Level-1 Networks Research Articles

Complex Hybridization in a Clade of Polytypic Salamanders (Plethodontidae: Desmognathus) Uncovered by Estimating Higher-Level Phylogenetic Networks.

Reticulation between incipient lineages is a common feature of diversification. We examine these phenomena in the Pisgah clade of Desmognathus salamanders from the southern Appalachian Mountains of the eastern United States. The group contains four to seven species exhibiting two discrete phenotypes, aquatic "shovel-nosed" and semi-aquatic "black-bellied" forms. These ecomorphologies are ancient and have apparently been transmitted repeatedly between lineages through introgression. Geographically proximate populations of both phenotypes exhibit admixture, and at least two black-bellied lineages have been produced via reticulations between shovel-nosed parentals, suggesting potential hybrid speciation dynamics. However, computational constraints currently limit our ability to reconstruct network radiations from gene-tree data. Available methods are limited to level-1 networks wherein reticulations do not share edges, and higher-level networks may be non-identifiable in many cases. We present a heuristic approach to recover information from higher-level networks across a range of potentially identifiable empirical scenarios, supported by theory and simulation. When extrinsic information indicates the location and direction of reticulations, our method can successfully estimate a reduced possible set of non-level-1 networks. Phylogenomic data support a single backbone topology with up to five overlapping hybrid edges in the Pisgah clade. These results suggest an unusual mechanism of ecomorphological hybrid speciation, wherein a binary threshold trait causes some hybrid populations to shift between microhabitat niches, promoting ecological divergence between sympatric hybrids and parentals. This contrasts with other well-known systems in which hybrids exhibit intermediate, novel, or transgressive phenotypes. The genetic basis of these phenotypes is unclear and further data are needed to clarify the evolutionary basis of morphological changes with ecological consequences.

Inference of Phylogenetic Networks from Sequence Data using Composite Likelihood.

While phylogenies have been essential in understanding how species evolve, they do not adequately describe some evolutionary processes. For instance, hybridization, a common phenomenon where interbreeding between two species leads to formation of a new species, must be depicted by a phylogenetic network, a structure that modifies a phylogenetic tree by allowing two branches to merge into one, resulting in reticulation. However, existing methods for estimating networks become computationally expensive as the dataset size and/or topological complexity increase. The lack of methods for scalable inference hampers phylogenetic networks from being widely used in practice, despite accumulating evidence that hybridization occurs frequently in nature. Here, we propose a novel method, PhyNEST (Phylogenetic Network Estimation using SiTe patterns), that estimates binary, level-1 phylogenetic networks with a fixed, user-specified number of reticulations directly from sequence data. By using the composite likelihood as the basis for inference, PhyNEST is able to use the full genomic data in a computationally tractable manner, eliminating the need to summarize the data as a set of gene trees prior to network estimation. To search network space, PhyNEST implements both hill climbing and simulated annealing algorithms. PhyNEST assumes that the data are composed of coalescent independent sites that evolve according to the Jukes-Cantor substitution model and that the network has a constant effective population size. Simulation studies demonstrate that PhyNEST is often more accurate than two existing composite likelihood summary methods (SNaQ and PhyloNet) and that it is robust to at least one form of model misspecification (assuming a less complex nucleotide substitution model than the true generating model). We applied PhyNEST to reconstruct the evolutionary relationships among Heliconius butterflies and Papionini primates, characterized by hybrid speciation and widespread introgression, respectively. PhyNEST is implemented in an open-source Julia package and is publicly available at https://github.com/sungsik-kong/PhyNEST.jl.

Identifiability of Level-1 Species Networks from Gene Tree Quartets

When hybridization or other forms of lateral gene transfer have occurred, evolutionary relationships of species are better represented by phylogenetic networks than by trees. While inference of such networks remains challenging, several recently proposed methods are based on quartet concordance factors—the probabilities that a tree relating a gene sampled from the species displays the possible 4-taxon relationships. Building on earlier results, we investigate what level-1 network features are identifiable from concordance factors under the network multispecies coalescent model. We obtain results on both topological features of the network, and numerical parameters, uncovering a number of failures of identifiability related to 3-cycles in the network. Addressing these identifiability issues is essential for designing statistically consistent inference methods.

