IDEAS home Printed from https://ideas.repec.org/a/eee/thpobi/v127y2019icp16-32.html
   My bibliography  Save this article

Phase-type distributions in population genetics

Author

Listed:
  • Hobolth, Asger
  • Siri-Jégousse, Arno
  • Bladt, Mogens

Abstract

Probability modelling for DNA sequence evolution is well established and provides a rich framework for understanding genetic variation between samples of individuals from one or more populations. We show that both classical and more recent models for coalescence (with or without recombination) can be described in terms of the so-called phase-type theory, where complicated and tedious calculations are circumvented by the use of matrix manipulations. The application of phase-type theory in population genetics consists of describing the biological system as a Markov model by appropriately setting up a state space and calculating the corresponding intensity and reward matrices. Formulae of interest are then expressed in terms of these aforementioned matrices. We illustrate this procedure by a number of examples: (a)Â Calculating the mean, (co)variance and even higher order moments of the site frequency spectrum in multiple merger coalescent models, (b)Â Analysing a sample of DNA sequences from the Atlantic Cod using the Beta-coalescent, and (c)Â Determining the correlation of the number of segregating sites for multiple samples in the two-locus ancestral recombination graph. We believe that phase-type theory has great potential as a tool for analysing probability models in population genetics. The compact matrix notation is useful for clarification of current models, and in particular their formal manipulation and calculations, but also for further development or extensions.

