Cell Population Dynamics: Its Relationship with Finite State Markov Chain and its Asymptotic Behavior

We consider a generalized model of cell population dynamics with n different phenotypes. Both the Markov branching process and the ODE system are presented, and exploited to investigate the dynamics of the phenotypic proportions. We will give a sufficient and necessary condition under which the phenotypic proportions satisfy the Kolmogorov forward equations of an n-state Markov chain. In general cases, we demonstrate that these proportions will tend to constants regardless of initial population states under weak conditions. These results explain the experimental phenomenon reported in Gupta et al.’s paper [9]. As an application, we will also give sufficient and necessary conditions under which the proportion of one phenotype tends to 0 or 1.