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November 2005 Power laws for family sizes in a duplication model
Rick Durrett, Jason Schweinsberg
Ann. Probab. 33(6): 2094-2126 (November 2005). DOI: 10.1214/009117905000000369


Qian, 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≪SN1−r, then the number of families of size at least S is approximately CNS−1/(1−r), while if N1−rS the distribution decays more rapidly than any power.


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Rick Durrett. Jason Schweinsberg. "Power laws for family sizes in a duplication model." Ann. Probab. 33 (6) 2094 - 2126, November 2005.


Published: November 2005
First available in Project Euclid: 7 December 2005

zbMATH: 1099.92055
MathSciNet: MR2184092
Digital Object Identifier: 10.1214/009117905000000369

Primary: 60J80
Secondary: 60J85 , 92D15 , 92D20

Keywords: genome sequencing , Multitype branching processes , power law , Yule processes

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 6 • November 2005
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