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Branching models with distributed ages for macroevolution

By Tugrul, M.; Hernandez-Garcia, E.; Eguiluz, V.M.; Keller-Schmidt, S.; Klemm, K.
Poster presented at the International Conference on Delayed Complex Systems 2012 (Palma de Mallorca, Spain) 2012

Phylogenetic trees summarize the evolutionary story of organisms on Earth. It has been known since long ago [1] that these trees are very unbalanced, meaning that some evolutionary lineages are extremely abundant whereas some others contain just a few taxa. A convenient way to quantify this is in terms of the dependence on the number of species of the tree depth, which is the mean distance (measured in number of branching events) from the leaves of the tree (the extant species) to the root (the last common ancestor). Recent large-scale analysis of species phylogenies [2,3] and also of phylogenies of protein families [4] indicate that the mean depth of the trees d scale with the number of species N faster than the logarithmic law d  ∼ logN that is obtained from most standard models of tree branching including the random branching tree. Either a power law or a square logarithmic behavior d  ∼ (logN)² [2,4] describe data more appropriately.

Here we introduce a model of branching that reproduces the d ∼ (logN)² scaling and provides a good fit to existing phylogenies databases [5]. The main ingredient is a dependence on the branching probabilities on the age of the nodes, i.e. on the time since last speciation. Analysis leads to difference and integral equations in which different branching events contribute with distributed delays. In fact this age model appears as a critical case within a more general family of models in which by varying the delay distribution there is a transition between the standard logarithmic behavior with all branches growing in a more or less symmetric way and an instability in which branching accumulates along a few branches.

References

[1] J. C. Willis. Age and Area: a study in geographical distribution and origin of species. Cambridge University Press, Cambridge, 1922.

[2] M.G. Blum and O. François. Which random processes describe the tree of life? a large-scale study of phylogenetic tree imbalance. Syst. Biol., 55:685-691, 2006.

[3] E. Alejandro Herrada, Claudio J. Tessone, Konstantin Klemm, Víctor M. Eguíluz, Emilio Hernández-García, and Carlos M. Duarte. Universal scaling in the branching of the tree of life. PLoS ONE, 3:e2757, 2008.

[4] A. Herrada, V.M. Eguíluz, E. Hernández-García, and C.M. Duarte. Scaling properties of protein family phylogenies. BMC Evol. Biol., 11:155, 2011.

[5] S. Keller-Schmidt, M. Tugrul, V. M. Eguíluz, E. Hernández-García, and K. Klemm. An age-dependent branching model for macroevolution. preprint arXiv:1012.3298.