Phenotypic Equilibrium as Probabilistic Convergence in Multi-phenotype Cell Population Dynamics

TL;DR

This paper demonstrates that in multi-phenotype cell populations, the proportions tend to stable constants over time, with proofs provided for both stochastic and deterministic models, explaining experimental observations and conditions for phenotype dominance or extinction.

Contribution

It establishes the phenotypic equilibrium as probabilistic convergence in both stochastic and deterministic models, extending results to non-Markovian cases and clarifying prior explanations.

Findings

Proportions tend to constants regardless of initial states.

Conditions for phenotype extinction or dominance are characterized.

Results extend to non-Markovian population dynamics.

Abstract

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.