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A coalescent dual process in a Moran model with genic selection

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  • Etheridge, A.M.
  • Griffiths, R.C.

Abstract

A coalescent dual process for a multi-type Moran model with genic selection is derived using a generator approach. This leads to an expansion of the transition functions in the Moran model and the Wright–Fisher diffusion process limit in terms of the transition functions for the coalescent dual. A graphical representation of the Moran model (in the spirit of Harris) identifies the dual as a strong dual process following typed lines backwards in time. An application is made to the harmonic measure problem of finding the joint probability distribution of the time to the first loss of an allele from the population and the distribution of the surviving alleles at the time of loss. Our dual process mirrors the Ancestral Selection Graph of [Krone, S. M., Neuhauser, C., 1997. Ancestral processes with selection. Theoret. Popul. Biol. 51, 210–237; Neuhauser, C., Krone, S. M., 1997. The genealogy of samples in models with selection. Genetics 145, 519–534], which allows one to reconstruct the genealogy of a random sample from a population subject to genic selection. In our setting, we follow [Stephens, M., Donnelly, P., 2002. Ancestral inference in population genetics models with selection. Aust. N. Z. J. Stat. 45, 395–430] in assuming that the types of individuals in the sample are known. There are also close links to [Fearnhead, P., 2002. The common ancestor at a nonneutral locus. J. Appl. Probab. 39, 38–54]. However, our methods and applications are quite different. This work can also be thought of as extending a dual process construction in a Wright–Fisher diffusion in [Barbour, A.D., Ethier, S.N., Griffiths, R.C., 2000. A transition function expansion for a diffusion model with selection. Ann. Appl. Probab. 10, 123–162]. The application to the harmonic measure problem extends a construction provided in the setting of a neutral diffusion process model in [Ethier, S.N., Griffiths, R.C., 1991. Harmonic measure for random genetic drift. In: Pinsky, M.A. (Ed.), Diffusion Processes and Related Problems in Analysis, vol. 1. In: Progress in Probability Series, vol. 22, Birkhäuser, Boston, pp. 73–81].

Suggested Citation

  • Etheridge, A.M. & Griffiths, R.C., 2009. "A coalescent dual process in a Moran model with genic selection," Theoretical Population Biology, Elsevier, vol. 75(4), pages 320-330.
    • Handle: RePEc:eee:thpobi:v:75:y:2009:i:4:p:320-330
    DOI: 10.1016/j.tpb.2009.03.004
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    References listed on IDEAS

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    1. Mano, Shuhei, 2009. "Duality, ancestral and diffusion processes in models with selection," Theoretical Population Biology, Elsevier, vol. 75(2), pages 164-175.
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    2. Carinci, Gioia & Giardinà, Cristian & Giberti, Claudio & Redig, Frank, 2015. "Dualities in population genetics: A fresh look with new dualities," Stochastic Processes and their Applications, Elsevier, vol. 125(3), pages 941-969.
    3. Griffiths, Robert C. & Jenkins, Paul A. & Lessard, Sabin, 2016. "A coalescent dual process for a Wright–Fisher diffusion with recombination and its application to haplotype partitioning," Theoretical Population Biology, Elsevier, vol. 112(C), pages 126-138.
    4. Frank Hollander & Shubhamoy Nandan, 2022. "Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering," Journal of Theoretical Probability, Springer, vol. 35(3), pages 1795-1841, September.
    5. Griffiths, Robert C. & Jenkins, Paul A. & Spanò, Dario, 2018. "Wright–Fisher diffusion bridges," Theoretical Population Biology, Elsevier, vol. 122(C), pages 67-77.
    6. Desai, Michael M. & Nicolaisen, Lauren E. & Walczak, Aleksandra M. & Plotkin, Joshua B., 2012. "The structure of allelic diversity in the presence of purifying selection," Theoretical Population Biology, Elsevier, vol. 81(2), pages 144-157.
    7. Etheridge, Alison M. & Griffiths, Robert C. & Taylor, Jesse E., 2010. "A coalescent dual process in a Moran model with genic selection, and the lambda coalescent limit," Theoretical Population Biology, Elsevier, vol. 78(2), pages 77-92.
    8. Corujo, Josué, 2021. "Dynamics of a Fleming–Viot type particle system on the cycle graph," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 57-91.
    9. Der, Ricky & Epstein, Charles L. & Plotkin, Joshua B., 2011. "Generalized population models and the nature of genetic drift," Theoretical Population Biology, Elsevier, vol. 80(2), pages 80-99.
    10. Lenz, Ute & Kluth, Sandra & Baake, Ellen & Wakolbinger, Anton, 2015. "Looking down in the ancestral selection graph: A probabilistic approach to the common ancestor type distribution," Theoretical Population Biology, Elsevier, vol. 103(C), pages 27-37.
    11. Vogl, Claus & Bergman, Juraj, 2015. "Inference of directional selection and mutation parameters assuming equilibrium," Theoretical Population Biology, Elsevier, vol. 106(C), pages 71-82.
    12. Vogl, Claus & Clemente, Florian, 2012. "The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates," Theoretical Population Biology, Elsevier, vol. 81(3), pages 197-209.
    13. Vogl, Claus & Mikula, Lynette Caitlin, 2021. "A nearly-neutral biallelic Moran model with biased mutation and linear and quadratic selection," Theoretical Population Biology, Elsevier, vol. 139(C), pages 1-17.
    14. Burden, Conrad J. & Tang, Yurong, 2017. "Rate matrix estimation from site frequency data," Theoretical Population Biology, Elsevier, vol. 113(C), pages 23-33.
    15. Malaguti, Giulia & Singh, Param Priya & Isambert, Hervé, 2014. "On the retention of gene duplicates prone to dominant deleterious mutations," Theoretical Population Biology, Elsevier, vol. 93(C), pages 38-51.
    16. Vogl, Claus & Mikula, Lynette C. & Burden, Conrad J., 2020. "Maximum likelihood estimators for scaled mutation rates in an equilibrium mutation–drift model," Theoretical Population Biology, Elsevier, vol. 134(C), pages 106-118.
    17. Schrempf, Dominik & Hobolth, Asger, 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," Theoretical Population Biology, Elsevier, vol. 114(C), pages 88-94.

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