Abstract
We have found, under assumption of existence of the limit distribution of allelic states at pairs of chromosomes, explicit forms of the limit. Depending on the limit behavior of population size, we obtained different limit joint probability distributions of pairs. An interesting example of application of expression for the constant population size limit of the joint distribution is the model of microsatellite mutation with lower and upper bounds on the microsatellite size. From the viewpoint of microsatellite models, this is an unusual situation, since most of them, with a notable exception of Durretts’s model do not have a limit distribution of repeat count.
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References
Hille E, Phillips RS (1957) Functional analysis and semi-groups. Am Math Soc Colloq Publ 31:26, 53, 61, 65 and 76 (Am Math Soc Providence RI)
Engel K-J, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, New York
Beaumont MA (1998) Detecting population expansion and decline using microsatellites. Genetics 153:2013–2029
Bobrowski A, Kimmel M, Chakraborty R, Arino O (2001) A semigroup representation and asymptotic behavior of the Fisher-Wright-Moran coalescent. In: Rao CR, Shanbhag DN (eds) Handbook of statistics 19: stochastic processes: theory and methods, Chapter 8. Elsevier Science, Amsterdam
Bobrowski A, Wang N, Chakraborty R, Kimmel M (2002) Non-homogeneous infinite sites model under demographic change: mathematical description and asymptotic behavior of pairwise distributions. Math Biosci 175(2):83–115
Deka R, Guangyun S, Smelser D, Zhong Y, Kimmel M, Chakraborty R (1999) Rate and directionality of mutations and effects of allele size constraints at anonymous, gene-associated, and disease-causing trinucleotide loci. Mol Biol Evol 16:1166–1177
Durrett R, Kruglyak S (1999) A new stochastic model of microsatellite evolution. J Appl Prob 36:621–631
Norris JR (1997) Markov chains. Cambridge University Press, Cambridge
Bobrowski A (2004) Quasi-stationary distributions of a pair of Markov chains related to time evolution of a DNA locus. Adv Appl Probab 36(1):57–77
Bobrowski A, Kimmel M (2004) Asymptotic behavior of joint distributions of characteristics of a pair of randomly chosen individuals in discrete - time fisher-wright models with mutations and drift. Theor Popul Biol 66(4):355–367
Bobrowski A, Kimmel M (2008) Asymptotic behavior of a Feller evolution family involved in the Fisher-Wright model. Adv Appl Probab 40(3):734–758
Feller W (1996) An introduction to probability theory and its applications, vol 2, 2nd edn. Wiley, New York (1971)
Bobrowski A, Kimmel M, Kubalińska M (2010) Non-homogeneous infinitely many sites discrete-time model with exact coalescent. Math Methods Appl Sci 33(6):713–732
Bobrowski A, Kimmel M (2003) A random evolution related to a Fisher-Wright-Moran model with mutation, recombination and drift. Math Methods Appl Sci 2003(26):1587–1599
Bobrowski A, Kimmel M, Wojdyła T (2010) Asymptotic behavior of a Moran model with mutations, drift and recombinations among multiple loci. J Math Biol 61:455–473
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Bobrowski, A., Kimmel, M. (2015). Master Equation and Asymptotic Behavior of Its Solutions. In: An Operator Semigroup in Mathematical Genetics. SpringerBriefs in Applied Sciences and Technology(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35958-3_5
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DOI: https://doi.org/10.1007/978-3-642-35958-3_5
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