  We show that simple stochastic models of genome evolution lead to power law
asymptotics of protein domain family size distribution. These models, called
Birth, Death and Innovation Models (BDIM), represent a special class of
balanced birth-and-death processes, in which domain duplication and deletion
rates are asymptotically equal up to the second order. The simplest, linear
BDIM shows an excellent fit to the observed distributions of domain family size
in diverse prokaryotic and eukaryotic genomes. However, the stochastic version
of the linear BDIM explored here predicts that the actual size of large
paralogous families is reached on an unrealistically long timescale. We show
that introduction of non-linearity, which might be interpreted as interaction
of a particular order between individual family members, allows the model to
achieve genome evolution rates that are much better compatible with the current
estimates of the rates of individual duplication/loss events.
