Combinatoric and probabilistic properties regarding the topology of genealogical trees and application in population genetics

• Applicant: Professor Dr. Thomas Wiehe
• Subject Area: Mathematics
• Term: from 2012 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 221524226

Bifurcating trees have extensively and succesfully been used as tools to model evolutionary dynamics. The genealogical tree of a set of alleles, genes, or species can be considered as a single realization of the evolutionary process. Yet, in models and their applications - for instance the coalescent model and the neutrality tests derived from it - it is typically assumed that samples are obtained under long-term average conditions. However, this assumption may not be appropriate when interpreting experimental data, and overlooking this important point may lead to severe mis-interpretations and mis-inferences. In order to gain a clearer understanding it is critically important to investigate the conditional sampling distribution of the tree properties. A comprehensive theory is however still missing. One goal of this proposal is to bridge this gap. We build on the results obtained during the first funding period, but emphasis will shift from combinatoric to probabilistic properties and from static to evolving trees. In particular, we will investigate how strongly a given population genealogy impinges on the genealogical properties of samples by studying the conditional sub-sampling distribution of tree properties, such as height, length and tree balance. Furthermore, we will study how strongly the contingent tree topology of a population leads to a bias in neutrality tests when applied to experimental data. Another key aspect is to integrate the fundamental evolutionary mechanism of recombination in this framework. We will use the ancestral recombination graph as a model of the spatial coalescent and study tree balance of samples and sub-samples as a stochastic process along the chromosome. Since recombination can be silent, i.e. not altering tree topology, it is essential to define topologically relevant recombination events and to quantify their rates. Complementing the spatial view, we will investigate tree balance as a process in time using the classical Moran model. In evolving trees lineages split or die and, as a consequence, tree balance changes over time. Critical times are those when the root jumps to younger tree nodes: in these events history is erased and new evolutionary episodes start. We will study the effect of this process on the sampling and conditional sub-sampling distributions of tree properties. Of particular interest are the rate of change and persistence times relative to generation time. Finally, on a slightly different tack, we will extend work from the first funding period and use tree topology and combinatoric properties of ordered trees to investigate the evolutionary mechanisms behind the distribution of large gene families along chromosomes. We will apply our theoretical results to the analysis of experimental data and their interpretation in the light of neutral vs. adaptive evolution.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1590: Probabilistic Structures in Evolution