Tree-child Networks Research Articles

The tree-child network inference problem for line trees and the shortest common supersequence problem for permutation strings

One strategy for inference of phylogenetic networks is to solve the phylogenetic network problem, which involves inferring phylogenetic trees first and subsequently computing the smallest phylogenetic network that displays all the trees. This approach capitalizes on exceptional tools available for inferring phylogenetic trees from biomolecular sequences. Since the vast space of phylogenetic networks poses difficulties in obtaining comprehensive sampling, the researchers switch their attention to inferring tree-child networks from multiple phylogenetic trees, where in a tree-child network each non-leaf node must have at least one child that is an indegree-one node. Three results are obtained in this work: (1) The shortest common supersequence problem remains NP-hard even for permutation strings. (2) Derived from the first result, the tree-child network inference problem is also established as NP-hard even for line trees (also known as caterpillar trees). (3) The parsimonious tree-child networks that display all the line trees are the same as those displaying all the binary trees and their hybridization number is Θ(n3) for n taxa.

Is this network proper forest-based?

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 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 is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network 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 in terms of certain m-colorings.

Comparison of Orchard Networks Using Their Extended μ-Representation.

Phylogenetic networks generalize phylogenetic trees in order to model reticulation events. Although the comparison of phylogenetic trees is well studied, and there are multiple ways to do it in an efficient way, the situation is much different for phylogenetic networks. Some classes of phylogenetic networks, mainly tree-child networks, are known to be classified efficiently by their μ-representation, which essentially counts, for every node, the number of paths to each leaf. In this article, we introduce the extended μ-representation of networks, where the number of paths to reticulations is also taken into account. This modification allows us to distinguish orchard networks and to define a metric on the space of such networks that can, moreover, be computed efficiently. The class of orchard networks, as well as being one of the classes with biological significance (one such network can be interpreted as a tree with extra arcs involving coexisting organisms), is one of the most generic ones (in mathematical terms) for which such a representation can (conjecturally) exist, since a slight relaxation of the definition leads to a problem that is Graph Isomorphism Complete.

Computing the Bounds of the Number of Reticulations in a Tree-Child Network That Displays a Set of Trees.

Phylogenetic network is an evolutionary model that uses a rooted directed acyclic graph (instead of a tree) to model an evolutionary history of species in which reticulate events (e.g., hybrid speciation or horizontal gene transfer) occurred. Tree-child network is a kind of phylogenetic network with structural constraints. Existing approaches for tree-child network reconstruction can be slow for large data. In this study, we present several computational approaches for bounding from below the number of reticulations in a tree-child network that displays a given set of rooted binary phylogenetic trees. In addition, we also present some theoretical results on bounding from above the number of reticulations. Through simulation, we demonstrate that the new lower bounds on the reticulation number for tree-child networks can practically be computed for large tree data. The bounds can provide estimates of reticulation for relatively large data.

Generation of Orchard and Tree-Child Networks

Phylogenetic networks are an extension of phylogenetic trees that allow for the representation of reticulate evolution events. One of the classes of networks that has gained the attention of the scientific community over the last years is the class of orchard networks, that generalizes tree-child networks, one of the most studied classes of networks. In this paper we focus on the combinatorial and algorithmic problem of the generation of binary orchard networks, and also of binary tree-child networks. To this end, we use that these networks are defined as those that can be recovered by reversing a certain reduction process. Then, we show how to choose a “minimum” reduction process among all that can be applied to a network, and hence we get a unique representation of the network that, in fact, can be given in terms of sequences of pairs of integers, whose length is related to the number of leaves and reticulations of the network. Therefore, the generation of networks is reduced to the generation of such sequences of pairs. Our main result is a recursive method for the efficient generation of all minimum sequences, and hence of all orchard (or tree-child) networks with a given number of leaves and reticulations. An implementation in C of the algorithms described in this paper, along with some computational experiments, can be downloaded from the public repository https://github.com/gerardet46/OrchardGenerator. Using this implementation, we have computed the number of binary orchard networks with at most 6 leaves and 8 reticulations.

Orienting undirected phylogenetic networks

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network N can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of N and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

A fast and scalable method for inferring phylogenetic networks from trees by aligning lineage taxon strings.