Exploring the Distribution of Phylogenetic Networks Generated Under a Birth-Death-Hybridization Process

Gene-flow processes such as hybridization and introgression play important roles in shaping diversity across the tree of life. Recent studies extending birth-death models have made it possible to investigate patterns of reticulation in a macroevolutionary context. These models allow for different macroevolutionary patterns of gene flow events that can either add, maintain, or remove lineages—with the gene flow itself possibly being dependent on the relatedness between species—thus creating complex diversification scenarios. Further, many reticulate phylogenetic inference methods assume specific reticulation structures or phylogenies belonging to certain network classes. However, the distributions of phylogenetic networks under reticulate birth-death processes are poorly characterized, and it is unknown whether they violate common methodological assumptions. We use simulation techniques to explore phylogenetic network space under a birth-death-hybridization process where the hybridization rate can have a linear dependence on genetic distance. Specifically, we measured the number of lineages through time and role of hybridization in diversification along with the proportion of phylogenetic networks that belong to commonly used network classes (e.g., tree-child, tree-based, or level-1 networks). We find that the growth of phylogenetic networks and class membership are largely affected by assumptions about macroevolutionary patterns of gene flow. In accordance with previous studies, a lower proportion of networks belonged to these classes based on type and density of reticulate events. However, under a birth-death-hybridization process, these factors form an antagonistic relationship; the type of reticulation events that cause high membership proportions also lead to the highest reticulation density, consequently lowering the overall proportion of phylogenies in some classes. Further, we observed that genetic distance–dependent gene flow and incomplete sampling increase the proportion of class membership, primarily due to having fewer reticulate events. Our results can inform studies if their biological expectations of gene flow are associated with evolutionary histories that satisfy the assumptions of current methodology and aid in finding phylogenetic classes that are relevant for methods development.

A Fixed-Parameter Tractable Algorithm for Finding Agreement Cherry-Reduced Subnetworks in Level-1 Orchard Networks.

Phylogenetic networks are increasingly being considered better suited to represent the complexity of the evolutionary relationships between species. One class of phylogenetic networks that have received a lot of attention recently is the class of orchard networks, which is composed of networks that can be reduced to a single leaf using cherry reductions. Cherry reductions, also called cherry-picking operations, remove either a leaf of a simple cherry (sibling leaves sharing a parent) or a reticulate edge of a reticulate cherry (two leaves whose parents are connected by a reticulate edge). In this article, we present a fixed-parameter tractable algorithm to solve the problem of finding a maximum agreement cherry-reduced subnetwork (MACRS) between two rooted binary level-1 networks. This is the first exact algorithm proposed to solve the MACRS problem. As proven in an earlier work, there is a direct relationship between finding an MACRS and calculating a distance based on cherry operations. As a result, the proposed algorithm also provides a distance that can be used for the comparison of level-1 networks.

Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman model

Phylogenetic networks model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models used for sequence evolution on trees can also be extended to networks and similar problems, such as determining if the network parameter is identifiable or finding the invariants of the model, can be studied. This paper focuses on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We determine all quadratic invariants for sunlet networks, and conjecture these generate the full sunlet ideal.

Exploring spaces of semi-directed level-1 networks

Semi-directed phylogenetic networks have recently emerged as a class of phylogenetic networks sitting between rooted (directed) and unrooted (undirected) phylogenetic networks as they contain both directed as well as undirected edges. While various spaces of rooted phylogenetic networks and unrooted phylogenetic networks have been analyzed in recent years and several rearrangement moves to traverse these spaces have been introduced, little is known about spaces of semi-directed phylogenetic networks. Here, we propose a simple rearrangement move for semi-directed phylogenetic networks, called cut edge transfer (CET), and show that the space of semi-directed level-1 networks with precisely k reticulations is connected under CET. This level-1 space is currently the predominantly used search space for most algorithms that reconstruct semi-directed phylogenetic networks. Our results imply that every semi-directed level-1 network with a fixed number of reticulations and leaf set can be reached from any other such network by a sequence of CETs. By introducing two additional moves, R^+ and R^-, that allow for the addition and deletion, respectively, of a reticulation, we then establish connectedness for the space of all semi-directed level-1 networks on a fixed leaf set. As a byproduct of our results for semi-directed phylogenetic networks, we also show that the space of rooted level-1 networks with a fixed number of reticulations and leaf set is connected under CET, when translated into the rooted setting.

