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

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.

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