Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics.
We consider the cell population dynamics with n different phenotypes. Both the Markovian branching process model (stochastic model) and the ordinary differential equation (ODE) system model (deterministic model) are presented, and exploited to investigate the dynamics of the phenotypic proportions. We will prove that in both models, these proportions will tend to constants regardless of initial population states ("phenotypic equilibrium") under weak conditions, which explains the experimental phenomenon in Gupta et al.'s paper. We also prove that Gupta et al.'s explanation is the ODE model under a special assumption. As an application, we will give sufficient and necessary conditions under which the proportion of one phenotype tends to 0 (die out) or 1 (dominate). We also extend our results to non-Markovian cases.
LMAM, School of Mathematical Sciences & Center for Statistical Science, Peking University, Beijing 100871, P.R. China.
Da-Quan
DQ
Jiang
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, United States of America.
Yue
http://orcid.org/0000-0001-5918-7525
Y
Wang
School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R. China.
Da
D
Zhou
D019143:Evolution, Molecular; D008957:Models, Genetic; D010641:Phenotype; D011336:Probability