Clustering systems of phylogenetic networks.

Rooted acyclic graphs appear naturally when the phylogenetic relationship of a set X of taxa involves not only speciations but also recombination, horizontal transfer, or hybridization that cannot be captured by trees. A variety of classes of such networks have been discussed in the literature, including phylogenetic, level-1, tree-child, tree-based, galled tree, regular, or normal networks as models of different types of evolutionary processes. Clusters arise in models of phylogeny as the sets [Formula: see text] of descendant taxa of a vertex v. The clustering system [Formula: see text] comprising the clusters of a network N conveys key information on N itself. In the special case of rooted phylogenetic trees, T is uniquely determined by its clustering system [Formula: see text]. Although this is no longer true for networks in general, it is of interest to relate properties of N and [Formula: see text]. Here, we systematically investigate the relationships of several well-studied classes of networks and their clustering systems. The main results are correspondences of classes of networks and clustering systems of the following form: If N is a network of type [Formula: see text], then [Formula: see text] satisfies [Formula: see text], and conversely if [Formula: see text] is a clustering system satisfying [Formula: see text] then there is network N of type [Formula: see text] such that [Formula: see text].This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.

Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks

In the context of reconstructing phylogenetic networks from a collection of phylogenetic trees, several characterisations and subsequently algorithms have been established to reconstruct a phylogenetic network that collectively embeds all trees in the input in some minimum way. For many instances however, the resulting network also embeds additional phylogenetic trees that are not part of the input. However, little is known about these inferred trees. In this paper, we explore the relationships among all phylogenetic trees that are embedded in a given phylogenetic network. First, we investigate some combinatorial properties of the collection P of all rooted binary phylogenetic trees that are embedded in a rooted binary phylogenetic network N. To this end, we associated a particular graph G, which we call rSPR graph, with the elements in P and show that, if |P|=2k, where k is the number of vertices with in-degree two in N, then G has a Hamilton cycle. Second, by exploiting rSPR graphs and properties of hypercubes, we turn to the well-studied class of rooted binary level-1 networks and give necessary and sufficient conditions for when a set of rooted binary phylogenetic trees can be embedded in a level-1 network without inferring any additional trees. Lastly, we show how these conditions translate into a polynomial-time algorithm to reconstruct such a network if it exists.

Polynomial invariants for cactuses

Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial invariants for graphs such as the well-known Tutte polynomial have been studied for several years, and recently there has been interest to also define such invariants for phylogenetic networks, a special type of graph that arises in the area of evolutionary biology. Recently Liu gave a complete invariant for (phylogenetic) trees. However, the polynomial invariants defined thus far for phylogenetic networks that are not trees require vertex labels and either contain a large number of variables, or they have exponentially many terms in the number of reticulations. This can make it difficult to compute these polynomials and to use them to analyse unlabelled networks. In this paper, we shall show how to circumvent some of these difficulties for rooted cactuses and cactuses. As well as being important in other areas such as operations research, rooted cactuses contain some common classes of phylogenetic networks such phylogenetic trees and level-1 networks. More specifically, we define a polynomial F that is a complete invariant for the class of rooted cactuses without vertices of indegree 1 and outdegree 1 that has 5 variables, and a polynomial Q that is a complete invariant for the class of rooted cactuses that has 6 variables whose degree can be bounded linearly in terms of the size of the rooted cactus. We also explain how to extend the Q polynomial to define a complete invariant for leaf-labelled rooted cactuses as well as (unrooted) cactuses.

Identifiability of local and global features of phylogenetic networks from average distances

Phylogenetic networks extend phylogenetic trees to model non-vertical inheritance, by which a lineage inherits material from multiple parents. The computational complexity of estimating phylogenetic networks from genome-wide data with likelihood-based methods limits the size of networks that can be handled. Methods based on pairwise distances could offer faster alternatives. We study here the information that average pairwise distances contain on the underlying phylogenetic network, by characterizing local and global features that can or cannot be identified. For general networks, we clarify that the root and edge lengths adjacent to reticulations are not identifiable, and then focus on the class of zipped-up semidirected networks. We provide a criterion to swap subgraphs locally, such as 3-cycles, resulting in indistinguishable networks. We propose the "distance split tree", which can be constructed from pairwise distances, and prove that it is a refinement of the network's tree of blobs, capturing the tree-like features of the network. For level-1 networks, this distance split tree is equal to the tree of blobs refined to separate polytomies from blobs, and we prove that the mixed representation of the network is identifiable. The information loss is localized around 4-cycles, for which the placement of the reticulation is unidentifiable. The mixed representation combines split edges for 4-cycles, regular tree and hybrid edges from the semidirected network, and edge parameters that encode all information identifiable from average pairwise distances.

