Phylogenetics beyond biology

Abstract

Evolutionary processes have been described not only in biology but also for a wide range of human cultural activities including languages and law. In contrast to the evolution of DNA or protein sequences, the detailed mechanisms giving rise to the observed evolution-like processes are not or only partially known. The absence of a mechanistic model of evolution implies that it remains unknown how the distances between different taxa have to be quantified. Considering distortions of metric distances, we first show that poor choices of the distance measure can lead to incorrect phylogenetic trees. Based on the well-known fact that phylogenetic inference requires additive metrics, we then show that the correct phylogeny can be computed from a distance matrix [Formula: see text] if there is a monotonic, subadditive function [Formula: see text] such that [Formula: see text] is additive. The required metric-preserving transformation [Formula: see text] can be computed as the solution of an optimization problem. This result shows that the problem of phylogeny reconstruction is well defined even if a detailed mechanistic model of the evolutionary process remains elusive.

Keywords: Additive metric; Cultural evolution; Metric-preserving functions; Phylogenetic tree.

Fig. 1

Metric-preserving transformations do not preserve the relation $\Vert$. The distance matrix $\mathbf{T}$ corresponds to the tree in the middle and, according to Eq. (1), satisfied $uv\Vert xy$. The function $\zeta$ satisfies (Z1), (Z2), (Z3) and is smooth. The transformed distance matrix $\mathbf{D}=\zeta (\mathbf{T})$ is presented by the networks shown on the r.h.s. (computed with SplitsTree (Huson and Bryant 2006). Here, $d(u,y)+d(x,v)$ is the distance pair with the shortest distance sum, i.e., it corresponds to the quadruple $uy\Vert xv$. This split corresponds to the longer one of the two side lengths of the box

Fig. 2

Empirical estimation of a transformation $\zeta$. Top: The relevant parameters $a$ and $c$ of the stretched exponential transform Eq. (5) can be estimated with the help of Eq. (4). Plotting $\Delta (\zeta )$ as a function of the parameters $a$ and $c$ in Eq. (5) shows that the minimal discrepancy is indeed found at the theoretical values $a=3/4$ and $c=1$ used to generate the transformed distance matrix $\mathbf{D}$ corresponding to a tree with 100 leaves. The color scale on the r.h.s. of the panel refers to ln$(1+\Delta (\zeta ))$. Below: The two small panels show the effect of increasing levels of measurement noise (left: $\epsilon =0.1$, right: $\epsilon =0.2$, see “Appendix 2” for details)