arXiv subjects

Peter F. Stadler

Publications and source records attributed to Peter F. Stadler.

At least 55 records · Page 3Linked to original sources

arXiv2021

The Past as a Stochastic Process

Historical processes manifest remarkable diversity. Nevertheless, scholars have long attempted to identify patterns and categorize historical actors and influences with some success. A stochastic process framework provides a structured approach for the analysis of large historical datasets that allows for detection of sometimes surprising patterns, identification of relevant causal actors both endogenous and exogenous to the process, and comparison between different historical cases. The combination of data, analytical tools and the organizing theoretical framework of stochastic processes complements traditional narrative approaches in history and archaeology.

stat.AP↗

arXiv2021

Orientation of Fitch Graphs and Detection of Horizontal Gene Transfer in Gene Trees

Horizontal gene transfer events partition a gene tree $T$ and thus, its leaf set into subsets of genes whose evolutionary history is described by speciation and duplication events alone. Indirect phylogenetic methods can be used to infer such partitions $\mathcal{P}$ from sequence similarity or evolutionary distances without any a priory knowledge about the underlying tree $T$. In this contribution, we assume that such a partition $\mathcal{P}$ of a set of genes $X$ is given and that, independently, an estimate $T$ of the original gene tree on $X$ has been derived. We then ask to what extent $T$ and the xenology information, i.e., $\mathcal{P}$ can be combined to determine the horizontal transfer edges in $T$. We show that for each pair of genes $x$ and $y$ with $x,y$ being in different parts of $\mathcal{P}$, it can be decided whether there always exists or never exists a horizontal gene transfer in $T$ along the path connecting $y$ and the most recent common ancestor of $x$ and $y$. This problem is equivalent to determining the presence or absence of the directed edge $(x,y)$ in so-called Fitch graphs; a more fine-grained version of graphs that represent the dependencies between the sets in $\mathcal{P}$. We then consider the generalization to insufficiently resolved gene trees and show that analogous results can be obtained. We show that the classification of $(x,y)$ can be computed in constant time after linear-time preprocessing. Using simulated gene family histories, we observe empirically that the vast majority of horizontal transfer edges in the gene tree $T$ can be recovered unambiguously.

cs.DM↗

arXiv2021

Compatibility of Partitions with Trees, Hierarchies, and Split Systems

The question whether a partition $\mathcal{P}$ and a hierarchy $\mathcal{H}$ or a tree-like split system $\mathfrak{S}$ are compatible naturally arises in a wide range of classification problems. In the setting of phylogenetic trees, one asks whether the sets of $\mathcal{P}$coincide with leaf sets of connected components obtained by deleting some edges from the tree $T$ that represents $\mathcal{H}$ or $\mathfrak{S}$, respectively. More generally, we ask whether a refinement $T^*$ of $T$ exists such that $T^*$ and $\mathcal{P}$ are compatible in this sense. The latter is closely related to the question as to whether there exists a tree at all that is compatible with $\mathcal{P}$. We report several characterizations for (refinements of) hierarchies and split systems that are compatible with (systems of) partitions. In addition, we provide a linear-time algorithm to check whether refinements of trees and a given partition are compatible. The latter problem becomes NP-complete but fixed-parameter tractable if a system of partitions is considered instead of a single partition. In this context, we also explore the close relationship of the concept of compatibility and so-called Fitch maps.

cs.DM↗

arXiv2021

Planar Median Graphs and Cubesquare-Graphs

Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median graphs. More specifically, we characterize when a planar graph $G$ is a median graph in terms of forbidden subgraphs and the structure of isometric cycles in $G$, and also in terms of subgraphs of $G$ that are contained inside and outside of 4-cycles with respect to an arbitrary planar embedding of $G$. These results lead us to a new characterization of planar median graphs in terms of cubesquare-graphs that is, graphs that can be obtained by starting with cubes and square graphs, and iteratively replacing 4-cycle boundaries (relative to some embedding) by cubes or square-graphs. As a corollary we also show that a graph is planar median if and only if it can be obtained from cubes and square-graphs by a sequence of ``square-boundary'' amalgamations. These considerations also lead to an $\mathcal{O}(n\log n)$-time recognition algorithm to compute a decomposition of a planar median graph with $n$ vertices into cubes and square-graphs.