The reconstruction of phylogenetic networks is an important but challenging problem in phylogenetics and genome evolution, as the space of phylogenetic networks is vast and cannot be sampled well. One approach to the problem is to solve the minimum phylogenetic network problem, in which phylogenetic trees are first inferred, and then the smallest phylogenetic network that displays all the trees is computed. The approach takes advantage of the fact that the theory of phylogenetic trees is mature, and there are excellent tools available for inferring phylogenetic trees from a large number of biomolecular sequences. A tree-child network is a phylogenetic network satisfying the condition that every nonleaf node has at least one child that is of indegree one. Here, we develop a new method that infers the minimum tree-child network by aligning lineage taxon strings in the phylogenetic trees. This algorithmic innovation enables us to get around the limitations of the existing programs for phylogenetic network inference. Our new program, named ALTS, is fast enough to infer a tree-child network with a large number of reticulations for a set of up to 50 phylogenetic trees with 50 taxa that have only trivial common clusters in about a quarter of an hour on average.

Counting phylogenetic networks with few reticulation vertices: A second approach

Tree-child networks, one of the prominent network classes in phylogenetics, have been introduced for the purpose of modeling reticulate evolution. Recently, the first author together with Gittenberger and Mansouri (2019) showed that the number TC ℓ , k of tree-child networks with ℓ leaves and k reticulation vertices has the first-order asymptotics TC ℓ , k ∼ c k 2 e ℓ ℓ ℓ + 2 k − 1 , ( ℓ → ∞ ) . Moreover, they also computed c k for k = 1 , 2 , and 3. In this short note, we give a second approach to the above result which is based on a recent (algorithmic) approach for the counting of tree-child networks due to Cardona and Zhang (2020). This second approach is also capable of giving a simple, closed-form expression for c k , namely, c k = 2 k − 1 2 / k ! for all k ≥ 0 .

Distinct-Cluster Tree-Child Phylogenetic Networks and Possible Uses to Study Polyploidy

As phylogenetic networks become more widely studied and the networks grow larger, it may be useful to “simplify” such networks into especially tractable networks. Recent results have found methods to simplify networks into normal networks. By definition, normal networks contain no redundant arcs. Nevertheless, there may be redundant arcs in networks where speciation events involving allopolyploidy occur. It is therefore desirable to find a different tractable class of networks that may contain redundant arcs. This paper proposes distinct-cluster tree-child networks as such a class, here abbreviated as DCTC networks. They are shown to have a number of useful properties, such as quadratic growth of the number of vertices with the number of leaves. A DCTC network is shown to be essentially a normal network to which some redundant arcs may have been added without losing the tree-child property. Every phylogenetic network can be simplified into a DCTC network depending only on the structure of the original network. There is always a CSD map from the original network to the resulting DCTC network. As a result, the simplified network can readily be interpreted via a “wired lift” in which the original network is redrawn with each arc represented in one of two ways.

Forest-Based Networks

In evolutionary studies, it is common to use phylogenetic trees to represent the evolutionary history of a set of species. However, in case the transfer of genes or other genetic information between the species or their ancestors has occurred in the past, a tree may not provide a complete picture of their history. In such cases, tree-based phylogenetic networks can provide a useful, more refined representation of the species’ evolution. Such a network is essentially a phylogenetic tree with some arcs added between the tree’s edges so as to represent reticulate events such as gene transfer, hybridization and recombination. Even so, this model does not permit the direct representation of evolutionary scenarios where reticulate events have taken place between different subfamilies or lineages of species. To represent such scenarios, in this paper we introduce the notion of a forest-based network, that is, a collection of leaf-disjoint phylogenetic trees on a set of species with arcs added between the edges of distinct trees within the collection. Forest-based networks include the recently introduced class of overlaid species forests which can be used to model introgression. As we shall see, even though the definition of forest-based networks is closely related to that of tree-based networks, they lead to new mathematical theory which complements that of tree-based networks. As well as studying the relationship of forest-based networks with other classes of phylogenetic networks, such as tree-child networks and universal tree-based networks, we present some characterizations of some special classes of forest-based networks. We expect that our results will be useful for developing new models and algorithms to understand reticulate evolution, such as introgression and gene transfer between species.

