A practical algorithm for reconstructing level-1 phylogenetic networks

Recently, much attention has been devoted to the construction of phylogenetic networks which generalize phylogenetic trees in order to accommodate complex evolutionary processes. Here, we present an efficient, practical algorithm for reconstructing level-1 phylogenetic networks--a type of network slightly more general than a phylogenetic tree--from triplets. Our algorithm has been made publicly available as the program LEV1ATHAN. It combines ideas from several known theoretical algorithms for phylogenetic tree and network reconstruction with two novel subroutines. Namely, an exponential-time exact and a greedy algorithm both of which are of independent theoretical interest. Most importantly, LEV1ATHAN runs in polynomial time and always constructs a level-1 network. If the data are consistent with a phylogenetic tree, then the algorithm constructs such a tree. Moreover, if the input triplet set is dense and, in addition, is fully consistent with some level-1 network, it will find such a network. The potential of LEV1ATHAN is explored by means of an extensive simulation study and a biological data set. One of our conclusions is that LEV1ATHAN is able to construct networks consistent with a high percentage of input triplets, even when these input triplets are affected by a low to moderate level of noise.     

Constructing a minimum phylogenetic network from a dense triplet set

For a given set L of species and a set T of triplets on L, we seek to construct a phylogenetic network which is consistent with T i.e. which represents all triplets of T. The level of a network is defined as the maximum number of hybrid vertices in its biconnected components. When T is dense, there exist polynomial time algorithms to construct level-0,1 and 2 networks (Aho et al., 1981; Jansson, Nguyen and Sung, 2006; Jansson and Sung, 2006; Iersel et al., 2009). For higher levels, partial answers were obtained in the paper by Iersel and Kelk (2008), with a polynomial time algorithm for simple networks. In this paper, we detail the first complete answer for the general case, solving a problem proposed in Jansson and Sung (2006) and Iersel et al. (2009). For any k fixed, it is possible to construct a level-k network having the minimum number of hybrid vertices and consistent with T, if there is any, in time O(T(k+1)n([4k/3]+1)).     

Trinets encode tree-child and level-2 phylogenetic networks

Phylogenetic networks generalize evolutionary trees, and are commonly used to represent evolutionary histories of species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in trying to develop methods to construct rooted phylogenetic networks from triplets, that is rooted trees on three species. However, although triplets determine or encode rooted phylogenetic trees, they do not in general encode rooted phylogenetic networks, which is a potential issue for any such method. Motivated by this fact, Huber and Moulton recently introduced trinets as a natural extension of rooted triplets to networks. In particular, they showed that $\text{ level-1 }$ phylogenetic networks are encoded by their trinets, and also conjectured that all “recoverable” rooted phylogenetic networks are encoded by their trinets. Here we prove that recoverable binary level-2 networks and binary tree-child networks are also encoded by their trinets. To do this we prove two decomposition theorems based on trinets which hold for all recoverable binary rooted phylogenetic networks. Our results provide some additional evidence in support of the conjecture that trinets encode all recoverable rooted phylogenetic networks, and could also lead to new approaches to construct phylogenetic networks from trinets.     

Binets: Fundamental Building Blocks for Phylogenetic Networks

Phylogenetic networks are a generalization of evolutionary trees that are used by biologists to represent the evolution of organisms which have undergone reticulate evolution. Essentially, a phylogenetic network is a directed acyclic graph having a unique root in which the leaves are labelled by a given set of species. Recently, some approaches have been developed to construct phylogenetic networks from collections of networks on 2- and 3-leaved networks, which are known as binets and trinets, respectively. Here we study in more depth properties of collections of binets, one of the simplest possible types of networks into which a phylogenetic network can be decomposed. More specifically, we show that if a collection of level-1 binets is compatible with some binary network, then it is also compatible with a binary level-1 network. Our proofs are based on useful structural results concerning lowest stable ancestors in networks. In addition, we show that, although the binets do not determine the topology of the network, they do determine the number of reticulations in the network, which is one of its most important parameters. We also consider algorithmic questions concerning binets. We show that deciding whether an arbitrary set of binets is compatible with some network is at least as hard as the well-known graph isomorphism problem. However, if we restrict to level-1 binets, it is possible to decide in polynomial time whether there exists a binary network that displays all the binets. We also show that to find a network that displays a maximum number of the binets is NP-hard, but that there exists a simple polynomial-time 1/3-approximation algorithm for this problem. It is hoped that these results will eventually assist in the development of new methods for constructing phylogenetic networks from collections of smaller networks.     

