Phenotype divergence and cooperation in isogenic multicellularity and in cancer.

We discuss the mathematical modelling of two of the main mechanisms that pushed forward the emergence of multicellularity: phenotype divergence in cell differentiation and between-cell cooperation. In line with the atavistic theory of cancer, this disease being specific of multicellular animals, we set special emphasis on how both mechanisms appear to be reversed, however not totally impaired, rather hijacked, in tumour cell populations. Two settings are considered: the completely innovating, tinkering, situation of the emergence of multicellularity in the evolution of species, which we assume to be constrained by external pressure on the cell populations, and the completely planned-in the body plan-situation of the physiological construction of a developing multicellular animal from the zygote, or of bet hedging in tumours, assumed to be of clonal formation, although the body plan is largely-but not completely-lost in its constituting cells. We show how cancer impacts these two settings and we sketch mathematical models for them. We present here our contribution to the question at stake with a background from biology, from mathematics and from philosophy of science.

Evolution of a structured cell population endowed with plasticity of traits under constraints on and between the traits.

Confronted with the biological problem of managing plasticity in cell populations, which is in particular responsible for transient and reversible drug resistance in cancer, we propose a rationale consisting of an integro-differential and a reaction-advection-diffusion equation, the properties of which are studied theoretically and numerically. By using a constructive finite volume method, we show the existence and uniqueness of a weak solution and illustrate by numerical approximations and their simulations the capacity of the model to exhibit divergence of traits. This feature may be theoretically interpreted as describing a physiological step towards multicellularity in animal evolution and, closer to present-day clinical challenges in oncology, as a possible representation of bet hedging in cancer cell populations.

Cell plasticity in cancer cell populations.

In this review, we propose a recension of biological observations on plasticity in cancer cell populations and discuss theoretical considerations about their mechanisms.

Identification of a transient state during the acquisition of temozolomide resistance in glioblastoma.

Drug resistance limits the therapeutic efficacy in cancers and leads to tumor recurrence through ill-defined mechanisms. Glioblastoma (GBM) are the deadliest brain tumors in adults. GBM, at diagnosis or after treatment, are resistant to temozolomide (TMZ), the standard chemotherapy. To better understand the acquisition of this resistance, we performed a longitudinal study, using a combination of mathematical models, RNA sequencing, single cell analyses, functional and drug assays in a human glioma cell line (U251). After an initial response characterized by cell death induction, cells entered a transient state defined by slow growth, a distinct morphology and a shift of metabolism. Specific genes expression associated to this population revealed chromatin remodeling. Indeed, the histone deacetylase inhibitor trichostatin (TSA), specifically eliminated this population and thus prevented the appearance of fast growing TMZ-resistant cells. In conclusion, we have identified in glioblastoma a population with tolerant-like features, which could constitute a therapeutic target.

Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis.

We propose a mathematical model to describe the evolution of hematopoietic stem cells (HSCs) and stromal cells in considering the bi-directional interaction between them. Cancerous cells are also taken into account in our model. HSCs are structured by a continuous phenotype characterising the population heterogeneity in a way relevant to the question at stake while stromal cells are structured by another continuous phenotype representing their capacity of support to HSCs. We then analyse the model in the framework of adaptive dynamics. More precisely, we study single Dirac mass steady states, their linear stability and we investigate the role of parameters in the model on the nature of the evolutionary stable distributions (ESDs) such as monomorphism, dimorphism and the uniqueness properties. We also study the dominant phenotypes by an asymptotic approach and we obtain the equation for dominant phenotypes. Numerical simulations are employed to illustrate our analytical results. In particular, we represent the case of the invasion of malignant cells as well as the case of co-existence of cancerous cells and healthy HSCs.

Control in dormancy or eradication of cancer stem cells: Mathematical modeling and stability issues.

