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Power laws for family sizes in a duplication model. (English) Zbl 1099.92055

Diversification of protein folds in a genome is formulated as a multitype Yule process, starting with one individual. 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 \(r\). Individuals of the same type are called families. Based on the connections between Yule processes and Pólya urns, an approximation is derived for the joint family-size distribution when the population size reaches \(N\). It is shown that if \(1\ll S\ll N^{1-r}\), then the number of families of size at least \(S\) is approximately \(\text{CNS}^{-1(1-r)}\). The limiting distribution of the size of the large families is derived and it is shown that if \(N^{1-r}\ll S\), the expected number of families of size at least \(S\) decays more rapidly than any power. It is shown that the model has a close relation to the ‘Chinese restaurant process’. Connections to preferential attachment models are discussed.

MSC:

92D15 Problems related to evolution
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J85 Applications of branching processes
92D20 Protein sequences, DNA sequences
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