Fourier Analysis and Phylogenetic Trees

Summary

We give an overview of phylogenetic invariants: a technique for reconstructing evolutionary family trees from DNA sequence data. This method is useful in practice and is based on a number of simple ideas from elementary group theory, probability, linear algebra, and commutative algebra.

1. Introduction

Phylogeny is the branch of biology that seeks to reconstruct evolutionary family trees. Such reconstruction can take place at various scales. For example, we could attempt to build the family tree for various present day indigenous populations in the Americas and Asia in order to glean information about the possible course of migration of humans into the Americas. At the level of species, we could seek to determine whether modern humans are more closely related to chimpanzees or to gorillas. Ultimately, we would like to be able to reconstruct the entire “tree of life” that describes the course of evolution leading to all present day species. Because the status of the “leaves” on which we wish to build a tree differs from instance to instance, biologists use the general term taxa (singular taxon) for the leaves in a general phylogenetic problem.

For example, for 4 taxa, we might seek to decide whether the tree describes the course of evolution. In such trees:

• • the arrow of time is down the page,

• • paths down through the tree represent lineages (lines of descent),

• • any point on a lineage corresponds to a point of time in the life of some ancestor of a taxon,

• • vertices other than leaves represent times at which lineages diverge,

• • the root corresponds to the most recent common ancestor of all the taxa.