On encodings of phylogenetic networks of bounded level

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? For the large class of level-1 (phylogenetic) networks we characterize those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. In addition, we show that three known distance measures for comparing phylogenetic networks are in fact metrics on the resulting subclass and give the diameter for two of them. Finally, we investigate the related concept of indistinguishability and also show that many properties enjoyed by level-1 networks are not satisfied by networks of higher level.     

TripNet: A Method for Constructing Rooted Phylogenetic Networks from Rooted Triplets

The problem of constructing an optimal rooted phylogenetic network from an arbitrary set of rooted triplets is an NP-hard problem. In this paper, we present a heuristic algorithm called TripNet, which tries to construct a rooted phylogenetic network with the minimum number of reticulation nodes from an arbitrary set of rooted triplets. Despite of current methods that work for dense set of rooted triplets, a key innovation is the applicability of TripNet to non-dense set of rooted triplets. We prove some theorems to clarify the performance of the algorithm. To demonstrate the efficiency of TripNet, we compared TripNet with SIMPLISTIC. It is the only available software which has the ability to return some rooted phylogenetic network consistent with a given dense set of rooted triplets. But the results show that for complex networks with high levels, the SIMPLISTIC running time increased abruptly. However in all cases TripNet outputs an appropriate rooted phylogenetic network in an acceptable time. Also we tetsed TripNet on the Yeast data. The results show that Both TripNet and optimal networks have the same clustering and TripNet produced a level-3 network which contains only one more reticulation node than the optimal network.     

When two trees go to war

Rooted phylogenetic networks are used to model non-treelike evolutionary histories. Such networks are often constructed by combining trees, clusters, triplets or characters into a single network that in some well-defined sense simultaneously represents them all. We review these four models and investigate how they are related. Motivated by the parsimony principle, one often aims to construct a network that contains as few reticulations (non-treelike evolutionary events) as possible. In general, the model chosen influences the minimum number of reticulation events required. However, when one obtains the input data from two binary (i.e. fully resolved) trees, we show that the minimum number of reticulations is independent of the model. The number of reticulations necessary to represent the trees, triplets, clusters (in the softwired sense) and characters (with unrestricted multiple crossover recombination) are all equal. Furthermore, we show that these results also hold when not the number of reticulations but the level of the constructed network is minimised. We use these unification results to settle several computational complexity questions that have been open in the field for some time. We also give explicit examples to show that already for data obtained from three binary trees the models begin to diverge.     

Comparing and displaying phylogenetic trees using edge union networks

The general problem of representing collections of trees as a single graph has led to many tree summary techniques. Many consensus approaches take sets of trees (either inferred as separate gene trees or gleaned from the posterior of a Bayesian analysis) and produce a single “best” tree. In scenarios where horizontal gene transfer or hybridization are suspected, networks may be preferred, which allow for nodes to have two parents, representing the fusion of lineages. One such construct is the cluster union network (CUN), which is constructed using the union of all clusters in the input trees. The CUN has a number of mathematically desirable properties, but can also present edges not observed in the input trees. In this paper we define a new network construction, the edge union network (EUN), which displays edges if and only if they are contained in the input trees. We also demonstrate that this object can be constructed with polynomial time complexity given arbitrary phylogenetic input trees, and so can be used in conjunction with network analysis techniques for further phylogenetic hypothesis testing.     