Embedding gene trees into phylogenetic networks by conflict resolution algorithms

BackgroundPhylogenetic networks are mathematical models of evolutionary processes involving reticulate events such as hybridization, recombination, or horizontal gene transfer. One of the crucial notions in phylogenetic network modelling is displayed tree, which is obtained from a network by removing a set of reticulation edges. Displayed trees may represent an evolutionary history of a gene family if the evolution is shaped by reticulation events.ResultsWe address the problem of inferring an optimal tree displayed by a network, given a gene tree G and a tree-child network N, under the deep coalescence and duplication costs. We propose an O(mn)-time dynamic programming algorithm (DP) to compute a lower bound of the optimal displayed tree cost, where m and n are the sizes of G and N, respectively. In addition, our algorithm can verify whether the solution is exact. Moreover, it provides a set of reticulation edges corresponding to the obtained cost. If the cost is exact, the set induces an optimal displayed tree. Otherwise, the set contains pairs of conflicting edges, i.e., edges sharing a reticulation node. Next, we show a conflict resolution algorithm that requires 2^{r+1}-1 invocations of DP in the worst case, where r is the number of reticulations. We propose a similar O(2^kmn)-time algorithm for level-k tree-child networks and a branch and bound solution to compute lower and upper bounds of optimal costs. We also extend the algorithms to a broader class of phylogenetic networks. Based on simulated data, the average runtime is Theta (2^{{0.543}k}mn) under the deep-coalescence cost and Theta (2^{{0.355}k}mn) under the duplication cost.ConclusionsDespite exponential complexity in the worst case, our algorithms perform significantly well on empirical and simulated datasets, due to the strategy of resolving internal dissimilarities between gene trees and networks. Therefore, the algorithms are efficient alternatives to enumeration strategies commonly proposed in the literature and enable analyses of complex networks with dozens of reticulations.

Bijections for ranked tree-child networks

The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.

New FPT Algorithms for Finding the Temporal Hybridization Number for Sets of Phylogenetic Trees

We study the problem of finding a temporal hybridization network containing at most k reticulations, for an input consisting of a set of phylogenetic trees. First, we introduce an FPT algorithm for the problem on an arbitrary set of m binary trees with n leaves each with a running time of O(5^kcdot ncdot m). We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most d and at most k reticulations in O((8k)^d5^ kcdot kcdot ncdot m) time. Lastly, we introduce an O(6^kk!cdot kcdot n^2) time algorithm for computing a temporal hybridization network for a set of two nonbinary trees. We also provide an implementation of all algorithms and an experimental analysis on their performance.

Non-essential arcs in phylogenetic networks

In the study of rooted phylogenetic networks, analyzing the set of rooted phylogenetic trees that are embedded in such a network is a recurring task. Algorithmically, this analysis requires a search of the exponential-sized multiset S of rooted phylogenetic trees that are embedded in a rooted phylogenetic network N. In this paper, we introduce the notion of a non-essential arc, which is an arc whose deletion from N results in a rooted phylogenetic network N′ that embeds the same set of rooted phylogenetic trees as N but whose associated multiset is smaller than S. We characterize non-essential arcs for the popular class of tree-child networks and show that identifying and deleting such arcs takes time that is cubic in the number of leaves of the network. Moreover, we show that deciding if a given arc of an arbitrary phylogenetic network is non-essential is Π2P-complete.

A Practical Fixed-Parameter Algorithm for Constructing Tree-Child Networks from Multiple Binary Trees

We present the first fixed-parameter algorithm for constructing a tree-child phylogenetic network that displays an arbitrary number of binary input trees and has the minimum number of reticulations among all such networks. The algorithm uses the recently introduced framework of cherry picking sequences and runs in \(O((8k)^k \mathrm {poly}(n, t))\) time, where n is the number of leaves of every tree, t is the number of trees, and k is the reticulation number of the constructed network. Moreover, we provide an efficient parallel implementation of the algorithm and show that it can deal with up to 100 input trees on a standard desktop computer, thereby providing a major improvement over previous phylogenetic network construction methods.

