Phylogenetic networks have been studied over the last years as a richer model of the evolutionary history of sets of organisms than phylogenetic trees, because they take into account not only mutation events but also evolutionary events acting at the population level, like recombination between genes, hybridization between lineages, and lateral gene transfer. The latter turn phylogenies into reticulate networks, which are best modeled as directed acyclic graphs 12. For instance, Figure 1 shows two phylogenies inferred from evolutionary distances among three species of frog: R. Aurora, R. Boylii and R. Temporaria 3, enriched with a hypothetical reticulation event (between the R. Amerana and R. Laurasiana groups), which turned them into phylogenetic networks.

We briefly recall below some definitions and results from 4 on phylogenetic networks. See 5 for an introduction to reticulation in phylogenetic analysis.

A phylogenetic network on a set S of taxa is any rooted directed acyclic graph whose leaves (those nodes without outgoing edges) are bijectively labeled by the set S.

Let N = (V, E) be a phylogenetic network on S. A node u ∈ V is said to be a tree node if it has, at most, one incoming edge; otherwise it is called a hybrid node. A phylogenetic network on S is a tree-child phylogenetic network if every node either is a leaf or has at least one child that is a tree node. Tree-child phylogenetic network include galled-trees 67 as a particular case.

Let S = {ℓ₁, ..., ℓ_n} be the set of leaves. We define the μ-vector of a node u ∈ V as the vector μ(u) = (m₁(u), ..., m_n(u)), where m_i(u) is the number of different paths from u to the leaf ℓ_i. The multiset μ(N) = {μ(v) | v ∈ V} is called the μ-representation of N and, provided that N is a tree-child phylogenetic network, it turns out to completely characterize N, up to isomorphisms, among all tree-child phylogenetic networks on S.

This allows us to define a distance on the set of tree-child phylogenetic networks on S: the μ-distance between two given networks N₁ and N₂ is the symmetric difference of their μ-representations,

d_μ(N₁, N₂) = |μ(N₁) Δ μ(N₂)|.

This defines a true distance, and when N₁ and N₂ are phylogenetic trees, it coincides with the well-known partition distance 8.

This representation also allows us to define an optimal alignment between two tree-child phylogenetic networks on S, say n = |S|. Given two such networks N₁ = (V₁, E₁) and N₂ = (V₂, E₂) (where, for the sake of simplicity, we assume |V₁| ≤ |V₂|), an alignment is just an injective mapping M : V₁ → V₂. The weight of this alignment is

w ( M ) = ∑ v ∈ V 1 ( | | μ ( v ) − μ ( M ( v ) ) | | + χ ( v , M ( v ) ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGaem4DaCNaeiikaGIaemyta0KaeiykaKIaeyypa0ZaaabuaeaadaqadaqaaiabcYha8jabcYha8jabeY7aTjabcIcaOiabdAha2jabcMcaPiabgkHiTiabeY7aTjabcIcaOiabd2eanjabcIcaOiabdAha2jabcMcaPiabcMcaPiabcYha8jabcYha8jabgUcaRiabeE8aJjabcIcaOiabdAha2jabcYcaSiabd2eanjabcIcaOiabdAha2jabcMcaPiabcMcaPaGaayjkaiaawMcaaaWcbaGaemODayNaeyicI4SaemOvay1aaSbaaWqaaiabigdaXaqabaaaleqaniabggHiLdGccqGGSaalaaa@59C7@

where || · || stands for the Manhattan norm of a vector and χ (u, v) is 0 if both u and v are tree nodes or hybrid nodes, and 1/(2n) if one of them is a tree node and the other one is a hybrid node. An optimal alignment is, then, an alignment with minimal weight, which can be computed using the Hungarian algorithm 9.

The Perl module Bio::PhyloNetwork relies on the BioPerl bundle and implements several algorithms on phylogenetic networks, from parsing and temporal representation to distances between phylogenetic networks and optimal alignments. The companion Java applet and web-based application make use of the Bio::PhyloNetwork module and allow the user to make simple experiments with phylogenetic networks without having to develop a program or Perl script by him or herself.

While the Bio::PhyloNetwork module computes distances between galled-trees and tree-child phylogenetic networks, it will also support the more general tree-sibling phylogenetic networks in a next release.

The Perl package is available as part of the BioPerl bundle, at the url http://www.bioperl.org/. It can also be downloaded from the url http://dmi.uib.es/~gcardona/BioInfo/Bio-PhyloNetwork.tgz (see Additional file 1). The web-based application is available at the url http://dmi.uib.es/~gcardona/BioInfo/. The Perl package includes full documentation of all its features.

All authors conceived the method, prepared the manuscript, contributed to the discussion, and have approved the final manuscript. GC implemented the software. GV also implemented part of the software.