Phylosymmetric algebras: mathematical properties of a new tool in phylogenetics

TL;DR

This paper introduces phylosymmetric algebras, a new mathematical framework for phylogenetics that ensures rate matrix sets are closed under multiplication, simplifying computations and enabling biologically meaningful models with few parameters.

Contribution

It develops a method to construct rate matrix sets from tree structures, proving their closure under multiplication and exploring implications for biological modeling.

Findings

Rate matrix sets generated from tree structures are closed under matrix multiplication.

Equalizing internal node rates simplifies models while maintaining biological relevance.

The methodology enables parameterized amino acid substitution models with fewer parameters.

Abstract

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.