×

A diploid population model for copy number variation of genetic elements. (English) Zbl 1519.92148

Summary: We study the following model for a diploid population of constant size \(N\): Every individual carries a random number of (genetic) elements. Upon a reproduction event each of the two parents passes each element independently with probability \(\frac{1}{2}\) on to the offspring. We study the process \(X^N=(X^N(1),X^N(2),\ldots)\), where \({X_t^N}(k)\) is the frequency of individuals at time \(t\) that carry \(k\) elements, and prove convergence (in some weak sense) of \({X^N}\) jointly with its empirical first moment \({Z^N}\) to the “slow-fast” system \((Z,X)\), where \({X_t}=\text{Poi}({Z_t})\) and \(Z\) evolves according to a critical Feller branching process. We discuss heuristics explaining this finding and some extensions and limitations.

MSC:

92D15 Problems related to evolution
60J85 Applications of branching processes
60G57 Random measures

Software:

SageMath
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Baird, S. J., N. H. Barton, and A. M. Etheridge (2003). The distribution of surviving blocks of an ancestral genome. Theoretical population biology 64(4), 451-471. · Zbl 1108.92030
[2] Berglund, N. and B. Gentz (2006). Noise-induced phenomena in slow-fast dynamical systems: a sample-paths approach. Springer Science & Business Media. · Zbl 1098.37049
[3] Bourque, G., K. H. Burns, M. Gehring, V. Gorbunova, A. Seluanov, M. Hammell, M. Imbeault, Z. Izsvák, H. L. Levin, T. S. Macfarlan, D. L. Mager, and C. Feschotte (2018, 11). Ten things you should know about transposable elements. Genome Biol 19(1), 199.
[4] Chang, J. (1999). Recent common ancestors of all present-day individuals. J. Appl. Probab. 31, 1002-1026. · Zbl 0979.92027
[5] Coron, C. and Y. Le Jan (2020). Genetics of the biparental moran model. https://arxiv.org/abs/2007.15479.
[6] Etheridge, A. (2001). An introduction to superprocesses. American Mathematical Society.
[7] Ethier, S. N. and T. G. Kurtz (1986). Markov Processes. Characterization and Convergence. John Wiley, New York. · Zbl 0592.60049
[8] Foutel-Rodier, F. and E. Schertzer (2022). Convergence of genealogies through spinal decomposition, with an application to population genetics. https://https://arxiv.org/abs/2201.12412.
[9] Griffiths, R. C. and P. Marjoram (1997). An ancestral recombination graph. In Progress in population genetics and human evolution (Minneapolis, MN, 1994), Volume 87 of IMA Vol. Math. Appl., pp. 257-270. Springer, New York. · Zbl 0893.92020
[10] Katzenberger, G. S. (1991). Solutions of a stochastic differential equation forced onto a manifold by a large drift. Ann. Probab. 19, 1587-1628. · Zbl 0749.60053
[11] Kurtz, T. G. (1991). Random time changes and convergence in distribution under the Meyer-Zheng conditions. Ann. Probab. 19, 1010-1034. · Zbl 0742.60036
[12] Kurtz, T. G. (1992). Averaging for martingale problems and stochastic approximation. In Applied stochastic analysis (New Brunswick, NJ, 1991), Volume 177 of Lecture Notes in Control and Inform. Sci., pp. 186-209. Berlin: Springer.
[13] Lambert, A., V. M. Pina, and E. Schertzer (2021). Chromosome painting: how recombination mixes ancestral colors. Ann. Appl. Probab. 31(2), 826-864. · Zbl 1476.60009
[14] Meyer, P.-A. and W. A. Zheng (1984). Tightness criteria for laws of semimartingeles. Ann. Inst. H. Poincaré Probab. Statist. 20, 353-372. · Zbl 0551.60046
[15] Pardoux, E. and A. Y. Veretennikov (2001). On Poisson equation and diffusion approximation 1. Ann. Probab. 29, 1061-1085. · Zbl 1029.60053
[16] Rohde, D. L., S. Olson, and J. T. Chang (2004, Sep). Modelling the recent common ancestry of all living humans. Nature 431(7008), 562-566.
[17] Sawyer, S. A. and D. L. Hartl (1992, Dec). Population genetics of polymorphism and divergence. Genetics 132(4), 1161-1176.
[18] Sethupathy, P. and S. Hannenhalli (2008). A tutorial of the poisson random field model in population genetics. Adv Bioinformatics 4, 257864.
[19] The Sage Developers (2020). SageMath, the Sage Mathematics Software System (Version 9.0). https://www.sagemath.org.
[20] Wakeley, J., L. King, B. S. Low, and S. Ramachandran (2012). Gene genealogies within a fixed pedigree, and the robustness of Kingman’s coalescent. Genetics 190(4), 1433-1445.
[21] Wakeley, J., L. King, and P. R. Wilton (2016, 07). Effects of the population pedigree on genetic signatures of historical demographic events. Proc Natl Acad Sci 113(29), 7994-8001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.