[Paper Review] Operads and Phylogenetic Trees

John C. Baez, Nina Otter|arXiv (Cornell University)|Dec 10, 2015

Business Strategy and Innovation4 citations

TL;DR

This paper introduces the operad $φ\mathrm{Phyl}$, which formalizes phylogenetic trees as operations using edge labels for time or branch lengths. It proves a homeomorphism between the space of $n$-ary operations in $φ\mathrm{Phyl}$ and $\mathscr{T}_n \times [0,\infty)^{n+1}$, where $\mathscr{T}_n$ is the Billera-Holmes-Vogtmann space of metric $n$-trees, and shows that Markov models in phylogenetics yield coalgebras over $\mathrm{Phyl}$, linking evolutionary biology with algebraic topology.

ABSTRACT

We construct an operad $\mathrm{Phyl}$ whose operations are the edge-labelled trees used in phylogenetics. This operad is the coproduct of $\mathrm{Com}$, the operad for commutative semigroups, and $[0,\infty)$, the operad with unary operations corresponding to nonnegative real numbers, where composition is addition. We show that there is a homeomorphism between the space of $n$-ary operations of $\mathrm{Phyl}$ and $\mathcal{T}_n imes [0,\infty)^{n+1}$, where $\mathcal{T}_n$ is the space of metric $n$-trees introduced by Billera, Holmes and Vogtmann. Furthermore, we show that the Markov models used to reconstruct phylogenetic trees from genome data give coalgebras of $\mathrm{Phyl}$. These always extend to coalgebras of the larger operad $\mathrm{Com} + [0,\infty]$, since Markov processes on finite sets converge to an equilibrium as time approaches infinity. We show that for any operad $O$, its coproduct with $[0,\infty]$ contains the operad $W(O)$ constucted by Boardman and Vogt. To prove these results, we explicitly describe the coproduct of operads in terms of labelled trees.

Motivation & Objective

• To formalize phylogenetic trees with edge lengths as operations in an algebraic structure.
• To construct an operad $\mathrm{Phyl}$ whose operations correspond to edge-labelled trees used in phylogenetics.
• To establish a topological equivalence between the space of $n$-ary operations in $\mathrm{Phyl}$ and $\mathscr{T}_n \times [0,\infty)^{n+1}$, where $\mathscr{T}_n$ is the space of metric $n$-trees.
• To show that Markov processes modeling DNA evolution give rise to coalgebras over $\mathrm{Phyl}$, linking stochastic models to operadic structures.
• To demonstrate that the coproduct of $\mathrm{Com}$ and $[0,\infty)$ yields the Boardman-Vogt operad $W(O)$ for any operad $O$.

Proposed method

• Construct the operad $\mathrm{Phyl}$ as the coproduct of $\mathrm{Com}$ (for commutative semigroups) and $[0,\infty)$ (for nonnegative real numbers under addition).
• Define $\mathrm{Phyl}_n$, the set of $n$-ary operations, as rooted trees with $n$ leaves labeled by $1,\dots,n$, edges labeled in $[0,\infty)$, no unary vertices, and positive internal edge lengths.
• Equip $\mathrm{Phyl}_n$ with a topology based on isomorphism classes of trees and continuous variation of edge lengths, including degenerations where internal edges collapse to zero.
• Establish a bijection $f: \mathrm{Phyl}_n \to \mathscr{T}_n \times [0,\infty)^{n+1}$, mapping each tree to its underlying metric tree and the $n+1$ external edge lengths.
• Prove that $f$ is a homeomorphism by showing it is continuous and open using basis elements from the topology of $\mathscr{T}_n$ and $[0,\infty)^{n+1}$.
• Show that Markov processes on finite state spaces converge to equilibrium, allowing their coalgebra structure to extend to $\mathrm{Com} + [0,\infty]$, and thus to $\mathrm{Phyl}$.

Experimental results

Research questions

• RQ1 How can phylogenetic trees with edge lengths be formalized as operations in an algebraic structure?
• RQ2 What is the topological structure of the space of $n$-ary operations in the operad $\mathrm{Phyl}$?
• RQ3 Do standard Markov models of DNA evolution give rise to coalgebras over $\mathrm{Phyl}$?
• RQ4 How does the coproduct of operads relate to the Boardman-Vogt construction $W(O)$?
• RQ5 Can the space of metric $n$-trees $\mathscr{T}_n$ be naturally identified with a component of the space of $n$-ary operations in $\mathrm{Phyl}$?

Key findings

• The space of $n$-ary operations in $\mathrm{Phyl}$ is homeomorphic to $\mathscr{T}_n \times [0,\infty)^{n+1}$, where $\mathscr{T}_n$ is the Billera-Holmes-Vogtmann space of metric $n$-trees.
• The operad $\mathrm{Phyl}$ is isomorphic to the coproduct $\mathrm{Com} + [0,\infty)$, with composition defined by addition of edge lengths.
• Markov processes modeling DNA evolution on finite state spaces give rise to coalgebras over $\mathrm{Phyl}$, due to convergence to equilibrium as time $\to \infty$.
• The coproduct of $\mathrm{Com}$ and $[0,\infty)$ contains the Boardman-Vogt operad $W(O)$ for any operad $O$, generalizing a known construction.
• The topology on $\mathrm{Phyl}_n$ is generated by open sets corresponding to isomorphism classes of trees and continuous variations of edge lengths, including degenerations to lower arity.