math.CO↗

arXiv2021

A Simple Linear-Time Algorithm for the Common Refinement of Rooted Phylogenetic Trees on a Common Leaf Set

Background. The supertree problem, i.e., the task of finding a common refinement of a set of rooted trees is an important topic in mathematical phylogenetics. The special case of a common leaf set $L$ is known to be solvable in linear time. Existing approaches refine one input tree using information of the others and then test whether the results are isomorphic. Results. A linear-time algorithm, LinCR, for constructing the common refinement $T$ of $k$ input trees with a common leaf set is proposed that explicitly computes the parent function of $T$ in a bottom-up approach. Conclusion. LinCR is simpler to implement than other asymptotically optimal algorithms for the problem and outperforms the alternatives in empirical comparisons. Availability. An implementation of LinCR in Python is freely available at https://github.com/david-schaller/tralda.

cs.DS↗

arXiv2021

Defining Autocatalysis in Chemical Reaction Networks

Autocatalysis is a deceptively simple concept, referring to the situation that a chemical species $X$ catalyzes its own formation. From the perspective of chemical kinetics, autocatalysts show a regime of super-linear growth. Given a chemical reaction network, however, it is not at all straightforward to identify species that are autocatalytic in the sense that there is a sub-network that takes $X$ as input and produces more than one copy of $X$ as output. The difficulty arises from the need to distinguish autocatalysis e.g. from the superposition of a cycle that consumes and produces equal amounts of $X$ and a pathway that produces $X$. To deal with this issue, a number of competing notions, such as exclusive autocatalysis and autocatalytic cycles, have been introduced. A closer inspection of concepts and their usage by different authors shows, however, that subtle differences in the definitions often makes conceptually matching ideas difficult to bring together formally. In this contribution we make some of the available approaches comparable by translating them into a common formal framework that uses integer hyperflows as a basis to study autocatalysis in large chemical reaction networks. As an application we investigate the prevalence of autocatalysis in metabolic networks.

q-bio.MN↗

arXiv2021

Combining Orthology and Xenology Data in a Common Phylogenetic Tree

A rooted tree $T$ with vertex labels $t(v)$ and set-valued edge labels $λ(e)$ defines maps $δ$ and $\varepsilon$ on the pairs of leaves of $T$ by setting $δ(x,y)=q$ if the last common ancestor $\text{lca}(x,y)$ of $x$ and $y$ is labeled $q$, and $m\in \varepsilon(x,y)$ if $m\inλ(e)$ for at least one edge $e$ along the path from $\text{lca}(x,y)$ to $y$. We show that a pair of maps $(δ,\varepsilon)$ derives from a tree $(T,t,λ)$ if and only if there exists a common refinement of the (unique) least-resolved vertex labeled tree $(T_δ,t_δ)$ that explains $δ$ and the (unique) least resolved edge labeled tree $(T_{\varepsilon},λ_{\varepsilon})$ that explains $\varepsilon$ (provided both trees exist). This result remains true if certain combinations of labels at incident vertices and edges are forbidden.

math.CO↗

arXiv2021

Generalized Fitch Graphs III: Symmetrized Fitch maps and Sets of Symmetric Binary Relations that are explained by Unrooted Edge-labeled Trees

Binary relations derived from labeled rooted trees play an import role in mathematical biology as formal models of evolutionary relationships. The (symmetrized) Fitch relation formalizes xenology as the pairs of genes separated by at least one horizontal transfer event. As a natural generalization, we consider symmetrized Fitch maps, that is, symmetric maps $\varepsilon$ that assign a subset of colors to each pair of vertices in $X$ and that can be explained by a tree $T$ with edges that are labeled with subsets of colors in the sense that the color $m$ appears in $\varepsilon(x,y)$ if and only if $m$ appears in a label along the unique path between $x$ and $y$ in $T$. We first give an alternative characterization of the monochromatic case and then give a characterization of symmetrized Fitch maps in terms of compatibility of a certain set of quartets. We show that recognition of symmetrized Fitch maps is NP-complete. In the restricted case where $|\varepsilon(x,y)|\leq 1$ the problem becomes polynomial, since such maps coincide with class of monochromatic Fitch maps whose graph-representations form precisely the class of complete multi-partite graphs.

