Clades and clans: a comparison study of two evolutionary models

TL;DR

This paper compares two evolutionary models, YHK and PDA, analyzing their distributions of clade and clan sizes, revealing log-convexity and a critical size threshold that influences the likelihood of certain clades under each model.

Contribution

It provides a mathematical comparison of YHK and PDA models, establishing log-convexity of distributions and identifying a critical size threshold for clade probabilities.

Findings

Distributions are log-convex under both models.

Existence of a critical size $(n)$ affecting clade probabilities.

Similar results extended to unrooted binary trees.

Abstract

The Yule-Harding-Kingman (YHK) model and the proportional to distinguishable arrangements (PDA) model are two binary tree generating models that are widely used in evolutionary biology. Understanding the distributions of clade sizes under these two models provides valuable insights into macro-evolutionary processes, and is important in hypothesis testing and Bayesian analyses in phylogenetics. Here we show that these distributions are log-convex, which implies that very large clades or very small clades are more likely to occur under these two models. Moreover, we prove that there exists a critical value $\kappa(n)$ for each $n\geqslant 4$ such that for a given clade with size $k$, the probability that this clade is contained in a random tree with $n$ leaves generated under the YHK model is higher than that under the PDA model if $1<k<\kappa(n)$, and lower if $\kappa(n)<k<n$. Finally, we extend our results to binary unrooted trees, and obtain similar results for the distributions of clan sizes.