Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T14:22:31.916Z Has data issue: false hasContentIssue false
  • English
  • Français

The shape of a seed bank tree

Part of: Combinatorial probability Markov processes

Published online by Cambridge University Press:  02 June 2022

Adrián González Casanova ,
Lizbeth Peñaloza  and
Arno Siri-Jégousse
Adrián González Casanova*
Affiliation:
Universidad Nacional Autónoma de México
Lizbeth Peñaloza*
Affiliation:
Universidad del Mar
Arno Siri-Jégousse*
Affiliation:
Universidad Nacional Autónoma de México
*
*Postal address: Av. Universidad 3000, Col. UNAM Ciudad Universitaria, Delegación Coyoacán C.P. 04510 Ciudad de México.
***Postal address: Ciudad Universitaria, Santa María Huatulco, Oaxaca, México C.P. 70989.
*Postal address: Av. Universidad 3000, Col. UNAM Ciudad Universitaria, Delegación Coyoacán C.P. 04510 Ciudad de México.
Get access Rights & Permissions [Opens in a new window]

Abstract

We derive the asymptotic behavior of the total, active, and inactive branch lengths of the seed bank coalescent when the initial sample size grows to infinity. These random variables have important applications for populations evolving under some seed bank effects, such as plants and bacteria, and for some cases of structured populations like metapopulations. The proof relies on the analysis of the tree at a stopping time corresponding to the first time a deactivated lineage is reactivated. We also give conditional sampling formulas for the random partition, and we study the system at the time of the first reactivation of a lineage. All these results provide a good picture of the different regimes and behaviors of the block-counting process of the seed bank coalescent.

Keywords

Seed bankstructured coalescentbranch lengthssampling formula

MSC classification

Secondary: 60C05: Combinatorial probability 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 3 , September 2022 , pp. 631 - 651
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blath, J., González-Casanova, A., Kurt, N. and Wilke-Berenguer, M. (2016). A new coalescent for seed bank models. Ann. Appl. Prob. 26, 857–891.10.1214/15-AAP1106CrossRefGoogle Scholar
Delmas, J. F., Dhersin, J. S. and Siri-Jégousse, A. (2008). Asymptotic results on the length of coalescent trees. Ann. Appl. Prob. 18, 997–1025.10.1214/07-AAP476CrossRefGoogle Scholar
Dhersin, J. S., Freund, F., Siri-Jégousse, A. and Yuan, L. (2013). On the length of an external branch in the Beta-coalescent. Stoch. Proc. Appl. 123, 1691–1715.10.1016/j.spa.2012.12.010CrossRefGoogle Scholar
Diehl, C. S. and Kersting, G. (2019). Tree lengths for general $\Lambda$ -coalescents and the asymptotic site frequency spectrum around the Bolthausen–Sznitman coalescent. Ann. Appl. Prob. 29, 2700–2743.Google Scholar
Drmota, M., Iksanov, A., Möhle, M. and Rösler, U. (2007). Asymptotic results concerning the total branch length of the Bolthausen–Sznitman coalescent. Stoch. Proc. Appl. 117, 1404–1421.10.1016/j.spa.2007.01.011CrossRefGoogle Scholar
Donelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19.10.2307/1427237CrossRefGoogle Scholar
Durrett, R. (2008). Probability Models for DNA Sequence Evolution. Springer, New York.10.1007/978-0-387-78168-6CrossRefGoogle Scholar
Freund, F. and Siri-Jégousse, A. (2021). The minimal observable clade size of exchangeable coalescents. Braz. J. Prob. Statist. 35, 281–292.10.1214/20-BJPS480CrossRefGoogle Scholar
Freund, F. and Siri-Jégousse, A. (2021). The impact of genetic diversity statistics on model selection between coalescents. Comput. Statist. Data Anal. 156, 107055.10.1016/j.csda.2020.107055CrossRefGoogle Scholar
González-Casanova, A. et al. (2014). Strong seed bank effects in bacterial evolution. J. Theor. Biol. 356, 62–70.10.1016/j.jtbi.2014.04.009CrossRefGoogle ScholarPubMed
Hobolth, A., Siri-Jégousse, A. and Bladt, M. (2019). Phase-type distributions in population genetics. Theor. Pop. Biol. 127, 16–32.10.1016/j.tpb.2019.02.001CrossRefGoogle ScholarPubMed
Kaj, I., Krone, S. and Lascoux, M. (2001). Coalescent theory for seed bank models. J. Appl. Prob. 38, 285–300.10.1239/jap/996986745CrossRefGoogle Scholar
Kersting, G. (2012). The asymptotic distribution of the length of beta-coalescent trees. Ann. Appl. Prob. 22, 2086–2107.10.1214/11-AAP827CrossRefGoogle Scholar
Lambert, A. and Ma, C. (2015). The coalescent in peripatric metapopulations. J. Appl. Prob. 52, 538–557.10.1239/jap/1437658614CrossRefGoogle Scholar
Lennon, J. T. and Jones, S. E. (2011). Microbial seed banks: The ecological and evolutionary implications of dormancy. Nat. Rev. Microbiol. 9, 119.10.1038/nrmicro2504CrossRefGoogle ScholarPubMed
Maughan, H. (2007). Rates of molecular evolution in bacteria are relatively constant despite spore dormancy. Evolution 61, 280–288.10.1111/j.1558-5646.2007.00026.xCrossRefGoogle ScholarPubMed
Tellier, A. et al. (2011). Inference of seed bank parameters in two wild tomato species using ecological and genetic data. Proc. Nat. Acad. Sci. 108, 17052–17057.10.1073/pnas.1111266108CrossRefGoogle Scholar
Watterson, G. A. (1984). Lines of descent and the coalescent. Theor. Pop. Biol. 26, 77–92.10.1016/0040-5809(84)90025-XCrossRefGoogle Scholar