cs.DM↗

arXiv2021

Indirect Identification of Horizontal Gene Transfer

Several implicit methods to infer Horizontal Gene Transfer (HGT) focus on pairs of genes that have diverged only after the divergence of the two species in which the genes reside. This situation defines the edge set of a graph, the later-divergence-time (LDT) graph, whose vertices correspond to genes colored by their species. We investigate these graphs in the setting of relaxed scenarios, i.e., evolutionary scenarios that encompass all commonly used variants of duplication-transfer-loss scenarios in the literature. We characterize LDT graphs as a subclass of properly vertex-colored cographs, and provide a polynomial-time recognition algorithm as well as an algorithm to construct a relaxed scenario that explains a given LDT. An edge in an LDT graph implies that the two corresponding genes are separated by at least one HGT event. The converse is not true, however. We show that the complete xenology relation is described by an rs-Fitch graph, i.e., a complete multipartite graph satisfying constraints on the vertex coloring. This class of vertex-colored graphs is also recognizable in polynomial time. We finally address the question "how much information about all HGT events is contained in LDT graphs" with the help of simulations of evolutionary scenarios with a wide range of duplication, loss, and HGT events. In particular, we show that a simple greedy graph editing scheme can be used to efficiently detect HGT events that are implicitly contained in LDT graphs.

q-bio.PE↗

arXiv2021

Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

cs.DS↗

arXiv2021

From Modular Decomposition Trees to Rooted Median Graphs

The modular decomposition of a symmetric map $δ\colon X\times X \to Υ$ (or, equivalently, a set of symmetric binary relations, a 2-structure, or an edge-colored undirected graph) is a natural construction to capture key features of $δ$ in labeled trees. A map $δ$ is explained by a vertex-labeled rooted tree $(T,t)$ if the label $δ(x,y)$ coincides with the label of the last common ancestor of $x$ and $y$ in $T$, i.e., if $δ(x,y)=t(\mathrm{lca}(x,y))$. Only maps whose modular decomposition does not contain prime nodes, i.e., the symbolic ultrametrics, can be exaplained in this manner. Here we consider rooted median graphs as a generalization to (modular decomposition) trees to explain symmetric maps. We first show that every symmetric map can be explained by "extended" hypercubes and half-grids. We then derive a a linear-time algorithm that stepwisely resolves prime vertices in the modular decomposition tree to obtain a rooted and labeled median graph that explains a given symmetric map $δ$. We argue that the resulting "tree-like" median graphs may be of use in phylogenetics as a model of evolutionary relationships.

math.CO↗

arXiv2021

Heuristic Algorithms for Best Match Graph Editing

Best match graphs (BMGs) are a class of colored digraphs that naturally appear in mathematical phylogenetics and can be approximated with the help of similarity measures between gene sequences, albeit not without errors. The corresponding graph editing problem can be used as a means of error correction. Since the arc set modification problems for BMGs are NP-complete, efficient heuristics are needed if BMGs are to be used for the practical analysis of biological sequence data. Since BMGs have a characterization in terms of consistency of a certain set of rooted triples, we consider heuristics that operate on triple sets. As an alternative, we show that there is a close connection to a set partitioning problem that leads to a class of top-down recursive algorithms that are similar to Aho's supertree algorithm and give rise to BMG editing algorithms that are consistent in the sense that they leave BMGs invariant. Extensive benchmarking shows that community detection algorithms for the partitioning steps perform best for BMG editing.

math.CO↗

arXiv2021

Best Match Graphs with Binary Trees

Best match graphs (BMG) are a key intermediate in graph-based orthology detection and contain a large amount of information on the gene tree. We provide a near-cubic algorithm to determine whether a BMG is binary-explainable, i.e., whether it can be explained by a fully resolved gene tree and, if so, to construct such a tree. Moreover, we show that all such binary trees are refinements of the unique binary-resolvable tree (BRT), which in general is a substantial refinement of the also unique least resolved tree of a BMG. Finally, we show that the problem of editing an arbitrary vertex-colored graph to a binary-explainable BMG is NP-complete and provide an integer linear program formulation for this task.

