A Note on Encodings of Phylogenetic Networks of Bounded Level

Driven by the need for better models that allow one to shed light into the question how life's diversity has evolved, phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does one of the above collections encode (i.e. uniquely describe) the network that induces it? In this note, we present a complete answer to this question for the special case of a level-1 (phylogenetic) network by characterizing those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. Given that this type of network forms the first layer of the rich hierarchy of level-k networks, k a non-negative integer, it is natural to wonder whether our arguments could be extended to members of that hierarchy for higher values for k. By giving examples, we show that this is not the case

Constructing liberal and conservative supertrees and exact solutions for reduced consensus problems

This thesis studies two different approaches to extracting information from collections of phylogenetic trees: supertrees and reduced consensus. Supertree methods combine the phylogenetic information from multiple partially-overlapping trees into a larger phylogenetic tree called a supertree. Several supertree construction methods have been proposed to date, but most of these are not designed with any specific properties in mind. Recently, Cotton and Wilkinson proposed extensions of the majority-rule consensus tree method to the supertree setting that inherit many of the appealing properties of the former. We study a variant of one of Cotton and Wilkinson\u27s methods, called majority-rule (+) supertrees. After proving that a key underlying problem for constructing majority-rule (+) supertrees is NP-hard, we develop a polynomial-size exact integer linear programming formulation of the problem. We then present a data reduction heuristic that identifies smaller subproblems that can be solved independently. While this technique is not guaranteed to produce optimal solutions, it can achieve substantial problem-size reduction. Finally, we report on a computational study of our approach on various real data sets, including the 121-taxon, 7-tree Seabirds data set of Kennedy and Page. The results indicate that our exact method is computationally feasible for moderately large inputs. For larger inputs, our data reduction heuristic makes it feasible to tackle problems that are well beyond the range of the basic integer programming approach. Comparisons between the results obtained by our heuristic and exact solutions indicate that the heuristic produces good answers. Our results also suggest that the majority-rule (+) approach, in both its basic form and with data reduction, yields biologically meaningful phylogenies. Generalizations of the strict and loose consensus methods to the supertree setting, recently introduced by McMorris and Wilkinson, are studied. The supertrees these methods produce are conservative in the sense that they only preserve information (in the form of splits) that is supported by at least one the input trees and that is not contradicted by any of the input trees. Alternative, equivalent, formulations of these supertrees are developed. These are used to prove the NP-completeness of the underlying optimization problems and to give exact integer linear programming solutions. For larger data sets, a divide and conquer approach is adopted, based on the structural properties of these supertrees. Experiments show that it is feasible to solve problems with several hundred taxa and several hundred trees in a reasonable amount of time. A rogue taxon in a collection of phylogenetic trees is one whose position varies drastically from tree to tree. The presence of such taxa can greatly reduce the resolution of the consensus tree (e.g., the majority-rule or strict consensus) for a collection. The reduced consensus approach aims to identify rogue taxa and to produce more informative consensus trees. Given a collection of phylogenetic trees over the same leaf set, the goal is to find a set of taxa whose removal maximizes the number of internal edges in the consensus tree of the collection. This problem is proven to be NP-hard for strict and majority-rule consensus. We describe exact integer linear programming formulations for computing reduced strict, majority and loose consensus trees. In experimental tests, our exact solutions show significant improvement over heuristic methods on several problem instances

An approximate search engine for structure

As the size of structural databases grows, the need for efficiently searching these databases arises. Thanks to previous and ongoing research, searching by attribute-value and by text has become commonplace in these databases. However, searching by topological or physical structure, especially for large databases and especially for approximate matches, is still an art. In this dissertation, efficient search techniques are presented for retrieving trees from a database that are similar to a given query tree. Rooted ordered labeled trees, rooted unordered labeled trees and free trees are considered. Ordered labeled trees are trees in which each node has a label and the left-to-right order among siblings matters. Unordered labeled trees are trees in which the parent-child relationship is significant, but the order among siblings is unimportant. Free trees (unrooted unordered trees) are acyclic graphs. These trees find many applications in bioinformatics, Web log analysis, phyloinformatics, XML processing, etc. Two types of similarity measures are investigated: (i) counting the mismatching paths in the query tree and a data tree, and (ii) measuring the topological relationship between the trees. The proposed approaches include storing the paths of trees in a suffix array, employing hashing techniques to speed up retrieval, and counting the number of up-down operations to move a token from one node to another node in a tree. Various filters for accelerating a search, different strategies for parallelizing these search algorithms and applications of these algorithms to XML and phylogenetic data management are discussed. The proposed techniques have been implemented into a phylogenetic search engine which is fully operational and is available on the World Wide Web. Experimental results on comparing the similarity measures with existing tree metrics and on evaluating the efficiency of the search techniques demonstrate the effectiveness of the search engine. Future work includes extending the techniques to other structural data, as well as developing new filters and algorithms for speeding up searching and mining in complex structures

Inferring phylogenetic trees under the general Markov model via a minimum spanning tree backbone

Phylogenetic trees are models of the evolutionary relationships among species, with species typically placed at the leaves of trees. We address the following problems regarding the calculation of phylogenetic trees. (1) Leaf-labeled phylogenetic trees may not be appropriate models of evolutionary relationships among rapidly evolving pathogens which may contain ancestor-descendant pairs. (2) The models of gene evolution that are widely used unrealistically assume that the base composition of DNA sequences does not evolve. Regarding problem (1) we present a method for inferring generally labeled phylogenetic trees that allow sampled species to be placed at non-leaf nodes of the tree. Regarding problem (2), we present a structural expectation maximization method (SEM-GM) for inferring leaf-labeled phylogenetic trees under the general Markov model (GM) which is the most complex model of DNA substitution that allows the evolution of base composition. In order to improve the scalability of SEM-GM we present a minimum spanning tree (MST) framework called MST-backbone. MST-backbone scales linearly with the number of leaves. However, the unrealistic location of the root as inferred on empirical data suggests that the GM model may be overtrained. MST-backbone was inspired by the topological relationship between MSTs and phylogenetic trees that was introduced by Choi et al. (2011). We discovered that the topological relationship does not necessarily hold if there is no unique MST. We propose so-called vertex-order based MSTs (VMSTs) that guarantee a topological relationship with phylogenetic trees.Phylogenetische Bäume modellieren evolutionäre Beziehungen zwischen Spezies, wobei die Spezies typischerweise an den Blättern der Bäume sitzen. Wir befassen uns mit den folgenden Problemen bei der Berechnung von phylogenetischen Bäumen. (1) Blattmarkierte phylogenetische Bäume sind möglicherweise keine geeigneten Modelle der evolutionären Beziehungen zwischen sich schnell entwickelnden Krankheitserregern, die Vorfahren-Nachfahren-Paare enthalten können. (2) Die weit verbreiteten Modelle der Genevolution gehen unrealistischerweise davon aus, dass sich die Basenzusammensetzung von DNA-Sequenzen nicht ändert. Bezüglich Problem (1) stellen wir eine Methode zur Ableitung von allgemein markierten phylogenetischen Bäumen vor, die es erlaubt, Spezies, für die Proben vorliegen, an inneren des Baumes zu platzieren. Bezüglich Problem (2) stellen wir eine strukturelle Expectation-Maximization-Methode (SEM-GM) zur Ableitung von blattmarkierten phylogenetischen Bäumen unter dem allgemeinen Markov-Modell (GM) vor, das das komplexeste Modell von DNA-Substitution ist und das die Evolution von Basenzusammensetzung erlaubt. Um die Skalierbarkeit von SEM-GM zu verbessern, stellen wir ein Minimale Spannbaum (MST)-Methode vor, die als MST-Backbone bezeichnet wird. MST-Backbone skaliert linear mit der Anzahl der Blätter. Die Tatsache, dass die Lage der Wurzel aus empirischen Daten nicht immer realistisch abgeleitet warden kann, legt jedoch nahe, dass das GM-Modell möglicherweise übertrainiert ist. MST-backbone wurde von einer topologischen Beziehung zwischen minimalen Spannbäumen und phylogenetischen Bäumen inspiriert, die von Choi et al. 2011 eingeführt wurde. Wir entdeckten, dass die topologische Beziehung nicht unbedingt Bestand hat, wenn es keinen eindeutigen minimalen Spannbaum gibt. Wir schlagen so genannte vertex-order-based MSTs (VMSTs) vor, die eine topologische Beziehung zu phylogenetischen Bäumen garantieren

Uniqueness, intractability and exact algorithms: reflections on level-k phylogenetic networks

Phylogenetic networks provide a way to describe and visualize evolutionary histories that have undergone so-called reticulate evolutionary events such as recombination, hybridization or horizontal gene transfer. The level k of a network determines how non-treelike the evolution can be, with level-0 networks being trees. We study the problem of constructing level-k phylogenetic networks from triplets, i.e. phylogenetic trees for three leaves (taxa). We give, for each k, a level-k network that is uniquely defined by its triplets. We demonstrate the applicability of this result by using it to prove that (1) for all k of at least one it is NP-hard to construct a level-k network consistent with all input triplets, and (2) for all k it is NP-hard to construct a level-k network consistent with a maximum number of input triplets, even when the input is dense. As a response to this intractability we give an exact algorithm for constructing level-1 networks consistent with a maximum number of input triplets