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Multitype infinite-allele branching processes in continuous time

Part of: Markov processes

Published online by Cambridge University Press:  22 June 2017

Thomas O. McDonald  and
Marek Kimmel
Thomas O. McDonald*
Affiliation:
Rice University
Marek Kimmel*
Affiliation:
Rice University and Silesian University of Technology
*
*Current address: Department of Biostatistics and Computational Biology, Mailstop CLS-11054, Dana-Farber Cancer Institute, 450 Brookline Avenue, Boston, MA 02115, USA. Email address: mcdonald@jimmy.harvard.edu
** Postal address: Department of Statistics, Rice University, MS-138, PO Box 1892, Houston, TX 77251-1892, USA. Email address: kimmel@rice.edu
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Abstract

We introduce extensions to an infinite-allele branching process that allows for multiple types to exist alongside labels. We consider a Markov branching process and general branching process under different assumptions, and show asymptotic results about the growth of the labels as well as the frequency spectrum. These results are motivated by two separate models. The Markov binary splitting results are motivated by a model of clonal evolution in cancer that considers the effect of both driver and passenger mutations on tumor growth. The general process has applications in viral reproduction and dynamics.

Keywords

Multitype branching processinfinite-allele model

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 92D25: Population dynamics (general)
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 550 - 568
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York. Google Scholar
[2] Baumgärtel, V. et al. (2011). Live-cell visualization of dynamics of HIV budding site interactions with an ESCRT component. Nature Cell Biol. 13, 469–474. Google Scholar
[3] Ewens, W. J. (2004). Mathematical Population Genetics. I. Theoretical Introduction, 2nd edn. Springer, New York. Google Scholar
[4] Freed, E. O. (2015). HIV-1 assembly, release and maturation. Nature Rev. Microbiol. 13, 484–496. Google Scholar
[5] Griffiths, R. C. and Pakes, A. G. (1988). An infinite-alleles version of the simple branching process. Adv. Appl. Prob. 20, 489–524. Google Scholar
[6] Iwasa, Y., Nowak, M. A. and Michor, F. (2006). Evolution of resistance during clonal expansion. Genetics 172, 2557–2566. Google Scholar
[7] Jagers, P. (1989). General branching processes as Markov fields. Stoch. Process. Appl. 32, 183–212. Google Scholar
[8] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259. Google Scholar
[9] Jagers, P. and Nerman, O. (1996). The asymptotic composition of supercritical, multi-type branching populations. In Séminaire de Probabilités XXX (Lecture Notes Math. 1626), Springer, Berlin, pp. 40–54. Google Scholar
[10] Kouyos, R. D. et al. (2012). Exploring the complexity of the HIV-1 fitness landscape. PLoS Genet. 8, 9pp. CrossRefGoogle ScholarPubMed
[11] McDonald, T. O. and Kimmel, M. http://www.stat.rice.edu/~mathbio/infallele_nomen.pdf. Google Scholar
[12] McDonald, T. O. and Kimmel, M. (2015). A multitype infinite-allele branching process with applications to cancer evolution. J. Appl. Prob. 52, 864–876. Google Scholar
[13] Mode, C. J. (1971). Multitype Branching Processes. Theory and Applications. American Elsevier, New York. Google Scholar
[14] Mode, C. J. (1971). Multitype age-dependent branching processes and cell cycle analysis. Math. Biosci. 10, 177–190. Google Scholar
[15] Pakes, A. G. (1989). An infinite alleles version of the Markov branching process. J. Austral. Math. Soc. Ser. A 46, 146–169. Google Scholar
[16] Sewastjanow, B. A. (1974). Verzweigungsprozesse [Branching Processes]. Akademie, Berlin. Google Scholar
[17] Stewart, S.A. et al. (2000). Human immunodeficiency virus type 1 vpr induces apoptosis through caspase activation. J. Virology 74, 3105–3111. Google Scholar
[18] Taïb, Z. (1992). Branching Processes and Neutral Evolution (Lecture Notes Biomath. 93). Springer, Berlin. Google Scholar