The Annals of Applied Probability

A new coalescent for seed-bank models

Jochen Blath, Adrián González Casanova, Noemi Kurt, and Maite Wilke-Berenguer

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Abstract

We identify a new natural coalescent structure, which we call the seed-bank coalescent, that describes the gene genealogy of populations under the influence of a strong seed-bank effect, where “dormant forms” of individuals (such as seeds or spores) may jump a significant number of generations before joining the “active” population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright–Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size $N$. This extends earlier results of Kaj, Krone and Lascoux [J. Appl. Probab. 38 (2011) 285–300] who show that the genealogy of a Wright–Fisher model in the presence of a “weak” seed-bank effect is given by a suitably time-changed Kingman coalescent. The qualitatively new feature of the seed-bank coalescent is that ancestral lineages are independently blocked at a certain rate from taking part in coalescence events, thus strongly altering the predictions of classical coalescent models. In particular, the seed-bank coalescent “does not come down from infinity,” and the time to the most recent common ancestor of a sample of size $n$ grows like $\log\log n$. This is in line with the empirical observation that seed-banks drastically increase genetic variability in a population and indicates how they may serve as a buffer against other evolutionary forces such as genetic drift and selection.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 2 (2016), 857-891.

Dates
Received: November 2014
Revised: February 2015
First available in Project Euclid: 22 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651822

Digital Object Identifier
doi:10.1214/15-AAP1106

Mathematical Reviews number (MathSciNet)
MR3476627

Zentralblatt MATH identifier
1339.60137

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D10: Genetics {For genetic algebras, see 17D92}

Keywords
Wright–Fisher model seed-bank coalescent coming down from infinity age structure

Citation

Blath, Jochen; González Casanova, Adrián; Kurt, Noemi; Wilke-Berenguer, Maite. A new coalescent for seed-bank models. Ann. Appl. Probab. 26 (2016), no. 2, 857--891. doi:10.1214/15-AAP1106. https://projecteuclid.org/euclid.aoap/1458651822


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References

  • [1] Berestycki, N. (2009). Recent Progress in Coalescent Theory. Ensaios Matemáticos 16. Sociedade Brasileira de Matemática, Rio de Janeiro.
  • [2] Blath, J., González Casanova, A., Eldon, B. and Kurt, N. (2014). Genealogy of a Wright Fisher model with strong seed-bank component. Preprint.
  • [3] Blath, J., González Casanova, A., Kurt, N. and Spanò, D. (2013). The ancestral process of long-range seed bank models. J. Appl. Probab. 50 741–759.
  • [4] Dong, R., Gnedin, A. and Pitman, J. (2007). Exchangeable partitions derived from Markovian coalescents. Ann. Appl. Probab. 17 1172–1201.
  • [5] Etheridge, A. (2011). Some Mathematical Models from Population Genetics. Lecture Notes in Math. 2012. Springer, Heidelberg.
  • [6] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence, 2nd ed. Wiley, New York.
  • [7] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed. Wiley, New York.
  • [8] Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Oxford Univ. Press, London.
  • [9] Goldschmidt, C. and Martin, J. B. (2005). Random recursive trees and the Bolthausen–Sznitman coalescent. Electron. J. Probab. 10 718–745 (electronic).
  • [10] González Casanova, A., Aguirre-von Wobeser, E., Espín, G., Servín-González, L., Kurt, N., Spanò, D., Blath, J. and Soberón-Chávez, G. (2014). Strong seed-bank effects in bacterial evolution. J. Theoret. Biol. 356 62–70.
  • [11] Herbots, H. M. (1994). Stochastic models in population genetics: Genealogical and genetic differentiation in structured populations. Ph.D. dissertation, Univ. London.
  • [12] Herbots, H. M. (1997). The structured coalescent. In Progress in Population Genetics and Human Evolution (Minneapolis, MN, 1994). IMA Vol. Math. Appl. 87 231–255. Springer, New York.
  • [13] Jansen, S. and Kurt, N. (2014). On the notion(s) of duality for Markov processes. Probab. Surv. 11 59–120.
  • [14] Jenkins, P. A., Fearnhead, P. and Song, Y. S. (2014). Tractable stochastic models of evolution for loosely linked loci. Available at arXiv:1405.6863.
  • [15] Kaj, I., Krone, S. M. and Lascoux, M. (2001). Coalescent theory for seed bank models. J. Appl. Probab. 38 285–300.
  • [16] Kingman, J. F. C. (1982). The coalescent. Stochastic Process. Appl. 13 235–248.
  • [17] Lennon, J. T. and Jones, S. E. (2011). Microbial seed banks: The ecological and evolutionary implications of dormancy. Nature Reviews Microbiology 9 119–130.
  • [18] Levin, D. A. (1990). The seed bank as a source of genetic novelty in plants. Amer. Nat. 135 563–572.
  • [19] Neuhauser, C. and Krone, S. M. (1997). The genealogy of samples in models with selection. Genetics 145 519–534.
  • [20] Notohara, M. (1990). The coalescent and the genealogical process in geographically structured population. J. Math. Biol. 29 59–75.
  • [21] Nunney, L. (2002). The effective size of annual plant populations: The interaction of a seed bank with fluctuating population size in maintaining genetic variation. Amer. Nat. 160 195–204.
  • [22] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.
  • [23] Schweinsberg, J. (2000). A necessary and sufficient condition for the $\Lambda$-coalescent to come down from infinity. Electron. Commun. Probab. 5 1–11 (electronic).
  • [24] Takahata, N. (1988). The coalescent in two partially isolated diffusion populations. Genet. Res. 53 213–222.
  • [25] Tellier, A., Laurent, S. J. Y., Lainer, H., Pavlidis, P. and Stephan, W. (2011). Inference of seed bank parameters in two wild tomato species using ecological and genetic data. Proc. Natl. Acad. Sci. USA 108 17052–17057.
  • [26] Templeton, A. R. and Levin, D. A. (1979). Evolutionary consequences of seed pools. Amer. Nat. 114 232–249.
  • [27] Vitalis, R., Glémin, S. and Oliviere, I. (2004). When genes got to sleep: The population genetic consequences of seed dormancy and monocarpic perenniality. Amer. Nat. 163 295–311.
  • [28] Wakeley, J. (2009). Coalescent Theory. Roberts and Co, Greenwood Village, Colorado.
  • [29] Wright, S. (1931). Evolution in Mendelian populations. Genetics 16 97–159.
  • [30] Živković, D. and Tellier, A. (2012). Germ banks affect the inference of past demographic events. Mol. Ecol. 21 5434–5446.