From modular decomposition trees to level-1 networks: Pseudo-cographs, polar-cats and prime polar-cats

The modular decomposition of a graph G is a natural construction to capture key features of G in terms of a labeled tree ( T , t ) whose vertices are labeled as “series” (1), “parallel” (0) or “prime”. However, full information of G is provided by its modular decomposition tree ( T , t ) only, if G does not contain prime modules. In this case, ( T , t ) explains G , i.e., { x , y } ∈ E ( G ) if and only if the lowest common ancestor lca T ( x , y ) of x and y has label “1”. This information, however, gets lost whenever ( T , t ) contains vertices with label “prime”. In this contribution, we aim at replacing “prime” vertices in ( T , t ) by simple 0/1-labeled cycles, which leads to the concept of rooted labeled level-1 networks ( N , t ) . We characterize graphs that can be explained by such level-1 networks ( N , t ) , which generalizes the concept of graphs that can be explained by labeled trees, that is, cographs. We provide three novel graph classes: polar-cats are a proper subclass of pseudo-cographs which forms a proper subclass of prime polar-cats . In particular, every cograph is a pseudo-cograph and prime polar-cats are precisely those graphs that can be explained by a labeled level-1 network. The class of prime polar-cats is defined in terms of the modular decomposition of graphs and the property that all prime modules “induce” polar-cats. We provide a plethora of structural results and characterizations for graphs of these new classes. In particular, Polar-cats are precisely those graphs that can be explained by an elementary level-1 network ( N , t ) , i.e., ( N , t ) contains exactly one cycle C that is rooted at the root ρ N of N and where ρ N has exactly two children while every vertex distinct from ρ N has a unique child that is not located in C . Pseudo-cographs are less restrictive and those graphs that can be explained by particular level-1 networks ( N , t ) that contain at most one cycle. These findings, eventually, help us to characterize the class of all graphs that can be explained by labeled level-1 networks, namely prime polar-cats. Moreover, we show under which conditions there is a unique least-resolved labeled level-1 network that explains a given graph. In addition, we provide linear-time algorithms to recognize all these types of graphs and to construct level-1 networks to explain them.

A Polynomial-Time Algorithm for Minimizing the Deep Coalescence Cost for Level-1 Species Networks.

Phylogenetic analyses commonly assume that the species history can be represented as a tree. However, in the presence of hybridization, the species history is more accurately captured as a network. Despite several advances in modeling phylogenetic networks, there is no known polynomial-time algorithm for parsimoniously reconciling gene trees with species networks while accounting for incomplete lineage sorting. To address this issue, we present a polynomial-time algorithm for the case of level-1 networks, in which no hybrid species is the direct ancestor of another hybrid species. This work enables more efficient reconciliation of gene trees with species networks, which in turn, enables more efficient reconstruction of species networks.

An algorithm for reconstructing level-2 phylogenetic networks from trinets

Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of a network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, level-1 networks may not be able to capture the complexity of some data sets. In this paper, we present a polynomial-time algorithm for constructing a rooted binary level-2 phylogenetic network from a collection of 3-leaf networks or trinets. Moreover, we prove that the algorithm will correctly reconstruct such a network if it is given all of the trinets in the network as input. The algorithm runs in time O(t⋅n+n4) with t the number of input trinets and n the number of leaves. We also show that there is a fundamental obstruction to constructing level-3 networks from trinets, and so new approaches will need to be developed for constructing level-3 and higher level-networks.

Distinguishing level-1 phylogenetic networks on the basis of data generated by Markov processes

Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based methods for reconstructing phylogenetic networks. For these methods to be consistent, the network parameter needs to be identifiable from data generated under the model. Here, we show that the semi-directed network parameter of a triangle-free, level-1 network model with any fixed number of reticulation vertices is generically identifiable under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints.

