Computing hybridization networks using agreement forests Fakultät für Mathematik, Informatik und Statistik - Digitale Hochschulschriften der LMU - Teil 02/02
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- Education
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
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
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