Combinatorial characterization of a certain class of words and a conjectured connection with general subclasses of phylogenetic tree-child networks

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

Trinets encode orchard phylogenetic networks.

Rooted triples, rooted binary phylogenetic trees on three leaves, are sufficient to encode rooted binary phylogenetic trees. That is, if [Formula: see text] and [Formula: see text] are rooted binary phylogenetic X-trees that infer the same set of rooted triples, then [Formula: see text] and [Formula: see text] are isomorphic. However, in general, this sufficiency does not extend to rooted binary phylogenetic networks. In this paper, we show that trinets, phylogenetic network analogues of rooted triples, are sufficient to encode rooted binary orchard networks. Rooted binary orchard networks naturally generalise rooted binary tree-child networks. Moreover, we present a polynomial-time algorithm for building a rooted binary orchard network from its set of trinets. As a consequence, this algorithm affirmatively answers a previously-posed question of whether there is a polynomial-time algorithm for building a rooted binary tree-child network from the set of trinets it infers.

A unifying characterization of tree-based networks and orchard networks using cherry covers

Phylogenetic networks are used to represent evolutionary relationships between species in biology. Such networks are often categorized into classes by their topological features, which stem from both biological and computational motivations. We study two network classes in this paper: tree-based networks and orchard networks. Tree-based networks are those that can be obtained by inserting edges between the edges of an underlying tree. Orchard networks are a recently introduced generalization of the class of tree-child networks. Structural characterizations have already been discovered for tree-based networks; this is not the case for orchard networks. In this paper, we introduce cherry covers—a unifying characterization of both network classes—in which we decompose the edges of the networks into so-called cherry shapes and reticulated cherry shapes. We show that cherry covers can be used to characterize the class of tree-based networks as well as the class of orchard networks. Moreover, we also generalize these results to non-binary networks.

The rigid hybrid number for two phylogenetic trees

Recently there has been considerable interest in the problem of finding a phylogenetic network with a minimum number of reticulation vertices which displays a given set of phylogenetic trees, that is, a network with minimum hybrid number. Such networks are useful for representing the evolution of species whose genomes have undergone processes such as lateral gene transfer and recombination that cannot be represented appropriately by a phylogenetic tree. Even so, as was recently pointed out in the literature, insisting that a network displays the set of trees can be an overly restrictive assumption when modeling certain evolutionary phenomena such as incomplete lineage sorting. In this paper, we thus consider the less restrictive notion of rigidly displaying which we introduce and study here. More specifically, we characterize when two trees can be rigidly displayed by a certain type of phylogenetic network called a temporal tree-child network in terms of fork-picking sequences. These are sequences of special subconfigurations of the two trees related to the well-studied cherry-picking sequences. We also show that, in case it exists, the rigid hybrid number for two phylogenetic trees is given by a minimum weight fork-picking sequence for the trees. Finally, we consider the relationship between the rigid hybrid number and three closely related numbers; the weak, beaded, and temporal hybrid numbers. In particular, we show that these numbers can all be different even for a fixed pair of trees, and also present an infinite family of pairs of trees which demonstrates that the difference between the rigid hybrid number and the temporal-hybrid number for two phylogenetic trees on the same set of n leaves can grow at least linearly with n.

Display Sets of Normal and Tree-Child Networks

Phylogenetic networks are leaf-labelled directed acyclic graphs that are used in computational biology to analyse and represent the evolutionary relationships of a set of species or viruses. In contrast to phylogenetic trees, phylogenetic networks have vertices of in-degree at least two that represent reticulation events such as hybridisation, lateral gene transfer, or reassortment. By systematically deleting various combinations of arcs in a phylogenetic network $\mathcal N$, one derives a set of phylogenetic trees that are embedded in $\mathcal N$. We recently showed that the problem of deciding if two binary phylogenetic networks embed the same set of phylogenetic trees is computationally hard, in particular, we showed it to be $\Pi^P_2$-complete. In this paper, we establish a polynomial-time algorithm for this decision problem if the initial two networks consist of a normal network and a tree-child network; two well-studied topologically restricted subclasses of phylogenetic networks, with normal networks being more structurally constrained than tree-child networks. The running time of the algorithm is quadratic in the size of the leaf sets.