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Quasi-stationary distributions of a pair of Markov chains related to time evolution of a DNA locus

Part of: Miscellaneous applications of operator theory

Published online by Cambridge University Press:  01 July 2016

A. Bobrowski
A. Bobrowski*
Affiliation:
Lublin University of Technology
*
∗ Postal address: Department of Mathematics, Faculty of Electrical Engineering, Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland. Email address: [email protected]
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Abstract

We consider a pair of Markov chains representing statistics of the Fisher-Wright-Moran model with mutations and drift. The chains have absorbing state at 0 and are related by the fact that some random time τ ago they were identical, evolving as a single Markov chain with values in {0,1,…}; from that time on they began to evolve independently, conditional on a state at the time of split, according to the same transition probabilities. The distribution of τ is a function of deterministic effective population size 2N(·). We study the impact of demographic history on the shape of the quasi-stationary distribution, conditional on nonabsorption at the margin (where one of the chains is at 0), and on the speed with which the probability mass escapes to the margin.

Keywords

Fisher-Wright-Moran modelquasi-stationary distributionMarkov chainmutationgenetic driftcoalescencemicrosatellite

MSC classification

Primary: 92D25: Population dynamics (general)
Secondary: 47N60: Applications in chemistry and life sciences
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 1 , March 2004 , pp. 57 - 77
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.CrossRefGoogle Scholar
[2] Beaumont, M. A. (1999). Detecting population expansion and decline using microsatellites. Genetics 153, 2013–2029.CrossRefGoogle ScholarPubMed
[3] Bobrowski, A. and Kimmel, M. (1999). Dynamics of the life history of a DNA-repeat sequence. Arch. Control Sci. 9, 57–67.Google Scholar
[4] Bobrowski, A., Kimmel, M., Chakraborty, R. and Arino, O. (2001). A semigroup representation and asymptotic behavior of certain statistics of the Fisher–Wright–Moran coalescent. In Handbook of Statistics, Vol. 19, Stochastic Processes: Theory and Methods, eds Rao, C. R. and Shanbhag, D. N., Elsevier Science, Amsterdam.Google Scholar
[5] Bobrowski, A., Wang, N., Kimmel, M. and Chakraborty, R. (2002). Nonhomogeneous infinite sites model under demographic change: mathematical description and asymptotic behavior of pairwise distributions. Math. Biosci. 175, 83–115.CrossRefGoogle ScholarPubMed
[6] Dunford, N. and Schwartz, J. T. (1958). Linear Operators Part I: General Theory. Wiley-Interscience, New York.Google Scholar
[7] Durrett, R. and Kruglyak, S. (1999). A new stochastic model of microsatellite evolution. J. Appl. Prob. 36, 621–631.Google Scholar
[8] Griffiths, R. C. and Tavaré, S. (1994). Sampling theory for neutral alleles in a varying environment. Phil. Trans. R. Soc. London B 344, 403–410.Google Scholar
[9] Kimmel, M. (1999). Population dynamics coded in DNA: genetic traces of the expansion of modern humans. Physica A 273, 158–168.CrossRefGoogle Scholar
[10] Kimmel, M. and Bobrowski, A. (2004). Long-time behavior of some statistics of the Fisher–Wright–Moran model with evolutionary implications. In preparation.Google Scholar
[11] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
[12] Kruglyak, S., Durrett, R. T., Schug, M. D. and Aquadro, C. F. (1998). Equilibrium distributions of microsatellite repeat length resulting from a balance between slippage events and point mutations. Proc. Nat. Acad. Sci. USA 95, 10774–10784.CrossRefGoogle ScholarPubMed
[13] Malécot, G., (1970). The Mathematics of Heredity. W. H. Freeman, San Francisco.Google Scholar
[14] Pakes, A. G. (1995). Quasi-stationary laws for Markov processes; examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.CrossRefGoogle Scholar
[15] Peng, N. F., Pearl, D. K., Chan, W. and Bartoszyński, R. (1993). Linear birth and death processes under the influence of disasters with time-dependent killing probabilities. Stoch. Process. Appl. 45, 243–258.CrossRefGoogle Scholar
[16] Rogers, A. R. and Harpending, H. C. (1992). Population growth makes waves in the distribution of pairwise genetic differences. Molec. Biol. Evol. 9, 552–569.Google Scholar
[17] Tajima, F. (1983). Evolutionary relationship of DNA sequences in finite populations. Genetics 105, 437–60.CrossRefGoogle ScholarPubMed
[18] Vere-Jones, D. (1969). Some limit theorems for evanescent processes. Austral. J. Statist. 11, 67–78.CrossRefGoogle Scholar