arXiv subjects

Marc Hellmuth

Publications and source records attributed to Marc Hellmuth.

At least 73 records · Page 4Linked to original sources

arXiv2019

Reconciling Event-Labeled Gene Trees with MUL-trees and Species Networks

Phylogenomics commonly aims to construct evolutionary trees from genomic sequence information. One way to approach this problem is to first estimate event-labeled gene trees (i.e., rooted trees whose non-leaf vertices are labeled by speciation or gene duplication events), and to then look for a species tree which can be reconciled with this tree through a \emph{reconciliation map} between the trees. In practice, however, it can happen that there is no such map from a given event-labeled tree to \emph{any} species tree. An important situation where this might arise is where the species evolution is better represented by a \emph{network} instead of a tree. In this paper, we therefore consider the problem of reconciling event-labeled trees with species networks. In particular, we prove that any event-labeled gene tree can be reconciled with some network and that, under certain mild assumptions on the gene tree, the network can even be assumed to be multi-arc free. To prove this result, we show that we can always reconcile the gene tree with some multi-labeled (MUL-)tree, which can then be "folded up" to produce the desired reconciliation and network. In addition, we study the interplay between reconciliation maps from event-labeled gene trees to MUL-trees and networks. Our results could be useful for understanding how genomes have evolved after undergoing complex evolutionary events such as polyploidy.

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arXiv2018

Alternative Characterizations of Fitch's Xenology Relation

According to Walter M. Fitch, two genes are xenologs if they are separated by at least one horizontal gene transfer. This concept is formalized through Fitch relations, which are defined as binary relations that comprise all pairs $(x,y)$ of genes $x$ and $y$ for which $y$ has been horizontally transferred at least once since it diverged from the least common ancestor of $x$ and $y$. This definition, in particular, preserves the directional character of the transfer. Fitch relations are characterized by a small set of forbidden induced subgraphs on three vertices and can be recognized in linear time. In this contribution, we provide two novel characterizations of Fitch relations and present an alternative, short and elegant proof of the characterization theorem established by Geiß et al.\ in \emph{J.\ Math.\ Bio 77(5), 2018}.

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arXiv2018

Simultaneous Credible Regions for Multiple Changepoint Locations

Within a Bayesian retrospective framework, we present a way of examining the distribution of \cps through a novel set estimator. For a given level, $α$, we aim at smallest sets that cover all \cps with a probability of at least $1-α$. These so-called smallest simultaneous credible regions, computed for certain values of $α$, provide parsimonious representations of the possible \cp locations. In addition, combining them for a range of different $α$'s enables very informative yet condensed visualisations. Therewith we allow for the evaluation of model choices and the analysis of \cp data to an unprecedented degree. This approach exhibits superior sensitivity, specificity and interpretability in comparison with highest density regions, marginal inclusion probabilities and confidence intervals inferred by \stepR. Whilst their direct construction is usually intractable, asymptotically correct solutions can be derived from posterior samples. This leads to a novel NP-complete problem. Through reformulations into an Integer Linear Program we show empirically that a fast greedy heuristic computes virtually exact solutions.

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arXiv2018

Partial Homology Relations - Satisfiability in terms of Di-Cographs

Directed cographs (di-cographs) play a crucial role in the reconstruction of evolutionary histories of genes based on homology relations which are binary relations between genes. A variety of methods based on pairwise sequence comparisons can be used to infer such homology relations (e.g.\ orthology, paralogy, xenology). They are \emph{satisfiable} if the relations can be explained by an event-labeled gene tree, i.e., they can simultaneously co-exist in an evolutionary history of the underlying genes. Every gene tree is equivalently interpreted as a so-called cotree that entirely encodes the structure of a di-cograph. Thus, satisfiable homology relations must necessarily form a di-cograph. The inferred homology relations might not cover each pair of genes and thus, provide only partial knowledge on the full set of homology relations. Moreover, for particular pairs of genes, it might be known with a high degree of certainty that they are not orthologs (resp.\ paralogs, xenologs) which yields forbidden pairs of genes. Motivated by this observation, we characterize (partial) satisfiable homology relations with or without forbidden gene pairs, provide a quadratic-time algorithm for their recognition and for the computation of a cotree that explains the given relations.

