Population dynamics of cancer cells with cell state conversions

Cell-state transitions and density-dependent interactions together explain the dynamics of spontaneous epithelial-mesenchymal heterogeneity

Cancer cell populations comprise phenotypes distributed among the epithelial-mesenchymal (E-M) spectrum. However, it remains unclear which population-level processes give rise to the observed experimental distribution and dynamical changes in E-M heterogeneity, including (1) differential growth, (2) cell-state switching, and (3) population density-dependent growth or state-transition rates. Here, we analyze the necessity of these three processes in explaining the dynamics of E-M population distributions as observed in PMC42-LA and HCC38 breast cancer cells. We find that, while cell-state transition is necessary to reproduce experimental observations of dynamical changes in E-M fractions, including density-dependent growth interactions (cooperation or suppression) better explains the data. Further, our models predict that treatment of HCC38 cells with transforming growth factor β (TGF-β) signaling and Janus kinase 2/signal transducer and activator of transcription 3 (JAK2/3) inhibitors enhances the rate of mesenchymal-epithelial transition (MET) instead of lowering that of E-M transition (EMT). Overall, our study identifies the population-level processes shaping the dynamics of spontaneous E-M heterogeneity in breast cancer cells.

A mathematical model for phenotypic heterogeneity in breast cancer with implications for therapeutic strategies

Inevitably, almost all cancer patients develop resistance to targeted therapy. Intratumour heterogeneity is a major cause of drug resistance. Mathematical models that explain experiments quantitatively are useful in understanding the origin of intratumour heterogeneity, which then could be used to explore scenarios for efficacious therapy. Here, we develop a mathematical model to investigate intratumour heterogeneity in breast cancer by exploiting the observation that HER2+ and HER2- cells could divide symmetrically or asymmetrically. Our predictions for the evolution of cell fractions are in quantitative agreement with single-cell experiments. Remarkably, the colony size of HER2+ cells emerging from a single HER2- cell (or vice versa), which occurs in about four cell doublings, also agrees with experimental results, without tweaking any parameter in the model. The theory explains experimental data on the responses of breast tumours under different treatment protocols. We then used the model to predict that, not only the order of two drugs, but also the treatment period for each drug and the tumour cell plasticity could be manipulated to improve the treatment efficacy. Mathematical models, when integrated with data on patients, make possible exploration of a broad range of parameters readily, which might provide insights in devising effective therapies.

Alternative models of cancer stem cells: The stemness phenotype model, 10 years later

The classical cancer stem cell (CSCs) theory proposed the existence of a rare but constant subpopulation of CSCs. In this model cancer cells are organized hierarchically and are responsible for tumor resistance and tumor relapse. Thus, eliminating CSCs will eventually lead to cure of cancer. This simplistic model has been challenged by experimental data. In 2010 we proposed a novel and controversial alternative model of CSC biology (the Stemness Phenotype Model, SPM). The SPM proposed a non-hierarchical model of cancer biology in which there is no specific subpopulation of CSCs in tumors. Instead, cancer cells are highly plastic in term of stemness and CSCs and non-CSCs can interconvert into each other depending on the microenvironment. This model predicts the existence of cancer cells ranging from a pure CSC phenotype to pure non-CSC phenotype and that survival of a single cell can originate a new tumor. During the past 10 years, a plethora of experimental evidence in a variety of cancer types has shown that cancer cells are indeed extremely plastic and able to interconvert into cells with different stemness phenotype. In this review we will (1) briefly describe the cumulative evidence from our laboratory and others supporting the SPM; (2) the implications of the SPM in translational oncology; and (3) discuss potential strategies to develop more effective therapeutic regimens for cancer treatment.

