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Simple Branching Models for Macroevolution

Tugrul, Murat (Directors: Emilio Hernandez-Garcia and Victor M. Eguiluz)
Master Thesis , (2009)

The theoretical tools of Statistical Physics offer powerful techniques to analyse problems in which chance and probabilities play a role. One of such subjects is the understanding of the rules of macroevolution, i.e., evolutionary development and diversification of species, which remains a not well developed part of evolutionary biology. Phylogenetic trees, describing the estimated evolutionary relationships between biological species, are obtained directly from molecular data and are an important indirect evidence for diversification patterns in macroevolution. Therefore, analysing the structures of such estimated trees and comparing to those obtained from branching models is an interesting approach to capture the rules of macroevolution.
In this thesis, we analyze the phylogenetic trees in the TREEBASE and PANDIT databases and characterize their topology (in particular their balance degree) via the mean depth, i.e. the average number of ancestor nodes from the tips to the root. A non-logarithmic scaling with tree size is found, which is not easy to get with branching models existing in the literature. With this motivation, we analyze analytically and numerically three simple branching models, two of which are proposed by us, and try to find their biological meaning if possible. The first is Ford alpha model; although a power law scaling of the mean depth with tree size was established analytically, our numerical results illustrate that the asymptotic regime is approached only at very large tree sizes. For the second model, named as activity model, we show analytically and numerically that it also displays a power law scaling of the mean depth with tree size at a critical parameter. Finally, we propose the so called age model in which the probability of branching depends on the age of the tips with a power parameter. The results of this model at a critical parameter value, which we are capable to express analytically, display a scaling behavior similar to the one obtained from databases analysed. In addition it is potentially open to biological interpretation.