Phylogenetics
Phylogenic trees are often used in biology to determine the close ancestor of different species. Typically, a tree “branches“when two species become distinguishable. For example, in the tree below, both lizards and snakes are reptiles, but their traits can be distinguished further from that. Species that are closely related are closer to each other and have fewer branches between them.
The first way topology connects to phylogenic trees is by determining which trees are the same. This is classified as two trees having the exact same set of “clades“ or groups at each branch. Clades can also be defined as “balanced“ or “unbalanced“ if they all have the same number of species. See examples below:
The first part of “math“ relating to phylogenic trees deals with statistics. This probably of certain trees and species relations is especially important when determining lineages and evolutionary changes. The simplest part of these statistics involves creating a metric for the tree’s “balance.” Known as “Colless’ Index (I),” the equation first finds the difference of all number of species/tips (N) on each side of the tree. If this value is 0, then the tree is known as balanced. If not, you divide this number by [(N-1)(N-2)/2]. This standardized the tree. If each clade or branch only has one species on it, the index will be 1. Any other form of a balanced tree will be between 0 and 1.
The example the referenced paper gives is using this index to compare birth/death rates. In theory, these values should be equal across all lineages, but it can also account for changes in rates over time.
There is one problem in the above method of defining trees. This is known as “endosymbiosis.” Specifically, in biology and evolution, one species from one branch will incorporate into another organism on a completely separate clade. This generally refers to eukorytes evolving from bacteria that lived symbiotically with them. For example, mitochondria and chloroplasts could have developed out of separate bacteria due to the eukaryotic cell not digesting the pathogen and instead incorporating them into its own life cycle. This creates many conflicted branches that cross over between clades.
The second paper “Operads and Phylogenetic Trees” takes a different approach to defining phylogenic trees. They made two main assumptions.
1) DNA in one species will randomly change over time, which each change equally likely.
2) Specieies will also undergo “random walks" where one species splits into two, which again happens at determinable times.
Each assumption allows for probable common ancestors to be determined and a phylogenetic tree to be constructed. They then go further to defining features of trees beyond number of clades. Although clades is one metric, they also define vertices and lengths with positive, finite numbers.
The way “mathematics” is done between trees mostly simplifies them. See the way the paper gives an example for an operad:
Technically, the tree is not changing since F still maintains the verticies of g1, g2, and g3. This allows topologists to find and classify homeomorphisms between phylogenic trees and implement “coalgebras,” something part of representation theory that I will try to cover in a future post.
References:
Libretexts. (2022, February 20). 10.5: Tree topology, tree shape, and tree balance under a birth-death model. Retrieved August 31, 2022, from https://bio.libretexts.org/Bookshelves/Evolutionary_Developmental_Biology/Phylogenetic_Comparative_Methods_(Harmon)/10%3A_Introduction_to_Birth-Death_Models/10.05%3A_Tree_topology%2C_tree_shape%2C_and_tree_balance_under_a_birth-death_model
Operads and phylogenetic trees - department of mathematics. (n.d.). Retrieved September 2, 2022, from https://math.ucr.edu/home/baez/phylo.pdf
Libretexts. (2022, February 20). 10.5: Tree topology, tree shape, and tree balance under a birth-death model. Retrieved August 31, 2022, from https://bio.libretexts.org/Bookshelves/Evolutionary_Developmental_Biology/Phylogenetic_Comparative_Methods_(Harmon)/10%3A_Introduction_to_Birth-Death_Models/10.05%3A_Tree_topology%2C_tree_shape%2C_and_tree_balance_under_a_birth-death_model