers, hybridizations or gene flow. Not all species follow the paradigm of vertical inheritance of their genetic material. While a great deal of research has flourished into the inference of phylogenetic trees, statistical methods to infer phylogenetic networks are still limited and under development. The main disadvantage of existing methods is a lack of scalability. Here, we present a statistical method to infer phylogenetic networks from multi-locus genetic data in a pseudolikelihood framework. Our model accounts for incomplete lineage sorting through the coalescent model, and for horizontal inheritance of genes through reticulation nodes in the network. Computation of the pseudolikelihood is fast and simple, and it avoids the burdensome calculation of the full likelihood which can be intractable with many species. Moreover, estimation at the quartet-level has the added computational benefit that it is easily parallelizable. Simulation studies comparing our method to a full likelihood approach show that our pseudolikelihood approach is much faster without compromising accuracy. We applied our method to reconstruct the evolutionary relationships among swordtails and platyfishes ($Xiphophorus$: Poeciliidae), which is characterized by widespread hybridizations