Barking up the wrong tree : some obstacles to phylogenetic reconstruction

Phylogenetics is the study of evolutionary relationships between entities, usually biological in nature. The primary aim of such study is to elucidate the structure of these evolutionary histories. Unfortunately, such study can run into a variety of obstacles, both practical and theoretical. In this thesis we explore theoretical obstacles to phylogenetic reconstruction, by examining several scenarios in which distinguishing between similar structures can become quite difficult. In Chapter 2, we consider when metrics on trees and metrics on networks can become indistinguishable, and present several novel results in this area, showing that it is possible for any tree metric to be represented on a non-trivial network, and provide early results on the possible structures of these networks. In Chapter 3, we consider tree-based networks - a phenomenon in which networks have a strong tree-like signal. We present the first findings on these networks in the context of unrooted non-binary networks. We characterise the circumstances under which such networks can become `saturated' by these signals, and provide some graph theoretical results in this area as well. In Chapter 4 we consider the scenario in which two trees can appear similar due to their hierarchical structure. We present a new metric to quantify this similarity, and use simulations to show several promising properties of the metric and the relative accuracy of a function that gives an upper bound to the metric

Doctor of Philosophy

dissertationAdvances in computer hardware have enabled routine MD simulations of systems with tens of thousands of atoms for up to microseconds (soon milliseconds). The key limiting factor in whether these simulations can advance hypothesis testing in active research is the accuracy of the force fields. In many ways, force fields for RNA are less mature than those for proteins. Yet even the current generation of force fields offers benefits to researchers as we demonstrate with our re-refinement effort on two RNA hairpins. Additionally, our simulation study of the binding of 2-aminobenzimidazole inhibitors to hepatitis C RNA offers a computational perspective on which of the two rather different published structures (one NMR, the other X-ray) is a more reasonable structure for future CADD efforts as well as which free energy methods are suited to these highly charged complexes. Finally, further effort on force field improvement is critical. We demonstrate an effective method to determine quantitative conformational population analysis of small RNAs using enhanced sampling methods. These efforts are allowing us to uncover force field pathologies and quickly test new modifications. In summary, this research serves to strengthen communication between experimental and theoretical methods in order produce mutual benefit

Minimal Permutation Representations of Classes of Semidirect Products of Groups

Given a finite group $G$, the smallest $n$ such that $G$ embeds into the symmetric group $S_n$ is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way

Minimal Permutation Representations of Classes of Semidirect Products of Groups

Given a finite group $G$, the smallest $n$ such that $G$ embeds into the symmetric group $S_n$ is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way

On the comparison of incompatibility of split systems across different taxa sizes

The concept of $k$-compatibility measures how many phylogenetic trees it would take to display all splits in a given set. A set of trees that display every single possible split is termed a \textit{universal tree set}. In this note, we find $A(n)$, the minimal size of a universal tree set for $n$ taxa. By normalising the $k$-compatibility using $A(n)$, one can then compare incompatibility of split systems across different taxa sizes. We demonstrate this application by comparing two SplitsTree networks of different sizes derived from archaeal genomes.Comment: 9 pages, 2 figure

Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics

In phylogenetics it is of interest for rate matrix sets to satisfy closure under matrix multiplication as this makes finding the set of corresponding transition matrices possible without having to compute matrix exponentials. It is also advantageous to have a small number of free parameters as this, in applications, will result in a reduction of computation time. We explore a method of building a rate matrix set from a rooted tree structure by assigning rates to internal tree nodes and states to the leaves, then defining the rate of change between two states as the rate assigned to the most recent common ancestor of those two states. We investigate the properties of these matrix sets from both a linear algebra and a graph theory perspective and show that any rate matrix set generated this way is closed under matrix multiplication. The consequences of setting two rates assigned to internal tree nodes to be equal are then considered. This methodology could be used to develop parameterised models of amino acid substitution which have a small number of parameters but convey biological meaning.Comment: 12 pages, 3 figure