Entanglement, Invariants, and Phylogenetics

TL;DR

This thesis applies mathematical physics and group theory to enhance the analysis of Markov models in phylogenetics, providing new invariant functions and a novel tree reconstruction method consistent with general models.

Contribution

It introduces a group theoretical approach to phylogenetic analysis, deriving polynomial invariants and a new quartet tree reconstruction technique for Markov models.

Findings

Derived polynomial group invariants for Markov models

Extended analysis of invariant functions in distance methods

Presented a new consistent quartet tree reconstruction method

Abstract

This thesis develops and expands upon known techniques of mathematical physics relevant to the analysis of the popular Markov model of phylogenetic trees required in biology to reconstruct the evolutionary relationships of taxonomic units from biomolecular sequence data. The techniques of mathematical physics are plethora and have been developed for some time. The Markov model of phylogenetics and its analysis is a relatively new technique where most progress to date has been achieved by using discrete mathematics. This thesis takes a group theoretical approach to the problem by beginning with a remarkable mathematical parallel to the process of scattering in particle physics. This is shown to equate to branching events in the evolutionary history of molecular units. The major technical result of this thesis is the derivation of existence proofs and computational techniques for calculating polynomial group invariant functions on a multi-linear space where the group action is that relevant to a Markovian time evolution. The practical results of this thesis are an extended analysis of the use of invariant functions in distance based methods and the presentation of a new reconstruction technique for quartet trees which is consistent with the most general Markov model of sequence evolution.