Suggested Citation

  • Hobolth, Asger & Siri-Jégousse, Arno & Bladt, Mogens, 2019. "Phase-type distributions in population genetics," Theoretical Population Biology, Elsevier, vol. 127(C), pages 16-32.
    • Handle: RePEc:eee:thpobi:v:127:y:2019:i:c:p:16-32
    DOI: 10.1016/j.tpb.2019.02.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0040580919300140
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.tpb.2019.02.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Hobolth, Asger & Jensen, Jens Ledet, 2014. "Markovian approximation to the finite loci coalescent with recombination along multiple sequences," Theoretical Population Biology, Elsevier, vol. 98(C), pages 48-58.
    2. Schweinsberg, Jason, 2003. "Coalescent processes obtained from supercritical Galton-Watson processes," Stochastic Processes and their Applications, Elsevier, vol. 106(1), pages 107-139, July.
    3. Klassmann, A. & Ferretti, L., 2018. "The third moments of the site frequency spectrum," Theoretical Population Biology, Elsevier, vol. 120(C), pages 16-28.
    4. Miroshnikov, Alexey & Steinrücken, Matthias, 2017. "Computing the joint distribution of the total tree length across loci in populations with variable size," Theoretical Population Biology, Elsevier, vol. 118(C), pages 1-19.
    5. Kumagai, Seiji & Uyenoyama, Marcy K., 2015. "Genealogical histories in structured populations," Theoretical Population Biology, Elsevier, vol. 102(C), pages 3-15.
    6. Sargsyan, Ori & Wakeley, John, 2008. "A coalescent process with simultaneous multiple mergers for approximating the gene genealogies of many marine organisms," Theoretical Population Biology, Elsevier, vol. 74(1), pages 104-114.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Miró Pina, Verónica & Joly, Émilien & Siri-Jégousse, Arno, 2023. "Estimating the Lambda measure in multiple-merger coalescents," Theoretical Population Biology, Elsevier, vol. 154(C), pages 94-101.
    2. Riccardo De Bin & Vegard Grødem Stikbakke, 2023. "A boosting first-hitting-time model for survival analysis in high-dimensional settings," Lifetime Data Analysis: An International Journal Devoted to Statistical Methods and Applications for Time-to-Event Data, Springer, vol. 29(2), pages 420-440, April.
    3. 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.
    4. Freund, Fabian & Siri-Jégousse, Arno, 2021. "The impact of genetic diversity statistics on model selection between coalescents," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).
    5. Blath, Jochen & Buzzoni, Eugenio & Koskela, Jere & Wilke Berenguer, Maite, 2020. "Statistical tools for seed bank detection," Theoretical Population Biology, Elsevier, vol. 132(C), pages 1-15.
    6. Hobolth, Asger & Rivas-González, Iker & Bladt, Mogens & Futschik, Andreas, 2024. "Phase-type distributions in mathematical population genetics: An emerging framework," Theoretical Population Biology, Elsevier, vol. 157(C), pages 14-32.
    7. Bo Friis Nielsen, 2022. "Characterisation of multivariate phase type distributions," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 229-231, April.
    8. Legried, Brandon & Terhorst, Jonathan, 2022. "Rates of convergence in the two-island and isolation-with-migration models," Theoretical Population Biology, Elsevier, vol. 147(C), pages 16-27.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Eldon, Bjarki, 2011. "Estimation of parameters in large offspring number models and ratios of coalescence times," Theoretical Population Biology, Elsevier, vol. 80(1), pages 16-28.
    2. Eldon, Bjarki & Stephan, Wolfgang, 2018. "Evolution of highly fecund haploid populations," Theoretical Population Biology, Elsevier, vol. 119(C), pages 48-56.
    3. Eldon, Bjarki & Degnan, James H., 2012. "Multiple merger gene genealogies in two species: Monophyly, paraphyly, and polyphyly for two examples of Lambda coalescents," Theoretical Population Biology, Elsevier, vol. 82(2), pages 117-130.
    4. Blath, Jochen & Cronjäger, Mathias Christensen & Eldon, Bjarki & Hammer, Matthias, 2016. "The site-frequency spectrum associated with Ξ-coalescents," Theoretical Population Biology, Elsevier, vol. 110(C), pages 36-50.
    5. 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.
    6. Bjarki Eldon, 2023. "Viability Selection at Linked Sites," Mathematics, MDPI, vol. 11(3), pages 1-23, January.
    7. Eldon, Bjarki, 2009. "Structured coalescent processes from a modified Moran model with large offspring numbers," Theoretical Population Biology, Elsevier, vol. 76(2), pages 92-104.
    8. Dhersin, Jean-Stéphane & Freund, Fabian & Siri-Jégousse, Arno & Yuan, Linglong, 2013. "On the length of an external branch in the Beta-coalescent," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1691-1715.
    9. Costa, Rui J. & Wilkinson-Herbots, Hilde M., 2021. "Inference of gene flow in the process of speciation: Efficient maximum-likelihood implementation of a generalised isolation-with-migration model," Theoretical Population Biology, Elsevier, vol. 140(C), pages 1-15.
    10. Hobolth, Asger & Rivas-González, Iker & Bladt, Mogens & Futschik, Andreas, 2024. "Phase-type distributions in mathematical population genetics: An emerging framework," Theoretical Population Biology, Elsevier, vol. 157(C), pages 14-32.
    11. Huillet, Thierry & Möhle, Martin, 2013. "On the extended Moran model and its relation to coalescents with multiple collisions," Theoretical Population Biology, Elsevier, vol. 87(C), pages 5-14.
    12. Birkner, Matthias & Blath, Jochen & Steinrücken, Matthias, 2011. "Importance sampling for Lambda-coalescents in the infinitely many sites model," Theoretical Population Biology, Elsevier, vol. 79(4), pages 155-173.
    13. Arbisser, Ilana M. & Jewett, Ethan M. & Rosenberg, Noah A., 2018. "On the joint distribution of tree height and tree length under the coalescent," Theoretical Population Biology, Elsevier, vol. 122(C), pages 46-56.
    14. Steinrücken, Matthias & Birkner, Matthias & Blath, Jochen, 2013. "Analysis of DNA sequence variation within marine species using Beta-coalescents," Theoretical Population Biology, Elsevier, vol. 87(C), pages 15-24.
    15. Carmi, Shai & Wilton, Peter R. & Wakeley, John & Pe’er, Itsik, 2014. "A renewal theory approach to IBD sharing," Theoretical Population Biology, Elsevier, vol. 97(C), pages 35-48.
    16. Ferretti, Luca & Klassmann, Alexander & Raineri, Emanuele & Ramos-Onsins, Sebastián E. & Wiehe, Thomas & Achaz, Guillaume, 2018. "The neutral frequency spectrum of linked sites," Theoretical Population Biology, Elsevier, vol. 123(C), pages 70-79.
    17. 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.
    18. Hadzibeganovic, Tarik & Liu, Chao & Li, Rong, 2021. "Effects of reproductive skew on the evolution of ethnocentrism in structured populations with variable size," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 568(C).
    19. Durrett, Rick & Schweinsberg, Jason, 2005. "A coalescent model for the effect of advantageous mutations on the genealogy of a population," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1628-1657, October.
    20. González Casanova, Adrián & Kurt, Noemi & Wakolbinger, Anton & Yuan, Linglong, 2016. "An individual-based model for the Lenski experiment, and the deceleration of the relative fitness," Stochastic Processes and their Applications, Elsevier, vol. 126(8), pages 2211-2252.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:thpobi:v:127:y:2019:i:c:p:16-32. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/intelligence .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.