Research Article: Phenotypic equilibrium as probabilistic convergence in multi-phenotype cell population dynamics

Date Published: February 9, 2017

Publisher: Public Library of Science

Author(s): Da-Quan Jiang, Yue Wang, Da Zhou, Zhen Jin.


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.

Partial Text

With the same genetic background, cell population may have different cellular phenotypes. This has been one of the major topics in the research of cell population dynamics [1, 2]. Very recently much attention has been paid to the stochastic conversions between different phenotypes [3, 4]. For example, we know that cancer stem cells can give rise to cancer non-stem cells, but cancer non-stem cells can also transform back to cancer stem cells [5, 6]. Generally, we can use a branching process (stochastic model) [7–11] or an ODE system (deterministic model) [12] to describe the dynamics of such cell population with multiple phenotypes. However, in many experimental settings, it is difficult or even impossible to count the total cell population [10, 11, 13]. Thus in the last fifty years, people began to consider the proportions of cell individuals with distinct phenotypes instead of the absolute numbers of cells of various phenotypes [7].

In the Markovian model [4], it is assumed that the population proportions p→ satisfies the Kolmogorov forward equations of an n-state Markov chain:
where Q is the transition rate matrix, satisfying 1→Q′=0→. In this section, we will discuss whether such assumption can be satisfied in the deterministic model.

In general cases, Eq (4) is not satisfied since different phenotypes may differ in cell cycling time [22, 23], then the Markovian model is invalid. Thus we need other methods to study the asymptotic behavior of the population dynamics. In this section, we will prove that under some mild conditions, the proportions of different phenotypes will tend to some constants regardless of initial population states.

In population dynamics, we are also concerned about when one phenotype dies out or dominates. In terms of the notations in this paper, we need to consider when Pi(t) → 0 or Pi(t) → 1 as t → ∞.

In the previous sections, we assumed that the lifetime of a cell is exponentially distributed and independent of the type and number of its descendants. However, in real biological system, the lifetime distribution should be more like lognormal, gamma, Weibull, or exponentially modified Gaussian distribution [31, 32]. Furthermore, the time needed for division and conversion have different distributions [32]. In this way the process is a multitype Bellman-Harris branching process (also called age-dependent branching process) [33], no longer Markovian.

We have presented a unified stochastic model for the population dynamics with cellular phenotypic conversions. We have given the sufficient and necessary condition under which the dynamical behavior of our model can be described by an n-state Markov chain. In general case, we have proved that the proportions of different phenotypes will tend to constants regardless of their initial values, and we have investigated the sufficient and necessary conditions under which one phenotype will die out or dominate. We also extend our model to non-Markovian case while keeping the above conclusions valid. In this way we explain experimental phenomenon in [4].




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