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Closed form modeling of evolutionary rates by exponential Brownian functionals

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Abstract

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.

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Authors and Affiliations

  1. Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Nicolas Privault

  2. Department of Statistics, The University of Auckland, Auckland, 1010, New Zealand

    Stéphane Guindon

Authors
  1. Nicolas Privault
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  2. Stéphane Guindon
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Correspondence to Nicolas Privault.

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Privault, N., Guindon, S. Closed form modeling of evolutionary rates by exponential Brownian functionals. J. Math. Biol. 71, 1387–1409 (2015). https://doi.org/10.1007/s00285-015-0863-6

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  • DOI: https://doi.org/10.1007/s00285-015-0863-6

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