Invariants for level-1 phylogenetic networks under the Cavendar-Farris-Neyman Model

Abstract

Phylogenetic networks can model more complicated evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. The same Markov models that are used to model evolution on trees can also be extended to networks and similar questions, such as the identifiability of the network parameter or the invariants of the model, can be asked. In this paper we focus on finding the invariants of the Cavendar-Farris-Neyman (CFN) model on level-1 phylogenetic networks. We do this by reducing the problem to finding invariants of sunlet networks, which are level-1 networks consisting of a single cycle with leaves at each vertex. We then determine all quadratic invariants in the sunlet network ideal which we conjecture generate the full ideal.torus action on them meaning they are T-varieties [22]. We use this torus action to break up the ideal of invariants of a n-leaf sunlet network into homogeneous graded pieces we call gloves. As a result, we arrive at the following theorem.Theorem. A quadratic f is an invariant of the n-sunlet network if and only if it is an invariant for both of the underlying trees obtained by deleting a reticulation edge.We then explicitly produce all quadratic generators of the sunlet network ideal that lie in a given graded piece which gives a complete set of quadratic generators of the sunlet network ideal under the CFN model. We conjecture that the sunlet network ideal is generated by quadratics which would imply our set of quadratic generators actually generate the entire ideal.We have also studied the 4-and 5-leaf sunlet networks in more detail. We have shown through explicit computation that their corresponding varieties are normal and Gorenstein. This means that any level-1 network that can be built by gluing together 4-and 5-leaf sunlets along trees is normal and Cohen-Macaulay since these properties are preserved by the toric fiber product. Level-1 networks built from gluing 4-and 5-sunlets along leaves that are not adjacent to the reticulation vertex of the respective networks are also Gorenstein for the same reason but this may not hold if networks are glued together along leaves adjacent to the reticulation vertex. Lastly, we compute the multigraded Hilbert function of the 4-leaf sunlet network. All of these computational results along with an implementation of our algorithm to find quadratic generators and computational evidence for our conjectures can be found at: https://github.com/bkholler/CFN Networks.This paper is organized as follows. In Section 2, we provide some background on phylogenetic models with a particular emphasis on the CFN model and the ideal of invariants for CFN tree models. We also describe the toric fiber product. In Section 3, we show that studying the CFN model on level-1 networks can be reduced to understanding the CFN model on n-sunlets. In Section 4, we give a complete description for quadratic invariants for any sunlet network. In Section 5, we focus on 4-and 5-leaf sunlet networks and describe some algebraic properties of their ...