Evolutionary history as a driver of ecological networks: a case study of plant–hummingbird interactions

Multiple factors drive species interactions in ecological networks, such as morphological barriers, spatio–temporal distribution, abundances and evolutionary histories of species. Novel methods are making it possible to evaluate the relative importance of each of these drivers. However, the lack of appropriate methods has prevented evaluating the extent to which interaction networks are shaped by species’ evolutionary histories. This study includes the evolutionary histories of species among the potential drivers of interactions, allowing the comparative analysis of its importance in structuring ecological networks. We hypothesized different possible phylogenetic scenarios to predict frequencies of interactions between species by combining concepts from the fields of ecological networks and ecophylogenetics. The usage of these scenarios is illustrated in a plant–hummingbird interaction network database from the Atlantic Forest, southeastern Brazil. We first evaluated which phylogenetic hypotheses better predict the observed network; subsequently, we evaluated the relative importance of species evolutionary histories, abundances, and matching on species morphologies and phenologies as drivers of their frequencies of interactions. The results suggest that the evolutionary histories of hummingbirds are more important than the species abundances in structuring the studied plant–hummingbird network but less important than the morphological and phenological matching among species. The approach developed here offers the potential to advance our understanding of the multiple factors structuring ecological networks.     

Using max cut to enhance rooted trees consistency

Supertree methods are used to construct a large tree over a large set of taxa from a set of small trees over overlapping subsets of the complete taxa set. Since accurate reconstruction methods are currently limited to a maximum of a few dozen taxa, the use of a supertree method in order to construct the tree of life is inevitable. Supertree methods are broadly divided according to the input trees: When the input trees are unrooted, the basic reconstruction unit is a quartet tree. In this case, the basic decision problem of whether there exists a tree that agrees with all quartets is NP-complete. On the other hand, when the input trees are rooted, the basic reconstruction unit is a rooted triplet and the above decision problem has a polynomial time algorithm. However, when there is no tree which agrees with all triplets, it would be desirable to find the tree that agrees with the maximum number of triplets. However, this optimization problem was shown to be NP-hard. Current heuristic approaches perform min cut on a graph representing the triplets inconsistency and return a tree that is guaranteed to satisfy some required properties. In this work, we present a different heuristic approach that guarantees the properties provided by the current methods and give experimental evidence that it significantly outperforms currently used methods. This method is based on a divide and conquer approach, where the min cut in the divide step is replaced by a max cut in a variant of the same graph. The latter is achieved by a lightweight semidefinite programming-like heuristic that leads to very fast running times     

Quartets and unrooted phylogenetic networks

Phylogenetic networks were introduced to describe evolution in the presence of exchanges of genetic material between coexisting species or individuals. Split networks in particular were introduced as a special kind of abstract network to visualize conflicts between phylogenetic trees which may correspond to such exchanges. More recently, methods were designed to reconstruct explicit phylogenetic networks (whose vertices can be interpreted as biological events) from triplet data. In this article, we link abstract and explicit networks through their combinatorial properties, by introducing the unrooted analog of level-k networks. In particular, we give an equivalence theorem between circular split systems and unrooted level-1 networks. We also show how to adapt to quartets some existing results on triplets, in order to reconstruct unrooted level-k phylogenetic networks. These results give an interesting perspective on the combinatorics of phylogenetic networks and also raise algorithmic and combinatorial questions.     

Genetic programming of polynomial harmonic networks using the discrete Fourier transform

This paper presents a genetic programming system that evolves polynomial harmonic networks. These are multilayer feed-forward neural networks with polynomial activation functions. The novel hybrids assume that harmonics with non-multiple frequencies may enter as inputs the activation polynomials. The harmonics with non-multiple, irregular frequencies are derived analytically using the discrete Fourier transform. The polynomial harmonic networks have tree-structured topology which makes them especially suitable for evolutionary structural search. Empirical results show that this hybrid genetic programming system outperforms an evolutionary system manipulating polynomials, the traditional Koza-style genetic programming, and the harmonic GMDH network algorithm on processing time series.     

