Journal of Mathematical Biology

, Volume 78, Issue 4, pp 1211–1224 | Cite as

The stationary distribution of a sample from the Wright–Fisher diffusion model with general small mutation rates

  • Conrad J. Burden
  • Robert C. GriffithsEmail author


The stationary distribution of a sample taken from a Wright–Fisher diffusion with general small mutation rates is found using a coalescent approach. The approximation is equivalent to having at most one mutation in the coalescent tree to the first order in the rates. The sample probabilities characterize an approximation for the stationary distribution from the Wright–Fisher diffusion. The approach is different from Burden and Tang (Theor Popul Biol 112:22–32, 2016; Theor Popul Biol 113:23–33, 2017) who use a probability flux argument to obtain the same results from a forward diffusion generator equation. The solution has interest because the solution is not known when rates are not small. An analogous solution is found for the configuration of alleles in a general exchangeable binary coalescent tree. In particular an explicit solution is found for a pure birth process tree when individuals reproduce at rate \(\lambda \).


Coalescent tree Small mutation rates Wright–Fisher diffusion 

Mathematics Subject Classification

92B99 92D15 



This research was done when Robert Griffiths visited the Mathematical Sciences Instutite, Australian National University in November and December 2017. He thanks the Instutite for their support and hospitality.


  1. Bhaskar A, Kamm JA, Song YS (2012) Approximate sampling formulae for general finite-alleles models of mutation. Adv Appl Probab 44:408–428MathSciNetCrossRefzbMATHGoogle Scholar
  2. Burden CJ, Tang Y (2016) An approximate stationary solution for multi-allele diffusion with low mutation rates. Theor Popul Biol 112:22–32CrossRefzbMATHGoogle Scholar
  3. Burden CJ, Tang Y (2017) Rate matrix estimation from site frequency data. Theor Popul Biol 113:23–33CrossRefzbMATHGoogle Scholar
  4. Burden CJ, Griffiths RC (2018) Stationary distribution of a 2-island 2-allele Wright–Fisher diffusion model with slow mutation and migration rates. Theor Popul Biol.
  5. De Maio N, Schrempf D, Kosiol C (2015) PoMo: an allele frequency based approach for species tree estimation. Syst. Biol. 64:1018–1031CrossRefGoogle Scholar
  6. De Iorio M, Griffiths RC (2004) Importance sampling on coalescent histories. I. Adv Appl Prob 36:417–433MathSciNetCrossRefzbMATHGoogle Scholar
  7. Etheridge A (2011) Some mathematical models from population genetics: École D’Été de Probabilits de Saint-Flour XXXIX-2009. Springer, BerlinCrossRefzbMATHGoogle Scholar
  8. Griffiths RC, Tavaré S (1998) The age of a mutation in a general coalescent tree. Stoch Models 14:273–295MathSciNetCrossRefzbMATHGoogle Scholar
  9. Griffiths RC, Tavaré S (2003) The genealogy of a neutral mutation. In: Green PJ, Hjort NL, Richardson S (eds) Highly structured stochastic systems. Oxford University Press, Oxford, pp 393–413Google Scholar
  10. Jenkins PA, Song YS (2010) An asymptotic sampling formula for the coalescent with recombination. Ann Appl Probab 20:1005–1028MathSciNetCrossRefzbMATHGoogle Scholar
  11. Jenkins PA, Song YS (2011) The effect of recurrent mutation on the frequency spectrum of a segregating site and the age of an allele. Theor Popul Biol 80:158–173CrossRefzbMATHGoogle Scholar
  12. Kimura M (1969) The number of heterozygous nucleotide sites maintained in a finite population due to steady flux of mutations. Genetics 61:893–903Google Scholar
  13. Kingman JFC (1982) The coalescent. Stoch Process Appl 13:235–248MathSciNetCrossRefzbMATHGoogle Scholar
  14. RoyChoudhury A, Wakeley J (2010) Sufficiency of the number of segregating sites in the limit under finite-sites mutation. Theor Popul Biol 78:118–122CrossRefzbMATHGoogle Scholar
  15. Schrempf D, Hobolth A (2017) An alternative derivation of the stationary distribution of the multivariate neutral Wright–Fisher model for low mutation rates with a view to mutation rate estimation from site frequency data. Theor Popul Biol 114:88–94CrossRefzbMATHGoogle Scholar
  16. Vogl C, Clemente F (2012) The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates. Theor Popul Biol 81:197–209CrossRefzbMATHGoogle Scholar
  17. Vogl C (2014) Estimating the scaled mutation rate and mutation bias with site frequency data. Theor Popul Biol 98:19–27CrossRefzbMATHGoogle Scholar
  18. Vogl C, Bergman J (2015) Inference of directional selection and mutation parameters assuming equilibrium. Theor Popul Biol 106:71–82CrossRefzbMATHGoogle Scholar
  19. Zeng K (2010) A simple multiallele model and its application to identifying preferred-unpreferred codons using polymorphism data. Mol Biol Evol 27:1327–1337CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical Sciences InstituteAustralian National UniversityCanberraAustralia
  2. 2.Research School of BiologyAustralian National UniversityCanberraAustralia
  3. 3.Department of StatisticsUniversity of OxfordOxfordUK

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