Power laws for family sizes in a duplication model

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dc.creatorDurrett, Rick
dc.creatorSchweinsberg, Jason
dc.date2004-06-10
dc.date2006-02-07
dc.date.accessioned2026-07-07T06:36:50Z
dc.date.available2026-07-07T06:36:50Z
dc.descriptionQian, Luscombe and Gerstein [J. Molecular Biol. 313 (2001) 673--681] introduced a model of the diversification of protein folds in a genome that we may formulate as follows. Consider a multitype Yule process starting with one individual in which there are no deaths and each individual gives birth to a new individual at rate 1. When a new individual is born, it has the same type as its parent with probability $1-r$ and is a new type, different from all previously observed types, with probability $r$. We refer to individuals with the same type as families and provide an approximation to the joint distribution of family sizes when the population size reaches $N$. We also show that if $1\ll S\ll N^{1-r}$, then the number of families of size at least $S$ is approximately $CNS^{-1/(1-r)}$, while if $N^{1-r}\ll S$ the distribution decays more rapidly than any power.
dc.descriptionPublished at http://dx.doi.org/10.1214/009117905000000369 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
dc.identifierhttps://arxiv.org/abs/math/0406216
dc.identifierhttp://arxiv.org/abs/math/0406216
dc.identifierAnnals of Probability 2005, Vol. 33, No. 6, 2094-2126
dc.identifierdoi:10.1214/009117905000000369
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/100219
dc.subjectProbability
dc.subjectPopulations and Evolution
dc.subject60J80 (Primary) 60J85, 92D15, 92D20 (Secondary)
dc.titlePower laws for family sizes in a duplication model
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