Electrical Impedance Spectroscopy for Monitoring Chemoresistance of Cancer Cells

Electrical impedance spectroscopy (EIS) is an electrokinetic method that allows for the characterization of intrinsic dielectric properties of cells. EIS has emerged in the last decade as a promising method for the characterization of cancerous cells, providing information on inductance, capacitance, and impedance of cells. The individual cell behavior can be quantified using its characteristic phase angle, amplitude, and frequency measurements obtained by fitting the input frequency-dependent cellular response to a resistor-capacitor circuit model. These electrical properties will provide important information about unique biomarkers related to the behavior of these cancerous cells, especially monitoring their chemoresistivity and sensitivity to chemotherapeutics. There are currently few methods to assess drug resistant cancer cells, and therefore it is difficult to identify and eliminate drug-resistant cancer cells found in static and metastatic tumors. Establishing techniques for the real-time monitoring of changes in cancer cell phenotypes is, therefore, important for understanding cancer cell dynamics and their plastic properties. EIS can be used to monitor these changes. In this review, we will cover the theory behind EIS, other impedance techniques, and how EIS can be used to monitor cell behavior and phenotype changes within cancerous cells.

Modeling differentiation-state transitions linked to therapeutic escape in triple-negative breast cancer

Drug resistance in breast cancer cell populations has been shown to arise through phenotypic transition of cancer cells to a drug-tolerant state, for example through epithelial-to-mesenchymal transition or transition to a cancer stem cell state. However, many breast tumors are a heterogeneous mixture of cell types with numerous epigenetic states in addition to stem-like and mesenchymal phenotypes, and the dynamic behavior of this heterogeneous mixture in response to drug treatment is not well-understood. Recently, we showed that plasticity between differentiation states, as identified with intracellular markers such as cytokeratins, is linked to resistance to specific targeted therapeutics. Understanding the dynamics of differentiation-state transitions in this context could facilitate the development of more effective treatments for cancers that exhibit phenotypic heterogeneity and plasticity. In this work, we develop computational models of a drug-treated, phenotypically heterogeneous triple-negative breast cancer (TNBC) cell line to elucidate the feasibility of differentiation-state transition as a mechanism for therapeutic escape in this tumor subtype. Specifically, we use modeling to predict the changes in differentiation-state transitions that underlie specific therapy-induced changes in differentiation-state marker expression that we recently observed in the HCC1143 cell line. We report several statistically significant therapy-induced changes in transition rates between basal, luminal, mesenchymal, and non-basal/non-luminal/non-mesenchymal differentiation states in HCC1143 cell populations. Moreover, we validate model predictions on cell division and cell death empirically, and we test our models on an independent data set. Overall, we demonstrate that changes in differentiation-state transition rates induced by targeted therapy can provoke distinct differentiation-state aggregations of drug-resistant cells, which may be fundamental to the design of improved therapeutic regimens for cancers with phenotypic heterogeneity.

A mathematical model of stem cell regeneration with epigenetic state transitions

In this paper, we study a mathematical model of stem cell regeneration with epigenetic state transitions. In the model, the heterogeneity of stem cells is considered through the epigenetic state of each cell, and each epigenetic state defines a subpopulation of stem cells. The dynamics of the subpopulations are modeled by a set of ordinary differential equations in which epigenetic state transition in cell division is given by the transition probability. We present analysis for the existence and linear stability of the equilibrium state. As an example, we apply the model to study the dynamics of state transition in breast cancer stem cells.

Epithelial/mesenchymal plasticity: how have quantitative mathematical models helped improve our understanding?

Phenotypic plasticity, the ability of cells to reversibly alter their phenotypes in response to signals, presents a significant clinical challenge to treating solid tumors. Tumor cells utilize phenotypic plasticity to evade therapies, metastasize, and colonize distant organs. As a result, phenotypic plasticity can accelerate tumor progression. A well‐studied example of phenotypic plasticity is the bidirectional conversions among epithelial, mesenchymal, and hybrid epithelial/mesenchymal (E/M) phenotype(s). These conversions can alter a repertoire of cellular traits associated with multiple hallmarks of cancer, such as metabolism, immune evasion, invasion, and metastasis. To tackle the complexity and heterogeneity of these transitions, mathematical models have been developed that seek to capture the experimentally verified molecular mechanisms and act as ‘hypothesis‐generating machines’. Here, we discuss how these quantitative mathematical models have helped us explain existing experimental data, guided further experiments, and provided an improved conceptual framework for understanding how multiple intracellular and extracellular signals can drive E/M plasticity at both the single‐cell and population levels. We also discuss the implications of this plasticity in driving multiple aggressive facets of tumor progression.

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.

The overshoot and phenotypic equilibrium in characterizing cancer dynamics of reversible phenotypic plasticity

The paradigm of phenotypic plasticity indicates reversible relations of different cancer cell phenotypes, which extends the cellular hierarchy proposed by the classical cancer stem cell (CSC) theory. Since it is still questionable if the phenotypic plasticity is a crucial improvement to the hierarchical model or just a minor extension to it, it is worthwhile to explore the dynamic behavior characterizing the reversible phenotypic plasticity. In this study we compare the hierarchical model and the reversible model in predicting the cell-state dynamics observed in biological experiments. Our results show that the hierarchical model shows significant disadvantages over the reversible model in describing both long-term stability (phenotypic equilibrium) and short-term transient dynamics (overshoot) in cancer cell populations. In a very specific case in which the total growth of population due to each cell type is identical, the hierarchical model predicts neither phenotypic equilibrium nor overshoot, whereas the reversible model succeeds in predicting both of them. Even though the performance of the hierarchical model can be improved by relaxing the specific assumption, its prediction to the phenotypic equilibrium strongly depends on a precondition that may be unrealistic in biological experiments. Moreover, it still does not show as rich dynamics as the reversible model in capturing the overshoots of both CSCs and non-CSCs. By comparison, it is more likely for the reversible model to correctly predict the stability of the phenotypic mixture and various types of overshoot behavior.

The phenotypic equilibrium of cancer cells: From average-level stability to path-wise convergence

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 the 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.

Overshoot in biological systems modelled by Markov chains: a non-equilibrium dynamic phenomenon

A number of biological systems can be modelled by Markov chains. Recently, there has been an increasing concern about when biological systems modelled by Markov chains will perform a dynamic phenomenon called overshoot. In this study, the authors found that the steady-state behaviour of the system will have a great effect on the occurrence of overshoot. They showed that overshoot in general cannot occur in systems that will finally approach an equilibrium steady state. They further classified overshoot into two types, named as simple overshoot and oscillating overshoot. They showed that except for extreme cases, oscillating overshoot will occur if the system is far from equilibrium. All these results clearly show that overshoot is a non-equilibrium dynamic phenomenon with energy consumption. In addition, the main result in this study is validated with real experimental data.

A multi-phenotypic cancer model with cell plasticity

The conventional cancer stem cell (CSC) theory indicates a hierarchy of CSCs and non-stem cancer cells (NSCCs), that is, CSCs can differentiate into NSCCs but not vice versa. However, an alternative paradigm of CSC theory with reversible cell plasticity among cancer cells has received much attention very recently. Here we present a generalized multi-phenotypic cancer model by integrating cell plasticity with the conventional hierarchical structure of cancer cells. We prove that under very weak assumption, the nonlinear dynamics of multi-phenotypic proportions in our model has only one stable steady state and no stable limit cycle. This result theoretically explains the phenotypic equilibrium phenomena reported in various cancer cell lines. Furthermore, according to the transient analysis of our model, it is found that cancer cell plasticity plays an essential role in maintaining the phenotypic diversity in cancer especially during the transient dynamics. Two biological examples with experimental data show that the phenotypic conversions from NCSSs to CSCs greatly contribute to the transient growth of CSCs proportion shortly after the drastic reduction of it. In particular, an interesting overshooting phenomenon of CSCs proportion arises in three-phenotypic example. Our work may pave the way for modeling and analyzing the multi-phenotypic cell population dynamics with cell plasticity.