A class of phylogenetic networks reconstructable from ancestral profiles

Phylogenetic network classes through the lens of expanding covers

It was recently shown that a large class of phylogenetic networks, the 'labellable' networks, is in bijection with the set of 'expanding' covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks can be characterised purely in terms of properties of their associated covers. These classes include the tree-based, tree-child, orchard, tree-sibling, and normal networks. In the opposite direction, we give an example of how a restriction on the set of expanding covers can define a new class of networks, which we call 'spinal' phylogenetic networks.

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.

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.

Labellable Phylogenetic Networks

Phylogenetic networks are mathematical representations of evolutionary history that are able to capture both tree-like evolutionary processes (speciations) and non-tree-like 'reticulate' processes such as hybridization or horizontal gene transfer. The additional complexity that comes with this capacity, however, makes networks harder to infer from data, and more complicated to work with as mathematical objects. In this paper, we define a new, large class of phylogenetic networks, that we call labellable, and show that they are in bijection with the set of 'expanding covers' of finite sets. This correspondence is a generalisation of the encoding of phylogenetic forests by partitions of finite sets. Labellable networks can be characterised by a simple combinatorial condition, and we describe the relationship between this large class and other commonly studied classes. Furthermore, we show that all phylogenetic networks have a quotient network that is labellable.

Autopolyploidy, Allopolyploidy, and Phylogenetic Networks with Horizontal Arcs

Polyploidization is an evolutionary process by which a species acquires multiple copies of its complete set of chromosomes. The reticulate nature of the signal left behind by it means that phylogenetic networks offer themselves as a framework to reconstruct the evolutionary past of species affected by it. The main strategy for doing this is to first construct a so-called multiple-labelled tree and to then somehow derive such a network from it. The following question therefore arises: How much can be said about that past if such a tree is not readily available? By viewing a polyploid dataset as a certain vector which we call a ploidy (level) profile, we show that among other results, there always exists a phylogenetic network in the form of a beaded phylogenetic tree with additional arcs that realizes a given ploidy profile. Intriguingly, the two end vertices of almost all of these additional arcs can be interpreted as having co-existed in time thereby adding biological realism to our network, a feature that is, in general, not enjoyed by phylogenetic networks. In addition, we show that our network may be viewed as a generator of ploidy profile space, a novel concept similar to phylogenetic tree space that we introduce to be able to compare phylogenetic networks that realize one and the same ploidy profile. We illustrate our findings in terms of a publicly available Viola dataset.

The tree of blobs of a species network: identifiability under the coalescent

Inference of species networks from genomic data under the Network Multispecies Coalescent Model is currently severely limited by heavy computational demands. It also remains unclear how complicated networks can be for consistent inference to be possible. As a step toward inferring a general species network, this work considers its tree of blobs, in which non-cut edges are contracted to nodes, so only tree-like relationships between the taxa are shown. An identifiability theorem, that most features of the unrooted tree of blobs can be determined from the distribution of gene quartet topologies, is established. This depends upon an analysis of gene quartet concordance factors under the model, together with a new combinatorial inference rule. The arguments for this theoretical result suggest a practical algorithm for tree of blobs inference, to be fully developed in a subsequent work.

The hybrid number of a ploidy profile

Polyploidization, whereby an organism inherits multiple copies of the genome of their parents, is an important evolutionary event that has been observed in plants and animals. One way to study such events is in terms of the ploidy number of the species that make up a dataset of interest. It is therefore natural to ask: How much information about the evolutionary past of the set of species that form a dataset can be gleaned from the ploidy numbers of the species? To help answer this question, we introduce and study the novel concept of a ploidy profile which allows us to formalize it in terms of a multiplicity vector indexed by the species the dataset is comprised of. Using the framework of a phylogenetic network, we present a closed formula for computing the hybrid number (i.e. the minimal number of polyploidization events required to explain a ploidy profile) of a large class of ploidy profiles. This formula relies on the construction of a certain phylogenetic network from the simplification sequence of a ploidy profile and the hybrid number of the ploidy profile with which this construction is initialized. Both of them can be computed easily in case the ploidy numbers that make up the ploidy profile are not too large. To help illustrate the applicability of our approach, we apply it to a simplified version of a publicly available Viola dataset.

Orchard Networks are Trees with Additional Horizontal Arcs

Phylogenetic networks are used in biology to represent evolutionary histories. The class of orchard phylogenetic networks was recently introduced for their computational benefits, without any biological justification. Here, we show that orchard networks can be interpreted as trees with additional horizontal arcs. Therefore, they are closely related to tree-based networks, where the difference is that in tree-based networks the additional arcs do not need to be horizontal. Then, we use this new characterization to show that the space of orchard networks on n leaves with k reticulations is connected under the rNNI rearrangement move with diameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(kn+n\log (n))$$\end{document}O(kn+nlog(n)).

