Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T23:46:00.878Z Has data issue: false hasContentIssue false
  • English
  • Français

The iterated Galton–Watson process

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Boleslaw Gawel  and
Marek Kimmel
Boleslaw Gawel*
Affiliation:
Silesian University
Marek Kimmel*
Affiliation:
Rice University
*
*Postal address: Department of Mathematics, Silesian University, Katowice, Poland.
**Postal address: Department of Statistics, Rice University, PO Box 1892, Houston, TX 77251, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the mathematical properties of a time-discrete stochastic process describing explosive proliferation of DNA repeats in human genetic diseases. The process is constructed using a cascade of Galton–Watson branching processes. The main results concern the probability of absorption and the supergeometric growth of the process in the supercritical case. Examples of simulations are provided.

Keywords

GALTON–WATSON PROCESSDNA

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 4 , December 1996 , pp. 949 - 959
Copyright
Copyright © Applied Probability Trust 1996 

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

[1] Athreya, K. B. and Ney, P. E. (1972) Branching Processes. Springer, New York.CrossRefGoogle Scholar
[2] Axelrod, D. E. and Kimmel, M. (1996) Stochastic models of expanding triplet repeats in hereditary diseases. In preparation.Google Scholar
[3] Caskey, C. T., Pizutti, A., Fu, Y.-H., Fenwick, R. G. Jr. and Nelson, D. L. (1992) Triplet repeat mutations in human disease. Science 256, 784–789.CrossRefGoogle ScholarPubMed
[4] Cavalier-Smith, T. (1985) Introduction: the evolutionary significance of genome size. In The Evolution of Genome Size. ed. Cavalier-Smith, T. Wiley, New York. pp 1–36.Google Scholar
[5] Chung, K. L. (1974) A Course in Probability Theory. 2nd edn. Academic Press, New York.Google Scholar
[6] Fu, Y.-H., Kuhl, D. P. A., Pizzutti, A., Pieretti, M., Sutcliffe, J. S., Richards, S., Verkeerk, A. J. M. H., Holden, J. J. A., Fenwick, R. G. Jr., Warren, S. T., Oostra, B. A., Nelson, D. L. and Caskey, C. T. (1992) Variation of the CGG repeat at the Fragile X site results in genetic instability: Resolution of the Sherman paradox. Cell 67, 1047–1058.CrossRefGoogle Scholar
[7] Imbert, D., Kretz, C., Johnson, K. and Mandel, J.-L. (1993) Origin of the expansion mutation in myotonic dystrophy. Nature Genet. 4, 72–76.CrossRefGoogle ScholarPubMed
[8] Kimmel, M., Stivers, D. N. (1994) Time-continuous branching walk models of unstable gene amplification. Bull. Math. Biol. 56, 337–358.CrossRefGoogle ScholarPubMed
[9] Kuhl, D. P. A. and Caskey, C. T. (1993) Trinucleotide repeats and genome variation. Current Opinion Genet. Develop. 3, 404–407.CrossRefGoogle ScholarPubMed
[10] Marx, J. (1993) New colon cancer gene discovered. Science 260, 751–752.CrossRefGoogle ScholarPubMed
[11] Mosig, G. (1987) The essential role of recombination in phage T4 growth. Ann. Rev. Genet. 21, 347–371.CrossRefGoogle ScholarPubMed
[12] Redman, J. B., Fenwick, R. G. Jr., Fu, Y.-H., Pizzutti, A. and Caskey, C. T. (1993) Relationship between parental trinucleotide GCT repeat length and severity of myotonic dystrophy in offspring. JAMA 269, 1960–1965.CrossRefGoogle ScholarPubMed
[13] Richards, R. I. and Sutherland, G. R. (1992) Dynamic mutations: A new class of mutations causing human disease. Cell 70, 709–712.CrossRefGoogle ScholarPubMed