Phylogenetic Networks as Circuits With Resistance Distance.

Phylogenetic networks are notoriously difficult to reconstruct. Here we suggest that it can be useful to view unknown genetic distance along edges in phylogenetic networks as analogous to unknown resistance in electric circuits. This resistance distance, well-known in graph theory, turns out to have nice mathematical properties which allow the precise reconstruction of networks. Specifically we show that the resistance distance for a weighted 1-nested network is Kalmanson, and that the unique associated circular split network fully represents the splits of the original phylogenetic network (or circuit). In fact, this full representation corresponds to a face of the balanced minimal evolution polytope for level-1 networks. Thus, the unweighted class of the original network can be reconstructed by either the greedy algorithm neighbor-net or by linear programming over a balanced minimal evolution polytope. We begin study of 2-nested networks with both minimum path and resistance distance, and include some counting results for 2-nested networks.

Counting phylogenetic networks of level 1 and 2.

Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as their number of cycles) are asymptotically normally distributed. These results are obtained by first providing a recursive description (also called combinatorial specification) of our networks, and by next applying classical methods of enumerative, symbolic and analytic combinatorics.

Reconstructibility of unrooted level-k phylogenetic networks from distances

A phylogenetic network is a graph-theoretical tool that is used by biologists to represent the evolutionary history of a collection of species. One potential way of constructing such networks is via a distance-based approach, where one is asked to find a phylogenetic network that in some way represents a given distance matrix, which gives information on the evolutionary distances between present-day taxa. Here, we consider the following question. For which k are unrooted level-k networks uniquely determined by their distance matrices? We consider this question for shortest distances as well as for the case that the multisets of all distances is given. We prove that level-1 networks and level-2 networks are reconstructible from their shortest distances and multisets of distances, respectively. Furthermore we show that, in general, networks of level higher than 1 are not reconstructible from shortest distances and that networks of level higher than 2 are not reconstructible from their multisets of distances.

NCHB: A method for constructing rooted phylogenetic networks from rooted triplets based on height function and binarization

Phylogenetics is a field that studies and models the evolutionary history of currently living species. The rooted phylogenetic network is an important approach that models non-tree-like events between currently living species. Rooted triplets are one type of inputs in constructing rooted phylogenetic networks. Constructing an optimal rooted phylogenetic network that contains all given rooted triplets is a NP-hard problem. To overcome this challenge efficiently, a novel heuristic method called NCHB is introduced in this paper. NCHB produces an optimal rooted phylogenetic network that covers all given rooted triplets. The NCHB optimality criterions in building a rooted phylogenetic network are minimizing the number of reticulation nodes, and minimizing the level of the final network. In NCHB, the two concepts: the height function and the binarization of a network are considered innovatively. In order to study the performance of NCHB, our proposed method is compared with the three state of the art algorithms that are LEV1ATHAN, SIMPLISTIC and TripNet in two scenarios. In the first scenario, triplet sets are generated under biological presumptions and our proposed method is compared with SIMPLISTIC and TripNet. The results show that NCHB outperforms TripNet and SIMPLISTIC according to the optimality criterions. In the second scenario, we designed a software for generating level-k networks. Then all triplets consistent with each network are obtained and are used as input for NCHB, LEV1THAN, SIMPLISTIC, and TripNet. LEV1ATHEN is just applicable for level-1 networks while the other algorithms can be performed to obtain higher level networks. The results show that the NCHB and LEV1ATHAN outputs are almost the same when we are restricted to level-1 networks. Also the results show that NCHB outperforms TripNet and SIMPLISTIC. Moreover NCHB outputs are very close to the generated networks (that are optimal) with respect to the criterions.

Quarnet Inference Rules for Level-1 Networks

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set X of species from a collection of trees, each having leaf-set some subset of X. In the 1980s, Colonius and Schulze gave certain inference rules for deciding when a collection of 4-leaved trees, one for each 4-element subset of X, can be simultaneously displayed by a single supertree with leaf-set X. Recently, it has become of interest to extend this and related results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has recently been shown that a certain type of phylogenetic network, called a (unrooted) level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here, we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of X, can all be simultaneously displayed by a level-1 network with leaf-set X. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of X from orderings on subsets of X of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics.