Power laws for family sizes in a duplication model

Abstract

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