Katharina T. Huber’s research while affiliated with University of East Anglia and other places

With the increasing availability of genomic data, biologists aim to find more accurate descriptions of evolutionary histories influenced by secondary contact, where diverging lineages reconnect before diverging again. Such reticulate evolutionary events can be more accurately represented in phylogenetic networks than in phylogenetic trees. Since the root location of phylogenetic networks can not be inferred from biological data under several evolutionary models, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. We demonstrate Squirrel's performance through simulations and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mbp long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user-interface that takes sequence alignments or quarnets as input, and is freely available at https://github.com/nholtgrefe/squirrel

Metric embeddings are central to metric theory and its applications. Here we consider embeddings of a different sort: maps from a set to subsets of a metric space so that distances between points are approximated by minimal distances between subsets. Our main result is a characterization of when a set of distances d(x,y) between elements in a set X have a subtree representation, a real tree T and a collection $\{S_x\}_{x \in X}$ of subtrees of~T such that d(x,y) equals the length of the shortest path in~T from a point in $S_x$ to a point in $S_y$ for all $x,y \in X$. The characterization was first established for {\em finite} X by Hirai (2006) using a tight span construction defined for distance spaces, metric spaces without the triangle inequality. To extend Hirai's result beyond finite X we establish fundamental results of tight span theory for general distance spaces, including the surprising observation that the tight span of a distance space is hyperconvex. We apply the results to obtain the first characterization of when a diversity -- a generalization of a metric space which assigns values to all finite subsets of X, not just to pairs -- has a tight span which is tree-like.

In evolutionary biology, phylogenetic trees are commonly inferred from a set of characters (partitions) of a collection of biological entities (e.g., species or individuals in a population). Such characters naturally arise from molecular sequences or morphological data. Interestingly, it has been known for some time that any binary phylogenetic tree can be (convexly) defined by a set of at most four characters, and that there are binary phylogenetic trees for which three characters are not enough. Thus, it is of interest to characterise those phylogenetic trees that are defined by a set of at most three characters. In this paper, we provide such a characterisation, in particular proving that a binary phylogenetic tree T is defined by a set of at most three dcharacters precisely if T has no internal subtree isomorphic to a certain tree.

Phylogenetic networks model the evolutionary history of taxa while allowing for reticulate events such as hybridization and horizontal gene transfer. As is the case for phylogenetic trees, it is often not possible to infer the root location of such a network directly from biological data for several evolutionary models. Hence, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. To build a network from quarnets, Squirrel first builds a tree, after which it repeatedly solves the Travelling Salesman Problem (TSP) to replace each high-degree vertex by a cycle. We demonstrate Squirrel's performance on randomly generated networks and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mpb long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user-interface that takes sequence alignments or quarnets as input, and is freely available at https://github.com/nholtgrefe/squirrel.

Phylogenetic networks are graphs that are used to represent evolutionary relationships between different taxa. They generalize phylogenetic trees since for example, unlike trees, they permit lineages to combine. Recently, there has been rising interest in semi-directed phylogenetic networks, which are mixed graphs in which certain lineage combination events are represented by directed edges coming together, whereas the remaining edges are left undirected. One reason to consider such networks is that it can be difficult to root a network using real data. In this paper, we consider the problem of when a semi-directed phylogenetic network is defined or encoded by the smaller networks that it induces on the 4-leaf subsets of its leaf set. These smaller networks are called quarnets. We prove that semi-directed binary level-2 phylogenetic networks are encoded by their quarnets, but that this is not the case for level-3. In addition, we prove that the so-called blob tree of a semi-directed binary network, a tree that gives the coarse-grained structure of the network, is always encoded by the quarnets of the network.

Convergent evolution is an important process in which independent species evolve similar features usually over a long period of time. It occurs with many different species across the tree of life, and is often caused by the fact that species have to adapt to similar environmental niches. In this paper, we introduce and study properties of a distance-based model for convergent evolution in which we assume that two ancestral species converge for a certain period of time within a collection of species that have otherwise evolved according to an evolutionary clock. Under these assumptions it follows that we obtain a distance on the collection that is a modification of an ultrametric distance arising from an equidistant phylogenetic tree. As well as characterising when this modified distance is a tree metric, we give conditions in terms of the model’s parameters for when it is still possible to recover the underlying tree and also its height, even in case the modified distance is not a tree metric.

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether or not a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it can be decided in polynomial time whether or not a binary, tree-child network with $m \ge 2$ roots is proper forest-based in case m=2, but that this problem is NP-complete for $m\ge 3$. We also give a fixed parameter tractable (FPT) algorithm for deciding whether or not a network in which every vertex has indegree at most 2 is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based which is given in terms of the existence of certain m-colorings of the vertices of the network.