Massive-Scale Gene Co-Expression Network Construction and Robustness Testing Using Random Matrix Theory

The study of gene relationships and their effect on biological function and phenotype is a focal point in systems biology. Gene co-expression networks built using microarray expression profiles are one technique for discovering and interpreting gene relationships. A knowledge-independent thresholding technique, such as Random Matrix Theory (RMT), is useful for identifying meaningful relationships. Highly connected genes in the thresholded network are then grouped into modules that provide insight into their collective functionality. While it has been shown that co-expression networks are biologically relevant, it has not been determined to what extent any given network is functionally robust given perturbations in the input sample set. For such a test, hundreds of networks are needed and hence a tool to rapidly construct these networks. To examine functional robustness of networks with varying input, we enhanced an existing RMT implementation for improved scalability and tested functional robustness of human (Homo sapiens), rice (Oryza sativa) and budding yeast (Saccharomyces cerevisiae). We demonstrate dramatic decrease in network construction time and computational requirements and show that despite some variation in global properties between networks, functional similarity remains high. Moreover, the biological function captured by co-expression networks thresholded by RMT is highly robust.     

Networks in phylogenetic analysis: new tools for population biology

Phylogenetic analysis has changed greatly in the past decade, including the more widespread appreciation of the idea that evolutionary histories are not always tree-like, and may, thus, be best represented as reticulated networks rather than as strictly dichotomous trees. Reconstructing such histories in the absence of a bifurcating speciation process is even more difficult than the usual procedure, and a range of alternative strategies have been developed. There seem to be two basic uses for a network model of evolution: the display of real but unobservable evolutionary events (i.e. a hypothesis of the true phylogenetic history), and the display of character conflict within the data itself (i.e. a summary of the data). These two general approaches are briefly reviewed here, and the strengths and weaknesses of the different implementations are compared and contrasted. Each network methodology seems to have limitations in terms of how it responds to increasing complexity (e.g. conflict) in the data, and therefore each is likely to be more appropriate for one of the two uses than for the other. Several examples using parasitological data sets illustrate the uses of networks within the context of population biology.     

All that glisters is not galled

Galled trees, evolutionary networks with isolated reticulation cycles, have appeared under several slightly different definitions in the literature. In this paper, we establish the actual relationships between the main four such alternative definitions: namely, the original galled trees, level-1 networks, nested networks with nesting depth 1, and evolutionary networks with arc-disjoint reticulation cycles.     

Do Triplets Have Enough Information to Construct the Multi-Labeled Phylogenetic Tree?

The evolutionary history of certain species such as polyploids are modeled by a generalization of phylogenetic trees called multi-labeled phylogenetic trees, or MUL trees for short. One problem that relates to inferring a MUL tree is how to construct the smallest possible MUL tree that is consistent with a given set of rooted triplets, or SMRT problem for short. This problem is NP-hard. There is one algorithm for the SMRT problem which is exact and runs in time, where is the number of taxa. In this paper, we show that the SMRT does not seem to be an appropriate solution from the biological point of view. Indeed, we present a heuristic algorithm named MTRT for this problem and execute it on some real and simulated datasets. The results of MTRT show that triplets alone cannot provide enough information to infer the true MUL tree. So, it is inappropriate to infer a MUL tree using triplet information alone and considering the minimum number of duplications. Finally, we introduce some new problems which are more suitable from the biological point of view.     

Detecting phylogenetic signal in mutualistic interaction networks using a Markov process model

Ecological interaction networks, such as those describing the mutualistic interactions between plants and their pollinators or between plants and their frugivores, exhibit non‐random structural properties that cannot be explained by simple models of network formation. One factor affecting the formation and eventual structure of such a network is its evolutionary history. We argue that this, in many cases, is closely linked to the evolutionary histories of the species involved in the interactions. Indeed, empirical studies of interaction networks along with the phylogenies of the interacting species have demonstrated significant associations between phylogeny and network structure. To date, however, no generative model explaining the way in which the evolution of individual species affects the evolution of interaction networks has been proposed. We present a model describing the evolution of pairwise interactions as a branching Markov process, drawing on phylogenetic models of molecular evolution. Using knowledge of the phylogenies of the interacting species, our model yielded a significantly better fit to 21% of a set of plant–pollinator and plant–frugivore mutualistic networks. This highlights the importance, in a substantial minority of cases, of inheritance of interaction patterns without excluding the potential role of ecological novelties in forming the current network architecture. We suggest that our model can be used as a null model for controlling evolutionary signals when evaluating the role of other factors in shaping the emergence of ecological networks.     