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arXiv2018

Generalized Fitch Graphs: Edge-labeled Graphs that are explained by Edge-labeled Trees

Fitch graphs $G=(X,E)$ are di-graphs that are explained by $\{\otimes,1\}$-edge-labeled rooted trees with leaf set $X$: there is an arc $xy\in E$ if and only if the unique path in $T$ that connects the least common ancestor $\textrm{lca}(x,y)$ of $x$ and $y$ with $y$ contains at least one edge with label $1$. In practice, Fitch graphs represent xenology relations, i.e., pairs of genes $x$ and $y$ for which a horizontal gene transfer happened along the path from $\textrm{lca}(x,y)$ to $y$. In this contribution, we generalize the concept of xenology and Fitch graphs and consider complete di-graphs $K_{|X|}$ with vertex set $X$ and a map $ε$ that assigns to each arc $xy$ a unique label $ε(x,y)\in M\cup \{\otimes\}$, where $M$ denotes an arbitrary set of symbols. A di-graph $(K_{|X|},ε)$ is a generalized Fitch graph if there is an $M\cup \{\otimes\}$-edge-labeled tree $(T,λ)$ that can explain $(K_{|X|},ε)$. We provide a simple characterization of generalized Fitch graphs $(K_{|X|},ε)$ and give an $O(|X|^2)$-time algorithm for their recognition as well as for the reconstruction of the unique least resolved phylogenetic tree that explains $(K_{|X|},ε)$.

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arXiv2018

Reconstructing Gene Trees From Fitch's Xenology Relation

Two genes are xenologs in the sense of Fitch if they are separated by at least one horizontal gene transfer event. Horizonal gene transfer is asymmetric in the sense that the transferred copy is distinguished from the one that remains within the ancestral lineage. Hence xenology is more precisely thought of as a non-symmetric relation: $y$ is xenologous to $x$ if $y$ has been horizontally transferred at least once since it diverged from the least common ancestor of $x$ and $y$. We show that xenology relations are characterized by a small set of forbidden induced subgraphs on three vertices. Furthermore, each xenology relation can be derived from a unique least-resolved edge-labeled phylogenetic tree. We provide a linear-time algorithm for the recognition of xenology relations and for the construction of its least-resolved edge-labeled phylogenetic tree. The fact that being a xenology relation is a heritable graph property, finally has far-reaching consequences on approximation problems associated with xenology relations.

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arXiv2018

The Matroid Structure of Representative Triple Sets and Triple-Closure Computation

The closure $\textrm{cl}(R)$ of a consistent set $R$ of triples (rooted binary trees on three leaves) provides essential information about tree-like relations that are shown by any supertree that displays all triples in $R$. In this contribution, we are concerned with representative triple sets, that is, subsets $R'$ of $R$ with $\textrm{cl}(R') = \textrm{cl}(R)$. In this case, $R'$ still contains all information on the tree structure implied by $R$, although $R'$ might be significantly smaller. We show that representative triple sets that are minimal w.r.t.\ inclusion form the basis of a matroid. This in turn implies that minimal representative triple sets also have minimum cardinality. In particular, the matroid structure can be used to show that minimum representative triple sets can be computed in polynomial time with a simple greedy approach. For a given triple set $R$ that "identifies" a tree, we provide an exact value for the cardinality of its minimum representative triple sets. In addition, we utilize the latter results to provide a novel and efficient method to compute the closure $\textrm{cl}(R)$ of a consistent triple set $R$ that improves the time complexity $\mathcal{O}(|R||L_R|^4)$ of the currently fastest known method proposed by Bryant and Steel (1995). In particular, if a minimum representative triple set for $R$ is given, it can be shown that the time complexity to compute $\textrm{cl}(R)$ can be improved by a factor up to $|R||L_R|$. As it turns out, collections of quartets (unrooted binary trees on four leaves) do not provide a matroid structure, in general.

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arXiv2017

Phylogenomics with Paralogs

Phylogenomics heavily relies on well-curated sequence data sets that consist, for each gene, exclusively of 1:1-orthologous. Paralogs are treated as a dangerous nuisance that has to be detected and removed. We show here that this severe restriction of the data sets is not necessary. Building upon recent advances in mathematical phylogenetics we demonstrate that gene duplications convey meaningful phylogenetic information and allow the inference of plausible phylogenetic trees, provided orthologs and paralogs can be distinguished with a degree of certainty. Starting from tree-free estimates of orthology, cograph editing can sufficiently reduce the noise in order to find correct event-annotated gene trees. The information of gene trees can then directly be translated into constraints on the species trees. While the resolution is very poor for individual gene families, we show that genome-wide data sets are sufficient to generate fully resolved phylogenetic trees, even in the presence of horizontal gene transfer. We demonstrate that the distribution of paralogs in large gene families contains in itself sufficient phylogenetic signal to infer fully resolved species phylogenies. This source of phylogenetic information is independent of information contained in orthologous sequences and is resilient against horizontal gene transfer. An important consequence is that phylogenomics data sets need not be restricted to 1:1 orthologs.

