Journal of Mathematical Biology

, Volume 71, Issue 6–7, pp 1387–1409 | Cite as

Closed form modeling of evolutionary rates by exponential Brownian functionals

  • Nicolas PrivaultEmail author
  • Stéphane Guindon


Accurate estimation of species divergence times from the analysis of genetic sequences relies on probabilistic models of evolution of the rate of molecular evolution. Importantly, while these models describe the sample paths of the substitution rates along a phylogenetic tree, only the (random) average rate can be estimated on each edge. For mathematical convenience, the stochastic nature of these averages is generally ignored. In this article we derive the probabilistic distribution of the average substitution rate assuming a geometric Brownian motion for the sample paths, and we investigate the corresponding error bounds via numerical simulations. In particular we confirm the validity of the gamma approximation proposed in Guindon (Syst Biol 62(1):22–34, 2013) for “small” values of the autocorrelation parameter.


Evolutionary rates Exponential Brownian functionals   Geometric Brownian bridge Molecular clocks Phylogenetics 

Mathematics Subject Classification

92D15 92D20 33C10 62J10 60J22 60J27 60J65 81S40 60H30 60H07 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Division of Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  2. 2.Department of StatisticsThe University of AucklandAucklandNew Zealand

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