Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations. We propose and study a novel framework to study sets of phylogenetic trees based on tropical geometry. In particular, we focus on characterizing our framework for statistical analyses of evolutionary biological processes represented by phylogenetic trees. Our setting exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics, as well as increased computational efficiency over the current state-of-the-art. We demonstrate our approach on seasonal influenza data.Comment: 28 pages, 5 figures, 1 tabl

Anytime Hierarchical Clustering

We propose a new anytime hierarchical clustering method that iteratively transforms an arbitrary initial hierarchy on the configuration of measurements along a sequence of trees we prove for a fixed data set must terminate in a chain of nested partitions that satisfies a natural homogeneity requirement. Each recursive step re-edits the tree so as to improve a local measure of cluster homogeneity that is compatible with a number of commonly used (e.g., single, average, complete) linkage functions. As an alternative to the standard batch algorithms, we present numerical evidence to suggest that appropriate adaptations of this method can yield decentralized, scalable algorithms suitable for distributed/parallel computation of clustering hierarchies and online tracking of clustering trees applicable to large, dynamically changing databases and anomaly detection.Comment: 13 pages, 6 figures, 5 tables, in preparation for submission to a conferenc

Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\log\Phi},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $\Phi$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \log\Phi)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log \Phi)$ and recourse $O(\log \Phi)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $\Omega(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on Discrete Algorithms (SODA24

The Orthology Road: Theory and Methods in Orthology Analysis

The evolution of biological species depends on changes in genes. Among these changes are the gradual accumulation of DNA mutations, insertions and deletions, duplication of genes, movements of genes within and between chromosomes, gene losses and gene transfer. As two populations of the same species evolve independently, they will eventually become reproductively isolated and become two distinct species. The evolutionary history of a set of related species through the repeated occurrence of this speciation process can be represented as a tree-like structure, called a phylogenetic tree or a species tree. Since duplicated genes in a single species also independently accumulate point mutations, insertions and deletions, they drift apart in composition in the same way as genes in two related species. The divergence of all the genes descended from a single gene in an ancestral species can also be represented as a tree, a gene tree that takes into account both speciation and duplication events. In order to reconstruct the evolutionary history from the study of extant species, we use sets of similar genes, with relatively high degree of DNA similarity and usually with some functional resemblance, that appear to have been derived from a common ancestor. The degree of similarity among different instances of the “same gene” in different species can be used to explore their evolutionary history via the reconstruction of gene family histories, namely gene trees. Orthology refers specifically to the relationship between two genes that arose by a speciation event, recent or remote, rather than duplication. Comparing orthologous genes is essential to the correct reconstruction of species trees, so that detecting and identifying orthologous genes is an important problem, and a longstanding challenge, in comparative and evolutionary genomics as well as phylogenetics. A variety of orthology detection methods have been devised in recent years. Although many of these methods are dependent on generating gene and/or species trees, it has been shown that orthology can be estimated at acceptable levels of accuracy without having to infer gene trees and/or reconciling gene trees with species trees. Therefore, there is good reason to look at the connection of trees and orthology from a different angle: How much information about the gene tree, the species tree, and their reconciliation is already contained in the orthology relation among genes? Intriguingly, a solution to the first part of this question has already been given by Boecker and Dress [Boecker and Dress, 1998] in a different context. In particular, they completely characterized certain maps which they called symbolic ultrametrics. Semple and Steel [Semple and Steel, 2003] then presented an algorithm that can be used to reconstruct a phylogenetic tree from any given symbolic ultrametric. In this thesis we investigate a new characterization of orthology relations, based on symbolic ultramterics for recovering the gene tree. According to Fitch’s definition [Fitch, 2000], two genes are (co-)orthologous if their last common ancestor in the gene tree represents a speciation event. On the other hand, when their last common ancestor is a duplication event, the genes are paralogs. The orthology relation on a set of genes is therefore determined by the gene tree and an “event labeling” that identifies each interior vertex of that tree as either a duplication or a speciation event. In the context of analyzing orthology data, the problem of reconciling event-labeled gene trees with a species tree appears as a variant of the reconciliation problem where genes trees have no labels in their internal vertices. When reconciling a gene tree with a species tree, it can be assumed that the species tree is correct or, in the case of a unknown species tree, it can be inferred. Therefore it is crucial to know for a given gene tree whether there even exists a species tree. In this thesis we characterize event-labelled gene trees for which a species tree exists and species trees to which event-labelled gene trees can be mapped. Reconciliation methods are not always the best options for detecting orthology. A fundamental problem is that, aside from multicellular eukaryotes, evolution does not seem to have conformed to the descent-with-modification model that gives rise to tree-like phylogenies. Examples include many cases of prokaryotes and viruses whose evolution involved horizontal gene transfer. To treat the problem of distinguishing orthology and paralogy within a more general framework, graph-based methods have been proposed to detect and differentiate among evolutionary relationships of genes in those organisms. In this work we introduce a measure of orthology that can be used to test graph-based methods and reconciliation methods that detect orthology. Using these results a new algorithm BOTTOM-UP to determine whether a map from the set of vertices of a tree to a set of events is a symbolic ultrametric or not is devised. Additioanlly, a simulation environment designed to generate large gene families with complex duplication histories on which reconstruction algorithms can be tested and software tools can be benchmarked is presented

Phylogenetic Inference via Sequential Monte Carlo

Bayesian inference provides an appealing general framework for phylogenetic analysis, able to incorporate a wide variety of modeling assumptions and to provide a coherent treatment of uncertainty. Existing computational approaches to Bayesian inference based on Markov chain Monte Carlo (MCMC) have not, however, kept pace with the scale of the data analysis problems in phylogenetics, and this has hindered the adoption of Bayesian methods. In this paper, we present an alternative to MCMC based on Sequential Monte Carlo (SMC). We develop an extension of classical SMC based on partially ordered sets and show how to apply this framework—which we refer to as PosetSMC—to phylogenetic analysis. We provide a theoretical treatment of PosetSMC and also present experimental evaluation of PosetSMC on both synthetic and real data. The empirical results demonstrate that PosetSMC is a very promising alternative to MCMC, providing up to two orders of magnitude faster convergence. We discuss other factors favorable to the adoption of PosetSMC in phylogenetics, including its ability to estimate marginal likelihoods, its ready implementability on parallel and distributed computing platforms, and the possibility of combining with MCMC in hybrid MCMC–SMC schemes. Software for PosetSMC is available at http://www.stat.ubc.ca/ bouchard/PosetSMC

Leaping through tree space: continuous phylogenetic inference for rooted and unrooted trees

Phylogenetics is now fundamental in life sciences, providing insights into the earliest branches of life and the origins and spread of epidemics. However, finding suitable phylogenies from the vast space of possible trees remains challenging. To address this problem, for the first time, we perform both tree exploration and inference in a continuous space where the computation of gradients is possible. This continuous relaxation allows for major leaps across tree space in both rooted and unrooted trees, and is less susceptible to convergence to local minima. Our approach outperforms the current best methods for inference on unrooted trees and, in simulation, accurately infers the tree and root in ultrametric cases. The approach is effective in cases of empirical data with negligible amounts of data, which we demonstrate on the phylogeny of jawed vertebrates. Indeed, only a few genes with an ultrametric signal were generally sufficient for resolving the major lineages of vertebrate. With cubic-time complexity and efficient optimisation via automatic differentiation, our method presents an effective way forwards for exploring the most difficult, data-deficient phylogenetic questions.Comment: 13 pages, 4 figures, 14 supplementary pages, 2 supplementary figure