Tracing evolutionary links between species

The idea that all life on earth traces back to a common beginning dates back at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists have tried to piece together parts of this `tree of life' based on what we can observe today: fossils, and the evolutionary signal that is present in the genomes and phenotypes of different organisms. Mathematics has played a key role in helping transform genetic data into phylogenetic (evolutionary) trees and networks. Here, I will explain some of the central concepts and basic results in phylogenetics, which benefit from several branches of mathematics, including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM

Phylogenetic Flexibility via Hall-Type Inequalities and Submodularity

Given a collection τ of subsets of a finite set X, we say that τ is phylogenetically flexible if, for any collection R of rooted phylogenetic trees whose leaf sets comprise the collection τ , R is compatible (i.e. there is a rooted phylogenetic X-tree that displays each tree in R). We show that τ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being ‘slim’. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not τ is slim. This ‘slim’ condition reduces to a simpler inequality in the case where all of the sets in τ have size 3, a property we call ‘thin’. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection τ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees can be applied to any input trees on given subsets of species

Slim sets of binary trees

AbstractA classical problem in phylogenetic tree analysis is to decide whether there is a phylogenetic tree T that contains all information of a given collection P of phylogenetic trees. If the answer is “yes” we say that P is compatible and T displays P. This decision problem is NP-complete even if all input trees are quartets, that is binary trees with exactly four leaves. In this paper, we prove a sufficient condition for a set of binary phylogenetic trees to be compatible. That result is used to give a short and self-contained proof of the known characterization of quartet sets of minimal cardinality which are displayed by a unique phylogenetic tree