Clades and clans: a comparison study of two evolutionary models

Area of Science:

• Evolutionary biology
• Phylogenetics
• Mathematical biology

Background:

• Binary tree generating models like Yule-Harding-Kingman (YHK) and proportional to distinguishable arrangements (PDA) are crucial in evolutionary biology.
• Understanding clade size distributions is vital for macro-evolutionary insights, hypothesis testing, and phylogenetic analyses.

Purpose of the Study:

• To analyze and compare clade size distributions under the YHK and PDA models.
• To determine the probabilistic behavior of clade sizes in random binary trees generated by these models.
• To extend these findings to binary unrooted trees.

Main Methods:

• Mathematical analysis of clade size distributions.
• Proof of log-convexity for distributions under YHK and PDA models.
• Derivation of a critical value to compare clade probabilities between the two models.

Main Results:

• Clade size distributions under both YHK and PDA models are proven to be log-convex.
• A critical value κ(n) is identified for n ≥ 4, indicating when YHK is more probable than PDA for a given clade size k (1 < k < κ(n)) and vice versa (κ(n) < k < n).
• Similar log-convexity and comparative results are established for clan sizes in binary unrooted trees.

Conclusions:

• The log-convexity of clade size distributions implies a higher likelihood of very large or very small clades.
• The identified critical value provides a quantitative basis for comparing evolutionary dynamics between the YHK and PDA models.
• The findings offer valuable tools for phylogenetic inference and understanding macro-evolutionary patterns.