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An explicit transition density expansion for a multi-allelic Wright–Fisher diffusion with general diploid selection

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  • Steinrücken, Matthias
  • Wang, Y.X. Rachel
  • Song, Yun S.

Abstract

Characterizing time-evolution of allele frequencies in a population is a fundamental problem in population genetics. In the Wright–Fisher diffusion, such dynamics is captured by the transition density function, which satisfies well-known partial differential equations. For a multi-allelic model with general diploid selection, various theoretical results exist on representations of the transition density, but finding an explicit formula has remained a difficult problem. In this paper, a technique recently developed for a diallelic model is extended to find an explicit transition density for an arbitrary number of alleles, under a general diploid selection model with recurrent parent-independent mutation. Specifically, the method finds the eigenvalues and eigenfunctions of the generator associated with the multi-allelic diffusion, thus yielding an accurate spectral representation of the transition density. Furthermore, this approach allows for efficient, accurate computation of various other quantities of interest, including the normalizing constant of the stationary distribution and the rate of convergence to this distribution.

Suggested Citation

  • Steinrücken, Matthias & Wang, Y.X. Rachel & Song, Yun S., 2013. "An explicit transition density expansion for a multi-allelic Wright–Fisher diffusion with general diploid selection," Theoretical Population Biology, Elsevier, vol. 83(C), pages 1-14.
    • Handle: RePEc:eee:thpobi:v:83:y:2013:i:c:p:1-14
    DOI: 10.1016/j.tpb.2012.10.006
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    References listed on IDEAS

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    1. Živković, Daniel & Stephan, Wolfgang, 2011. "Analytical results on the neutral non-equilibrium allele frequency spectrum based on diffusion theory," Theoretical Population Biology, Elsevier, vol. 79(4), pages 184-191.
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    3. Lukić, Sergio & Hey, Jody & Chen, Kevin, 2011. "Non-equilibrium allele frequency spectra via spectral methods," Theoretical Population Biology, Elsevier, vol. 79(4), pages 203-219.
    4. Buzbas, Erkan Ozge & Joyce, Paul & Rosenberg, Noah A., 2011. "Inference on the strength of balancing selection for epistatically interacting loci," Theoretical Population Biology, Elsevier, vol. 79(3), pages 102-113.
    5. Buzbas Erkan Ozge & Joyce Paul & Abdo Zaid, 2009. "Estimation of Selection Intensity under Overdominance by Bayesian Methods," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 8(1), pages 1-24, June.
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    Cited by:

    1. Mikula, Lynette Caitlin & Vogl, Claus, 2024. "The expected sample allele frequencies from populations of changing size via orthogonal polynomials," Theoretical Population Biology, Elsevier, vol. 157(C), pages 55-85.

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