The Cavender-Farris-Neyman Model with a Molecular Clock

TL;DR

This paper provides a combinatorial and polyhedral analysis of the invariants of the Cavender-Farris-Neyman model with a molecular clock on rooted binary trees, revealing structural properties and invariance features related to the tree's leaves.

Contribution

It introduces a combinatorial description of the toric ideal for the CFN-MC model, characterizes the associated polytope's structure, and shows invariance of certain algebraic properties with respect to tree topology.

Findings

Number of vertices of the polytope is a Fibonacci number.

Facets are described by the tree's cluster structure.

Ehrhart polynomial depends only on the number of leaves.

Abstract

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial "cluster" structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic Groebner basis with squarefree initial terms. Finally, we show that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself. These results are analogous to classic results for the Cavender-Farris-Neyman model without a molecular clock. However, new techniques are required because the molecular clock assumption destroys the toric fiber product structure that governs group-based models without the molecular clock.