Inferring phylogenetic relationships avoiding forbidden rooted triplets

To construct a phylogenetic tree or phylogenetic network for describing the evolutionary history of a set of species is a well-studied problem in computational biology. One previously proposed method to infer a phylogenetic tree/network for a large set of species is by merging a collection of known smaller phylogenetic trees on overlapping sets of species so that no (or as little as possible) branching information is lost. However, little work has been done so far on inferring a phylogenetic tree/network from a specified set of trees when in addition, certain evolutionary relationships among the species are known to be highly unlikely. In this paper, we consider the problem of constructing a phylogenetic tree/network which is consistent with all of the rooted triplets in a given set C and none of the rooted triplets in another given set F. Although NP-hard in the general case, we provide some efficient exact and approximation algorithms for a number of biologically meaningful variants of the problem.     

Application of phylogenetic networks in evolutionary studies

The evolutionary history of a set of taxa is usually represented by a phylogenetic tree, and this model has greatly facilitated the discussion and testing of hypotheses. However, it is well known that more complex evolutionary scenarios are poorly described by such models. Further, even when evolution proceeds in a tree-like manner, analysis of the data may not be best served by using methods that enforce a tree structure but rather by a richer visualization of the data to evaluate its properties, at least as an essential first step. Thus, phylogenetic networks should be employed when reticulate events such as hybridization, horizontal gene transfer, recombination, or gene duplication and loss are believed to be involved, and, even in the absence of such events, phylogenetic networks have a useful role to play. This article reviews the terminology used for phylogenetic networks and covers both split networks and reticulate networks, how they are defined, and how they can be interpreted. Additionally, the article outlines the beginnings of a comprehensive statistical framework for applying split network methods. We show how split networks can represent confidence sets of trees and introduce a conservative statistical test for whether the conflicting signal in a network is treelike. Finally, this article describes a new program, SplitsTree4, an interactive and comprehensive tool for inferring different types of phylogenetic networks from sequences, distances, and trees.     

Reconstructible Phylogenetic Networks: Do Not Distinguish the Indistinguishable

Phylogenetic networks represent the evolution of organisms that have undergone reticulate events, such as recombination, hybrid speciation or lateral gene transfer. An important way to interpret a phylogenetic network is in terms of the trees it displays, which represent all the possible histories of the characters carried by the organisms in the network. Interestingly, however, different networks may display exactly the same set of trees, an observation that poses a problem for network reconstruction: from the perspective of many inference methods such networks are indistinguishable. This is true for all methods that evaluate a phylogenetic network solely on the basis of how well the displayed trees fit the available data, including all methods based on input data consisting of clades, triples, quartets, or trees with any number of taxa, and also sequence-based approaches such as popular formalisations of maximum parsimony and maximum likelihood for networks. This identifiability problem is partially solved by accounting for branch lengths, although this merely reduces the frequency of the problem. Here we propose that network inference methods should only attempt to reconstruct what they can uniquely identify. To this end, we introduce a novel definition of what constitutes a uniquely reconstructible network. For any given set of indistinguishable networks, we define a canonical network that, under mild assumptions, is unique and thus representative of the entire set. Given data that underwent reticulate evolution, only the canonical form of the underlying phylogenetic network can be uniquely reconstructed. While on the methodological side this will imply a drastic reduction of the solution space in network inference, for the study of reticulate evolution this is a fundamental limitation that will require an important change of perspective when interpreting phylogenetic networks.