Algebraic tools for evolutionary biology
type: Item
instance of: scholarly article
title: Algebraic tools for evolutionary biology
zbMATH Open document ID: 1316.13041
published in: European Mathematical Society Newsletter
publication date: 2013-05-23T00:00:00Z
review text: The purpose of this article is to provide an overview of the uses of algebraic statistics and algebraic geometry in studying problems arising in evolutionary biology, particularly phylogenitics. The aim of phylogenetics is to `reconstruct the ancestral relations among species, i.e., the phylogenetic tree, from given DNA sequences.' A phylogenetic tree is an acyclic connected graph \(T\) with the data of: (1) a labelling on the leaves of \(T\) by contemporary species names (recorded as a DNA sequence) and (2) a length for every edge, called a branch length, representing the evolutionary distance between both ends of the edge. The topology of a phylogenetic tree is the topology of the labelled tree (disregarding branch lengths). Hence a phylogenetic tree is determined by its topology and branch lengths. Given a group of contemporary species, there are three key questions to answer when reconstructing the phylogenetic tree: (1) What evolutionary model is best suited to the given DNA sequences? (2) What tree topology best fits the data and what are its branch lengths? (3) Given the choice of model, is it possible to identify the phylogenetic tree from the given DNA sequences? Evolutionary models are presented in \S~2 as discrete-time Markov processes on trees. Choosing a model (question one above) amounts to placing restrictions on the general Markov model [\textit{D. Barry} and \textit{J. A. Hartigan}, Biometrics 43, 261--276 (1987; Zbl 0622.92012)]; [\textit{M. Steel}, Appl. Math. Lett. 7, No. 2, 19--23 (1994; Zbl 0794.60071)]. For instance, in this way one recovers models such as the Strand symmetric model [\textit{M. Casanellas} and \textit{S. Sullivant}, ``The Strand Symmetric Model,'' in: L. Pachter and B. Sturmfels, editors, Algebraic statistics for computational biology, chapter 16, Cambridge University Press (2005)], the Kimura \(3\)-parameter model [\textit{M. Kimura}, Proc. Natl. Acad. Sci. USA 78, 454--458 (1981; Zbl 0511.92013)], and the Jukes-Cantor model …
Mathematics Subject Classification ID: 13P25
Mathematics Subject Classification ID: 62H99
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 92D15
zbMATH DE Number: 6165376
zbMATH Keywords: algebraic statistics
zbMATH Keywords: phylogenetics
zbMATH Keywords: phylogenetic tree
zbMATH Keywords: phylogenetic invariant
zbMATH Keywords: algebraic variety
zbMATH Keywords: Markov model
zbMATH Keywords: Markov process
zbMATH Keywords: maximum likelihood
zbMATH Keywords: neighbor-joining
author: Marta Casanellas
reviewed by: Michael R. DiPasquale
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6165376