Using Generating Functions to Prove Additivity of Gene-Neighborhood Based Phylogenetics - Extended Abstract.

Prokaryotic evolution is often described as the Spaghetti of Life due to massive genome dynamics (GD) events of gene gain and loss, resulting in different evolutionary histories for the set of genes comprising the organism. These different histories, dubbed as gene trees provide confounding signals, hampering the attempt to reconstruct the species tree describing the main trend of evolution of the species under study. The synteny index (SI) between a pair of genomes combines gene order and gene content information, allowing comparison of unequal gene content genomes, together with order considerations of their common genes. Recently, GD has been modelled as a continuous-time Markov process. Under this formulation, the distance between genes along the chromosome was shown to follow a birth-death-immigration process. Using classical results from birth-death theory, we recently showed that the SI measure is consistent under that formulation. In this work, we provide an alternative, stand alone combinatorial proof of the same result. By using generating function techniques we derive explicit expressions of the system’s probabilistic dynamics in the form of rational functions of the model parameters. This, in turn, allows us to infer analytically the expected distances between organisms based on a transformation of their SI. Although the expressions obtained are rather complex, we establish additivity of this estimated evolutionary distance (a desirable property yielding phylogenetic consistency). This approach relies on holonomic functions and the Zeilberger Algorithm in order to establish additivity of the transformation of SI.