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arXiv2017

A Short Note on Undirected Fitch Graphs

The symmetric version of Fitch's xenology relation coincides with class of complete multipartite graph and thus cannot convey any non-trivial phylogenetic information.

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arXiv2017

Inference of Phylogenetic Trees from the Knowledge of Rare Evolutionary Events

Rare events have played an increasing role in molecular phylogenetics as potentially homoplasy-poor characters.In this contribution we analyze the phylogenetic information content from a combinatorial point of view by consid-ering the binary relation on the set of taxa defined by the existence of a single event separating two taxa. We showthat the graph-representation of this relation must be a tree. Moreover, we characterize completely the relationshipbetween the tree of such relations and the underlying phylogenetic tree. With directed operations such as tandem-duplication-random-loss events in mind we demonstrate how non-symmetric information constrains the position ofthe root in the partially reconstructed phylogeny.

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arXiv2017

Biologically Feasible Gene Trees, Reconciliation Maps and Informative Triples

The history of gene families - which are equivalent to \emph{event-labeled} gene trees - can be reconstructed from empirically estimated evolutionary event-relations containing pairs of orthologous, paralogous or xenologous genes. The question then arises as whether inferred event-labeled gene trees are \emph{biologically feasible}, that is, if there is a possible true history that would explain a given gene tree. In practice, this problem is boiled down to finding a reconciliation map - also known as DTL-scenario - between the event-labeled gene trees and a (possibly unknown) species tree. In this contribution, we first characterize whether there is a valid reconciliation map for binary event-labeled gene trees $T$ that contain speciation, duplication and horizontal gene transfer events and some unknown species tree $S$ in terms of "informative" triples that are displayed in $T$ and provide information of the topology of $S$. These informative triples are used to infer the unknown species tree $S$ for $T$. We obtain a similar result for non-binary gene trees. To this end, however, the reconciliation map needs to be further restricted. We provide a polynomial-time algorithm to decide whether there is a species tree for a given event-labeled gene tree, and in the positive case, to construct the species tree and the respective (restricted) reconciliation map. However, informative triples as well as DTL-scenarios have its limitations when they are used to explain the biological feasibility of gene trees. While reconciliation maps imply biological feasibility, we show that the converse is not true in general. Moreover, we show that informative triples do neither provide enough information to characterize "relaxed" DTL-scenarios nor non-restricted reconciliation maps for non-binary biologically feasible gene trees.

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arXiv2017

A Survey on Hypergraph Products (Erratum)

A surprising diversity of different products of hypergraphs have been discussed in the literature. Most of the hypergraph products can be viewed as generalizations of one of the four standard graph products. The most widely studied variant, the so-called square product, does not have this property, however. Here we survey the literature on hypergraph products with an emphasis on comparing the alternative generalizations of graph products and the relationships among them. In this context the so-called 2-sections and L2-sections are considered. These constructions are closely linked to related colored graph structures that seem to be a useful tool for the prime factor decompositions w.r.t.\ specific hypergraph products. We summarize the current knowledge on the propagation of hypergraph invariants under the different hypergraph multiplications. While the overwhelming majority of the material concerns finite (undirected) hypergraphs, the survey also covers a summary of the few results on products of infinite and directed hypergraphs.

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arXiv2017

A Local Prime Factor Decomposition Algorithm for Strong Product Graphs

This work is concerned with the prime factor decomposition (PFD) of strong product graphs. A new quasi-linear time algorithm for the PFD with respect to the strong product for arbitrary, finite, connected, undirected graphs is derived. Moreover, since most graphs are prime although they can have a product-like structure, also known as approximate graph products, the practical application of the well-known "classical" prime factorization algorithm is strictly limited. This new PFD algorithm is based on a local approach that covers a graph by small factorizable subgraphs and then utilizes this information to derive the global factors. Therefore, we can take advantage of this approach and derive in addition a method for the recognition of approximate graph products.

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arXiv2017

Local algorithms for the prime factorization of strong product graphs

The practical application of graph prime factorization algorithms is limited in practice by unavoidable noise in the data. A first step towards error-tolerant "approximate" prime factorization, is the development of local approaches that cover the graph by factorizable patches and then use this information to derive global factors. We present here a local, quasi-linear al- gorithm for the prime factorization of "locally unrefined" graphs with respect to the strong product. To this end we introduce the backbone B(G) for a given graph G and show that the neighborhoods of the backbone vertices provide enough information to determine the global prime factors.

