  The phenotypic equilibrium, i.e. heterogeneous population of cancer cells
tending to a fixed equilibrium of phenotypic proportions, has received much
attention in cancer biology very recently. In previous literature, some
theoretical models were used to predict the experimental phenomena of the
phenotypic equilibrium, which were often explained by different concepts of
stabilities of the models. Here we present a stochastic multi-phenotype
branching model by integrating conventional cellular hierarchy with phenotypic
plasticity mechanisms of cancer cells. Based on our model, it is shown that:
(i) our model can serve as a framework to unify the previous models for the
phenotypic equilibrium, and then harmonizes the different kinds of
average-level stabilities proposed in these models; and (ii) path-wise
convergence of our model provides a deeper understanding to the phenotypic
equilibrium from stochastic point of view. That is, the emergence of the
phenotypic equilibrium is rooted in the stochastic nature of (almost) every
sample path, the average-level stability just follows from it by averaging
stochastic samples.