cs.DS↗

arXiv2021

Least resolved trees for two-colored best match graphs

2-colored best match graphs (2-BMGs) form a subclass of sink-free bi-transitive graphs that appears in phylogenetic combinatorics. There, 2-BMGs describe evolutionarily most closely related genes between a pair of species. They are explained by a unique least resolved tree (LRT). Introducing the concept of support vertices we derive an $O(|V|+|E|\log^2|V|)$-time algorithm to recognize 2-BMGs and to construct its LRT. The approach can be extended to also recognize binary-explainable 2-BMGs with the same complexity. An empirical comparison emphasizes the efficiency of the new algorithm.

q-bio.PE↗

arXiv2020

Complexity of modification problems for best match graphs

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors usually result in empirically determined graphs that in general violate characteristic properties of BMGs. The arc modification problems for BMGs aim at correcting such violations and thus provide a means to improve the initial estimates of best match data. We show here that the arc deletion, arc completion and arc editing problems for BMGs are NP-complete and that they can be formulated and solved as integer linear programs. To this end, we provide a novel characterization of BMGs in terms of triples (binary trees on three leaves) and a characterization of BMGs with two colors in terms of forbidden subgraphs.

cs.CC↗

arXiv2020

Complete Characterization of Incorrect Orthology Assignments in Best Match Graphs

Genome-scale orthology assignments are usually based on reciprocal best matches. In the absence of horizontal gene transfer (HGT), every pair of orthologs forms a reciprocal best match. Incorrect orthology assignments therefore are always false positives in the reciprocal best match graph. We consider duplication/loss scenarios and characterize unambiguous false-positive (u-fp) orthology assignments, that is, edges in the best match graphs (BMGs) that cannot correspond to orthologs for any gene tree that explains the BMG. Moreover, we provide a polynomial-time algorithm to identify all u-fp orthology assignments in a BMG. Simulations show that at least $75\%$ of all incorrect orthology assignments can be detected in this manner. All results rely only on the structure of the BMGs and not on any a priori knowledge about underlying gene or species trees.

q-bio.PE↗

arXiv2020

Best Match Graphs

THIS IS A CORRECTED VERSION INCLUDING AN APPENDED CORRIGENDUM. Best match graphs arise naturally as the first processing intermediate in algorithms for orthology detection. Let $T$ be a phylogenetic (gene) tree $T$ and $σ$ an assignment of leaves of $T$ to species. The best match graph $(G,σ)$ is a digraph that contains an arc from $x$ to $y$ if the genes $x$ and $y$ reside in different species and $y$ is one of possibly many (evolutionary) closest relatives of $x$ compared to all other genes contained in the species $σ(y)$. Here, we characterize best match graphs and show that it can be decided in cubic time and quadratic space whether $(G,σ)$ derived from a tree in this manner. If the answer is affirmative, there is a unique least resolved tree that explains $(G,σ)$, which can also be constructed in cubic time.

math.CO↗

arXiv2020

Exact-$2$-Relation Graphs

Pairwise compatibility graphs (PCGs) with non-negative integer edge weights recently have been used to describe rare evolutionary events and scenarios with horizontal gene transfer. Here we consider the case that vertices are separated by exactly two discrete events: Given a tree $T$ with leaf set $L$ and edge-weights $λ: E(T)\to\mathbb{N}_0$, the non-negative integer pairwise compatibility graph $\textrm{nniPCG}(T,λ,2,2)$ has vertex set $L$ and $xy$ is an edge whenever the sum of the non-negative integer weights along the unique path from $x$ to $y$ in $T$ equals $2$. A graph $G$ has a representation as $\textrm{nniPCG}(T,λ,2,2)$ if and only if its point-determining quotient $G/\!\rthin$ is a block graph, where two vertices are in relation $\rthin$ if they have the same neighborhood in $G$. If $G$ is of this type, a labeled tree $(T,λ)$ explaining $G$ can be constructed efficiently. In addition, we consider an oriented version of this class of graphs.

math.CO↗