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arXiv2017

From event labeled gene trees to species trees

Background: Tree reconciliation problems have long been studied in phylogenetics. A particular variant of the reconciliation problem for a gene tree T and a species tree S assumes that for each interior vertex x of T it is known whether x represents a speciation or a duplication. This problem appears in the context of analyzing orthology data. Results: We show that S is a species tree for T if and only if S displays all rooted triples of T that have three distinct species as their leaves and are rooted in a speciation vertex. A valid reconciliation map can then be found in polynomial time. Simulated data shows that the event-labeled gene trees convey a large amount of information on underlying species trees, even for a large percentage of losses. Conclusions: The knowledge of event labels in a gene tree strongly constrains the possible species tree and, for a given species tree, also the possible reconciliation maps. Nevertheless, many degrees of freedom remain in the space of feasible solutions. In order to disambiguate the alternative solutions additional external constraints as well as optimization criteria could be employed.

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arXiv2017

On the Complexity of Recognizing S-composite and S-prime Graphs

S-prime graphs are graphs that cannot be represented as nontrivial subgraphs of nontrivial Cartesian products of graphs, i.e., whenever it is a subgraph of a nontrivial Cartesian product graph it is a subgraph of one the factors. A graph is S-composite if it is not S-prime. Although linear time recognition algorithms for determining whether a graph is prime or not with respect to the Cartesian product are known, it remained unknown if a similar result holds also for the recognition of S-prime and S-composite graphs. In this contribution the computational complexity of recognizing S-composite and S-prime graphs is considered. Klav{ž}ar \emph{et al.} [\emph{Discr.\ Math.} \textbf{244}: 223-230 (2002)] proved that a graph is S-composite if and only if it admits a nontrivial path-$k$-coloring. The problem of determining whether there exists a path-$k$-coloring for a given graph is shown to be NP-complete even for $k=2$. This in turn is utilized to show that determining whether a graph is S-composite is NP-complete and thus, determining whether a graph is S-prime is CoNP-complete. Many other problems are shown to be NP-hard, using the latter results.

cs.CC↗

arXiv2017

Forbidden Time Travel: Characterization of Time-Consistent Tree Reconciliation Maps

In the absence of horizontal gene transfer it is possible to reconstruct the history of gene families from empirically determined orthology relations, which are equivalent to event-labeled gene trees. Knowledge of the event labels considerably simplifies the problem of reconciling a gene tree T with a species trees S, relative to the reconciliation problem without prior knowledge of the event types. It is well-known that optimal reconciliations in the unlabeled case may violate time-consistency and thus are not biologically feasible. Here we investigate the mathematical structure of the event labeled reconciliation problem with horizontal transfer. We investigate the issue of time-consistency for the event-labeled version of the reconciliation problem, provide a convenient axiomatic framework, and derive a complete characterization of time-consistent reconciliations. This characterization depends on certain weak conditions on the event-labeled gene trees that reflect conditions under which evolutionary events are observable at least in principle. We give an O(|V(T)|log(|V(S)|))-time algorithm to decide whether a time-consistent reconciliation map exists. It does not require the construction of explicit timing maps, but relies entirely on the comparably easy task of checking whether a small auxiliary graph is acyclic. The combinatorial characterization of time consistency and thus biologically feasible reconciliation is an important step towards the inference of gene family histories with horizontal transfer from orthology data, i.e., without presupposed gene and species trees. The fast algorithm to decide time consistency is useful in a broader context because it constitutes an attractive component for all tools that address tree reconciliation problems.

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arXiv2016

The Mathematics of Xenology: Di-cographs, Symbolic Ultrametrics, 2-structures and Tree-representable Systems of Binary Relations

The concepts of orthology, paralogy, and xenology play a key role in molecular evolution. Orthology and paralogy distinguish whether a pair of genes originated by speciation or duplication. The corresponding binary relations on a set of genes form complementary cographs. Allowing more than two types of ancestral event types leads to symmetric symbolic ultrametrics. Horizontal gene transfer, which leads to xenologous gene pairs, however, is inherent asymmetric since one offspring copy "jumps" into another genome, while the other continues to be inherited vertically. We therefore explore here the mathematical structure of the non-symmetric generalization of symbolic ultrametrics. Our main results tie non-symmetric ultrametrics together with di-cographs (the directed generalization of cographs), so-called uniformly non-prime 2-structures, and hierarchical structures on the set of strong modules. This yields a characterization of relation structures that can be explained in terms of trees and types of ancestral events. This framework accommodates a horizontal-transfer relation in terms of an ancestral event and thus, is slightly different from the the most commonly used definition of xenology.

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