Family size distributions for multitype Yule processes
Markov Chains in Algorithms and Statistical Physics
February 04, 2005 11:00 AM to 11:45 AM
Speakers:
|
 |
Abstract: |
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 << 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. |
|
|
