arXiv subjects

David Schaller

Publications and source records attributed to David Schaller.

17 recordsLinked to original sources

arXiv2023

Relative Timing Information and Orthology in Evolutionary Scenarios

Evolutionary scenarios describing the evolution of a family of genes within a collection of species comprise the mapping of the vertices of a gene tree $T$ to vertices and edges of a species tree $S$. The relative timing of the last common ancestors of two extant genes (leaves of $T$) and the last common ancestors of the two species (leaves of $S$) in which they reside is indicative of horizontal gene transfers (HGT) and ancient duplications. Orthologous gene pairs, on the other hand, require that their last common ancestors coincides with a corresponding speciation event. The relative timing information of gene and species divergences is captured by three colored graphs that have the extant genes as vertices and the species in which the genes are found as vertex colors: the equal-divergence-time (EDT) graph, the later-divergence-time (LDT) graph and the prior-divergence-time (PDT) graph, which together form an edge partition of the complete graph. Here we give a complete characterization in terms of informative and forbidden triples that can be read off the three graphs and provide a polynomial time algorithm for constructing an evolutionary scenario that explains the graphs, provided such a scenario exists. We show that every EDT graph is perfect. While the information about LDT and PDT graphs is necessary to recognize EDT graphs in polynomial-time for general scenarios, this extra information can be dropped in the HGT-free case. However, recognition of EDT graphs without knowledge of putative LDT and PDT graphs is NP-complete for general scenarios. In contrast, PDT graphs can be recognized in polynomial-time. We finally connect the EDT graph to the alternative definitions of orthology that have been proposed for scenarios with horizontal gene transfer. With one exception, the corresponding graphs are shown to be colored cographs.

q-bio.PE↗

arXiv2023

Quasi-Best Match Graphs

Quasi-best match graphs (qBMGs) are a hereditary class of directed, properly vertex-colored graphs. They arise naturally in mathematical phylogenetics as a generalization of best match graphs, which formalize the notion of evolutionary closest relatedness of genes (vertices) in multiple species (vertex colors). They are explained by rooted trees whose leaves correspond to vertices. In contrast to BMGs, qBMGs represent only best matches at a restricted phylogenetic distance. We provide characterizations of qBMGs that give rise to polynomial-time recognition algorithms and identify the BMGs as the qBMGs that are color-sink-free. Furthermore, two-colored qBMGs are characterized as directed graphs satisfying three simple local conditions, two of which have appeared previously, namely bi-transitivity in the sense of Das et al. (2021) and a hierarchy-like structure of out-neighborhoods, i.e., $N(x)\cap N(y)\in\{N(x),N(y),\emptyset\}$ for any two vertices $x$ and $y$. Further results characterize qBMGs that can be explained by binary phylogenetic trees.

math.CO↗

arXiv2022

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 $\mathtt{C}(v)$ of descendant taxa of a vertex $v$. The clustering system $\mathscr{C}_N$ 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 $\mathscr{C}_T$. Although this is no longer true for networks in general, it is of interest to relate properties of $N$ and $\mathscr{C}_N$. 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 system of the following form: If $N$ is a network of type $\mathbb{X}$, then $\mathcal{C}_N$ satisfies $\mathbb{Y}$, and conversely if $\mathscr{C}$ is a clustering system satisfying $\mathbb{Y}$ then there is network $N$ of type $\mathbb{X}$ such that $\mathscr{C}\subseteq\mathscr{C}_N$.This, in turn, allows us to investigate the mutual dependencies between the distinct types of networks in much detail.

q-bio.PE↗

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

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

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

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

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

From Best Hits to Best Matches

Many of the commonly used methods for orthology detection start from mutually most similar pairs of genes (reciprocal best hits) as an approximation for evolutionary most closely related pairs of genes (reciprocal best matches). This approximation of best matches by best hits becomes exact for ultrametric dissimilarities, i.e., under the Molecular Clock Hypothesis. It fails, however, whenever there are large lineage specific rate variations among paralogous genes. In practice, this introduces a high level of noise into the input data for best-hit-based orthology detection methods. If additive distances between genes are known, then evolutionary most closely related pairs can be identified by considering certain quartets of genes provided that in each quartet the outgroup relative to the remaining three genes is known. \emph{A priori} knowledge of underlying species phylogeny greatly facilitates the identification of the required outgroup. Although the workflow remains a heuristic since the correct outgroup cannot be determined reliably in all cases, simulations with lineage specific biases and rate asymmetries show that nearly perfect results can be achieved. In a realistic setting, where distances data have to be estimated from sequence data and hence are noisy, it is still possible to obtain highly accurate sets of best matches. Improvements of tree-free orthology assessment methods can be expected from a combination of the accurate inference of best matches reported here and recent mathematical advances in the understanding of (reciprocal) best match graphs and orthology relations.

q-bio.PE↗

arXiv2019

PyRod -- Tracing Water Molecules in Molecular Dynamics Simulations

Ligands entering a protein binding pocket essentially compete with water molecules for binding to the protein. Hence, the location and thermodynamic properties of water molecules in protein structures have gained increased attention in the drug design community. Including corresponding data into 3D pharmacophore modeling is essential for efficient high throughput virtual screening. Here, we present PyRod, a free and open-source python software that allows for visualization of pharmacophoric binding pocket characteristics, identification of hot spots for ligand binding and subsequent generation of pharmacophore features for virtual screening. The implemented routines analyze the protein environment of water molecules in molecular dynamics (MD) simulations and can differentiate between hydrogen bonded waters as well as waters in a protein environment of hydrophobic, charged or aromatic atom groups. The gathered information is further processed to generate dynamic molecular interaction fields (dMIFs) for visualization and pharmacophoric features for virtual screening. The described software was applied to 5 therapeutically relevant drug targets and generated pharmacophores were evaluated using DUD-E benchmarking sets. The best performing pharmacophore was found for the HIV1 protease with an early enrichment factor of 54.6. PyRod adds a new perspective to structure-based screening campaigns by providing easy-to-interpret dMIFs and purely protein-based 3D pharmacophores that are solely based on tracing water molecules in MD simulations. Since structural information about co-crystallized ligands is not needed, screening campaigns can be followed, for which less or no ligand information is available. PyRod is freely available at https://github.com/schallerdavid/pyrod.

q-bio.BM↗

arXiv2006

Gravity's Relentless Pull: An interactive, multimedia website about black holes for Education and Public Outreach

We have created a website, called "Black Holes: Gravity's Relentless Pull", which explains the physics and astronomy of black holes for a general audience. The site emphasizes user participation and is rich in animations and astronomical imagery. It won the top prize of the 2005 Pirelli INTERNETional Awards competition for the best communication of science and technology using the internet. This article provides a brief overview of the site. The site starts with an opening animation that introduces the basic concept of a black hole. The user is then invited to embark on a journey from a backyard view of the night sky to a personal encounter with a singularity. This journey proceeds through three modules, which allow the user to: find black holes in the night sky; travel to a black hole in an animated starship; and explore a black hole from up close. There are also five "experiments" that allow the user to: create a black hole; orbit around a black hole; weigh a black hole; drop a clock into a black hole; or fall into a black hole. The modules and experiments offer goal-based scenarios tailored for novices and children. The site also contains an encyclopedia of frequently asked questions and a detailed glossary that are targeted more at experts and adults. The overall result is a website where scientific knowledge, learning theory, and fun converge. Despite its focus on black holes, the site also teaches many other concepts of physics, astronomy and scientific thought. The site aims to instill an appreciation for learning and an interest in science, especially in the younger users. It can be used as an aid in teaching introductory astronomy at the undergraduate level.

astro-ph↗