Level-1 Networks Research Articles

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.

A cubic-time algorithm for computing the trinet distance between level-1 networks

In evolutionary biology, phylogenetic networks are constructed to represent the evolution of species in which reticulate events are thought to have occurred, such as recombination and hybridization. It is therefore useful to have efficiently computable metrics with which to systematically compare such networks. Through developing an optimal algorithm to enumerate all trinets displayed by a level-1 network (a type of network that is slightly more general than an evolutionary tree), here we propose a cubic-time algorithm to compute the trinet distance between two level-1 networks. Employing simulations, we also present a comparison between the trinet metric and the so-called Robinson–Foulds phylogenetic network metric restricted to level-1 networks. The algorithms described in this paper have been implemented in JAVA and are freely available at https://www.uea.ac.uk/computing/TriLoNet.

On the challenge of reconstructing level-1 phylogenetic networks from triplets and clusters

Phylogenetic networks have gained prominence over the years due to their ability to represent complex non-treelike evolutionary events such as recombination or hybridization. Popular combinatorial objects used to construct them are triplet systems and cluster systems, the motivation being that any network N induces a triplet system mathcal R(N) and a softwired cluster system mathcal S(N). Since in real-world studies it cannot be guaranteed that all triplets/softwired clusters induced by a network are available, it is of particular interest to understand whether subsets of mathcal R(N) or mathcal S(N) allow one to uniquely reconstruct the underlying network N. Here we show that even within the highly restricted yet biologically interesting space of level-1 phylogenetic networks it is not always possible to uniquely reconstruct a level-1 network N, even when all triplets in mathcal R(N) or all clusters in mathcal S(N) are available. On the positive side, we introduce a reasonably large subclass of level-1 networks the members of which are uniquely determined by their induced triplet/softwired cluster systems. Along the way, we also establish various enumerative results, both positive and negative, including results which show that certain special subclasses of level-1 networks N can be uniquely reconstructed from proper subsets of mathcal R(N) and mathcal S(N). We anticipate these results to be of use in the design of algorithms for phylogenetic network inference.

TriLoNet: Piecing Together Small Networks to Reconstruct Reticulate Evolutionary Histories.

Phylogenetic networks are a generalization of evolutionary trees that can be used to represent reticulate processes such as hybridization and recombination. Here, we introduce a new approach called TriLoNet (Trinet Level- one Network algorithm) to construct such networks directly from sequence alignments which works by piecing together smaller phylogenetic networks. More specifically, using a bottom up approach similar to Neighbor-Joining, TriLoNet constructs level-1 networks (networks that are somewhat more general than trees) from smaller level-1 networks on three taxa. In simulations, we show that TriLoNet compares well with Lev1athan, a method for reconstructing level-1 networks from three-leaved trees. In particular, in simulations we find that Lev1athan tends to generate networks that overestimate the number of reticulate events as compared with those generated by TriLoNet. We also illustrate TriLoNet's applicability using simulated and real sequence data involving recombination, demonstrating that it has the potential to reconstruct informative reticulate evolutionary histories. TriLoNet has been implemented in JAVA and is freely available at https://www.uea.ac.uk/computing/TriLoNet.

Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets

Binets and trinets are phylogenetic networks with two and three leaves, respectively. Here we consider the problem of deciding if there exists a binary level-1 phylogenetic network displaying a given set mathbb {T} of binary binets or trinets over a taxon set X, and constructing such a network whenever it exists. We show that this is NP-hard for trinets but polynomial-time solvable for binets. Moreover, we show that the problem is still polynomial-time solvable for inputs consisting of binets and trinets as long as the cycles in the trinets have size three. Finally, we present an O(3^{|X|} poly(|X|)) time algorithm for general sets of binets and trinets. The latter two algorithms generalise to instances containing level-1 networks with arbitrarily many leaves, and thus provide some of the first supernetwork algorithms for computing networks from a set of rooted phylogenetic networks.

Spaces of phylogenetic networks from generalized nearest-neighbor interchange operations.

Phylogenetic networks are a generalization of evolutionary or phylogenetic trees that are used to represent the evolution of species which have undergone reticulate evolution. In this paper we consider spaces of such networks defined by some novel local operations that we introduce for converting one phylogenetic network into another. These operations are modeled on the well-studied nearest-neighbor interchange operations on phylogenetic trees, and lead to natural generalizations of the tree spaces that have been previously associated to such operations. We present several results on spaces of some relatively simple networks, called level-1 networks, including the size of the neighborhood of a fixed network, and bounds on the diameter of the metric defined by taking the smallest number of operations required to convert one network into another. We expect that our results will be useful in the development of methods for systematically searching for optimal phylogenetic networks using, for example, likelihood and Bayesian approaches.

Reconstructing a phylogenetic level-1 network from quartets.

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on [Formula: see text] taxa from the set of all quartets containing a certain fixed taxon, in [Formula: see text] time. We also present a more general method which can handle more diverse quartet data, but which takes [Formula: see text] time. Both methods proceed by solving a certain system of linear equations over the two-element field [Formula: see text]. For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our [Formula: see text] algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an [Formula: see text]-sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.

