Power laws for family sizes in a duplication model

Qian, Luscombe, and Gerstein (2001) 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 one. 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 N1−r, then the number of families of size at least S is approximately CNS−1/(1−r), while if N1−r S the distribution decays more rapidly than any power. Running head: Power laws for gene family sizes. ∗Partially supported by NSF grants from the probability program (0202935) and from a joint DMS/NIGMS initiative to support research in mathematical biology (0201037). †Supported by an NSF Postdoctoral Fellowship. AMS 2000 subject classifications. Primary 60J80; Secondary 60J85, 92D15, 92D20.