A polynomial invariant for a new class of phylogenetic networks

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this paper, we generalize a complete polynomial invariant on trees to a class of phylogenetic networks called separable networks, which will include orchard networks. Networks are becoming increasingly important for their ability to represent reticulation events, such as hybridization, in evolutionary history. We provide a function from the space of internally multi-labelled phylogenetic networks, a more generic graph structure than phylogenetic networks where the reticulations are also labelled, to a polynomial ring. We prove that the separability condition allows us to characterize, via the polynomial, the phylogenetic networks with the same number of leaves and same number of reticulations by considering their internally labelled versions. While the invariant for trees is a polynomial in Z[x1,…,xn,y] where n is the number of leaves, the invariant for internally multi-labelled phylogenetic networks is an element of Z[x1,…,xn,λ1,…,λr,y], where r is the number of reticulations in the network. When the networks are considered without leaf labels the number of variables reduces to r + 2.

Classes of explicit phylogenetic networks and their biological and mathematical significance

The evolutionary relationships among organisms have traditionally been represented using rooted phylogenetic trees. However, due to reticulate processes such as hybridization or lateral gene transfer, evolution cannot always be adequately represented by a phylogenetic tree, and rooted phylogenetic networks that describe such complex processes have been introduced as a generalization of rooted phylogenetic trees. In fact, estimating rooted phylogenetic networks from genomic sequence data and analyzing their structural properties is one of the most important tasks in contemporary phylogenetics. Over the last two decades, several subclasses of rooted phylogenetic networks (characterized by certain structural constraints) have been introduced in the literature, either to model specific biological phenomena or to enable tractable mathematical and computational analyses. In the present manuscript, we provide a thorough review of these network classes, as well as provide a biological interpretation of the structural constraints underlying these networks where possible. In addition, we discuss how imposing structural constraints on the network topology can be used to address the scalability and identifiability challenges faced in the estimation of phylogenetic networks from empirical data.

Comparing the topology of phylogenetic network generators

Phylogenetic networks represent evolutionary history of species and can record natural reticulate evolutionary processes such as horizontal gene transfer and gene recombination. This makes phylogenetic networks a more comprehensive representation of evolutionary history compared to phylogenetic trees. Stochastic processes for generating random trees or networks are important tools in evolutionary analysis, especially in phylogeny reconstruction where they can be utilized for validation or serve as priors for Bayesian methods. However, as more network generators are developed, there is a lack of discussion or comparison for different generators. To bridge this gap, we compare a set of phylogenetic network generators by profiling topological summary statistics of the generated networks over the number of reticulations and comparing the topological profiles.

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.

Defining phylogenetic networks using ancestral profiles

Rooted phylogenetic networks provide a more complete representation of the ancestral relationship between species than phylogenetic trees when reticulate evolutionary processes are at play. One way to reconstruct a phylogenetic network is to consider its 'ancestral profile' (the number of paths from each ancestral vertex to each leaf). In general, this information does not uniquely determine the underlying phylogenetic network. A recent paper considered a new class of phylogenetic networks called 'orchard networks' where this uniqueness was claimed to hold. Here we show that an additional restriction on the network, that of being 'stack-free', is required in order for the original uniqueness claim to hold. On the other hand, if the additional stack-free restriction is lifted, we establish an alternative result; namely, there is uniqueness within the class of orchard networks up to the resolution of vertices of high in-degree.

Combining Networks Using Cherry Picking Sequences

Phylogenetic networks are important for the study of evolution. The number of methods to find such networks is increasing, but most such methods can only reconstruct small networks. To find bigger networks, one can attempt to combine small networks. In this paper, we study the Network Hybridization problem, a problem of combining networks into another network with low complexity. We characterize this complexity via a restricted problem, Tree-child Network Hybridization, and we present an FPT algorithm to efficiently solve this restricted problem.

Generation of Binary Tree-Child phylogenetic networks

Phylogenetic networks generalize phylogenetic trees by allowing the modelization of events of reticulate evolution. Among the different kinds of phylogenetic networks that have been proposed in the literature, the subclass of binary tree-child networks is one of the most studied ones. However, very little is known about the combinatorial structure of these networks. In this paper we address the problem of generating all possible binary tree-child (BTC) networks with a given number of leaves in an efficient way via reduction/augmentation operations that extend and generalize analogous operations for phylogenetic trees, and are biologically relevant. Since our solution is recursive, this also provides us with a recurrence relation giving an upper bound on the number of such networks. We also show how the operations introduced in this paper can be employed to extend the evolutive history of a set of sequences, represented by a BTC network, to include a new sequence. An implementation in python of the algorithms described in this paper, along with some computational experiments, can be downloaded from https://github.com/bielcardona/TCGenerators.