A SHORT INTRODUCTION TO SOME RECENT PROGRESS IN PHYLOGENETIC NETWORK RECONSTRUCTION, GENOME MAPPING, GENE EXPRESSION ANALYSIS, MOLECULAR DYNAMIC SIMULATION, AND OTHER PROBLEMS IN BIOINFORMATICS

Journal of Bioinformatics and Computational BiologyVol. 10, No. 04, 1203002 (2012) Free AccessA SHORT INTRODUCTION TO SOME RECENT PROGRESS IN PHYLOGENETIC NETWORK RECONSTRUCTION, GENOME MAPPING, GENE EXPRESSION ANALYSIS, MOLECULAR DYNAMIC SIMULATION, AND OTHER PROBLEMS IN BIOINFORMATICSLIMSOON WONGLIMSOON WONG Search for more papers by this author https://doi.org/10.1142/S0219720012030023Cited by:1 Next AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail References Y.-J. Heet al., J. Bioinform. Comput. Biol. 4(1), 59 (2006). Link, Google ScholarL. van Iersel, S. Kelk and M. Mnich, J. Bioinform. Comput. Biol. 7(4), 597 (2009). Link, Google ScholarP. Gambette, B. Berry and C. Paul, J. Bioinform. Comput. Biol. 10(4), 1250004 (2012). Link, Google ScholarR. Pokrzywa, J. Bioinform. Comput. Biol. 6(2), 403 (2008). Link, Google ScholarK. Kimura, A. Koike and K. Nakai, J. Bioinform. Comput. Biol. 10(4), 1250002 (2012). Link, Google ScholarL. Licamele and L. Getoor, J. Bioinform. Comput. Biol. 9(3), 431 (2011). Link, Google ScholarO. Obidat and C. K. Reddy, J. Bioinform. Comput. Biol. 10(1), 1240002 (2012). Medline, Google ScholarV. Olmanet al., J. Bioinform. Comput. Biol. 4(5), 999 (2006). Link, Google ScholarQ. Han and G. Dong, J. Bioinform. Comput. Biol. 10(4), 1250005 (2012). Link, Google ScholarJ. Xu, J. Bioinform. Comput. Biol. 10(4), 1250007 (2012). Link, Google ScholarR. Kanagasabaiet al., J. Bioinform. Comput. Biol. 5(6), 1319 (2007). Link, Google ScholarC. H. Kohet al., J. Bioinform. Comput. Biol. 7(6), 973 (2009). Link, Google ScholarB. Limayeet al., J. Bioinform. Comput. Biol. 10(4), 1250006 (2012). Link, Google ScholarA. A. Terentievet al., J. Bioinform. Comput. Biol. 10(2), 1241012 (2012). Link, Google ScholarS. Hwanget al., J. Bioinform. Comput. Biol. 9(), 15 (2011). Link, Google ScholarN. C. Benson and V. Daggett, J. Bioinform. Comput. Biol. 10(4), 1250008 (2012). Link, Google ScholarR. Butawita and V. Palade, J. Bioinform. Comput. Biol. 10(4), 1250003 (2012). Link, Google ScholarC. Liet al., J. Bioinform. Comput. Biol. 4(5), 1119 (2006). Link, Google ScholarS. Ramseyet al., J. Bioinform. Comput. Biol. 3(2), 415 (2005). Link, Google ScholarD. Calcadaet al., J. Bioinform. Comput. Biol. 9(5), 613 (2011). Link, Google ScholarB. Liu and P. S. Thiagarajan, J. Bioinform. Comput. Biol. 10(4), 1231001 (2012). Link, Google Scholar FiguresReferencesRelatedDetailsCited By 1A study of the lipid-mediated dimerization of the RAGE TM+JM domains by molecular dynamic simulationsJialin Chen, Fude Sun, Peng Chen, Mengya Chai and Lida Xu et al.1 Jul 2018 | Chinese Chemical Letters, Vol. 29, No. 7 Recommended Vol. 10, No. 04 Metrics History PDF download

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.

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.

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.

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 > or = 1 it is NP-hard to construct a level-k network consistent with all input triplets, and (2) for all k > or = 0 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.

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.

Constructing Level-2 Phylogenetic Networks from Triplets

Jansson and Sung showed that, given a dense set of input triplets T (representing hypotheses about the local evolutionary relationships of triplets of taxa), it is possible to determine in polynomial time whether there exists a level-1 network consistent with T, and if so, to construct such a network [24]. Here, we extend this work by showing that this problem is even polynomial time solvable for the construction of level-2 networks. This shows that, assuming density, it is tractable to construct plausible evolutionary histories from input triplets even when such histories are heavily nontree-like. This further strengthens the case for the use of triplet-based methods in the construction of phylogenetic networks. We also implemented the algorithm and applied it to yeast data.

Integrated Narrowband/Broadband Access Architecture

Southwestern Bell Telephone (SWBT) Technology Planning chose a digital fiber-optic, integrated narrowband/broadband fiber-to-the-curb (FTTC) access architecture for deployment in the Richardson, Texas, video trial. The trial architecture consists of a two-level network: Level 1 and Level 2. The Level-1 network is the infrastructure provided by SWBT. The Level-2 network is that portion which provides the digital video services and the analog broadcast services. This paper describes the network designed for SWBT that will be used to replace and extend the existing telephony transport and access network in Richardson with a state-of-the-art network that will provide telephony and other narrowband services, analog broadcast video services, digital broadcast services, and interactive switched digital video services. Also discussed are some of the new operations, administration, maintenance, and provisioning systems (OAM&P) required in this new environment. The characteristics and challenges of this new network, new technology, and the Richardson video trial are examined in detail.

The Pathwidth and Treewidth of Cographs

It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.