OBJECTIVE: Modeling and analysis of cell population dynamics enhance our understanding of cancer. Here we introduce and explore a new model that may apply to many tissues. ANALYSES: An age-structured model describing coexistence between mutated and ordinary stem cells is developed and explored. The model is transformed into a nonlinear time-delay system governing the dynamics of healthy cells, coupled to a nonlinear differential-difference system describing dynamics of unhealthy cells. Its main features are highlighted and an advanced stability analysis of several steady states is performed, through specific Lyapunov-like functionals for descriptor-type systems. RESULTS: We propose a biologically based model endowed with rich dynamics. It incorporates a new parameter representing immunoediting processes, including the case where proliferation of cancer cells is locally kept under check by the immune cells. It also considers the overproliferation of cancer stem cells, modeled as a subpopulation of mutated cells that is constantly active in cell division. The analysis that we perform here reveals the conditions of existence of several steady states, including the case of cancer dormancy, in the coupled model of interest. Our study suggests that cancer dormancy may result from a plastic sensitivity of mutated cells to their shared environment, different from that - fixed - of healthy cells, and this is related to an action (or lack of action) of the immune system. Next, the stability analysis that we perform is essentially oriented towards the determination of sufficient conditions, depending on all the model parameters, that ensure either a regionally (i.e., locally) stable dormancy steady state or eradication of unhealthy cells. Finally, we discuss some biological interpretations, with regards to our findings, in light of current and emerging therapeutics. These final insights are particularly formulated in the paradigmatic case of hematopoiesis and acute leukemia, which is one of the best known malignancies for which it is always hard, in presence of a clinical and histological remission, to decide between cure and dormancy of a tumoral clone.

Genotype- or Phenotype-Targeting Anticancer Therapies? Lessons from Tumor Evolutionary Biology.

Despite the efficacy of most cancer therapies, drug resistance remains a major problem in the clinic. The eradication of the entire tumor and the cure of the patient by chemotherapy alone are rare, in particular for advanced disease. From an evolutionary perspective, the selective pressure exerted by chemotherapy leads to the emergence of resistant clones where resistance can be associated with many different functional mechanisms at the single cell level or can involve changes in the tumor micro-environment. In the last decade, tumor genomics has contributed to the improvement of our understanding of tumorigenesis and has led to the identification of numerous cellular targets for the development of novel therapies. However, since tumors are by nature extremely heterogeneous, the drug efficacy and economical sustainability of this approach is now debatable. Importantly, tumor cell heterogeneity depends not only on genetic modifications but also on non-genetic processes involving either stochastic events or epigenetic modifications making genetic biomarkers of uncertain utility. In this review, we wish to highlight how evolutionary biology can impact our understanding of carcinogenesis and resistance to therapies. We will discuss new approaches based on applied ecology and evolution dynamics that can be used to convert the cancer into a chronic disease where the drugs would control tumor growth. Finally, we will discuss the way metabolic dysfunction or phenotypic changes can help developing new delivery systems or phenotypetargeted drugs and how exploring new sources of active compounds can conduct to the development of drugs with original mechanisms of action.

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations.

BACKGROUND: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. RESULTS: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. CONCLUSIONS: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the 'maximum-tolerated-dose paradigm', as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones. REVIEWERS: This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling.

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation.

BACKGROUND: Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level. SCOPE OF REVIEW: We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies. MAJOR CONCLUSIONS: Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies. GENERAL SIGNIFICANCE: Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled "System Genetics" Guest Editor: Dr. Yudong Cai and Dr. Tao Huang.

Emergence of drug tolerance in cancer cell populations: an evolutionary outcome of selection, nongenetic instability, and stress-induced adaptation.

In recent experiments on isogenetic cancer cell lines, it was observed that exposure to high doses of anticancer drugs can induce the emergence of a subpopulation of weakly proliferative and drug-tolerant cells, which display markers associated with stem cell-like cancer cells. After a period of time, some of the surviving cells were observed to change their phenotype to resume normal proliferation and eventually repopulate the sample. Furthermore, the drug-tolerant cells could be drug resensitized following drug washout. Here, we propose a theoretical mechanism for the transient emergence of such drug tolerance. In this framework, we formulate an individual-based model and an integro-differential equation model of reversible phenotypic evolution in a cell population exposed to cytotoxic drugs. The outcomes of both models suggest that nongenetic instability, stress-induced adaptation, selection, and the interplay between these mechanisms can push an actively proliferating cell population to transition into a weakly proliferative and drug-tolerant state. Hence, the cell population experiences much less stress in the presence of the drugs and, in the long run, reacquires a proliferative phenotype, due to phenotypic fluctuations and selection pressure. These mechanisms can also reverse epigenetic drug tolerance following drug washout. Our study highlights how the transient appearance of the weakly proliferative and drug-tolerant cells is related to the use of high-dose therapy. Furthermore, we show how stem-like characteristics can act to stabilize the transient, weakly proliferative, and drug-tolerant subpopulation for a longer time window. Finally, using our models as in silico laboratories, we propose new testable hypotheses that could help uncover general principles underlying the emergence of cancer drug tolerance.

Modeling the effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors.

Histopathological evidence supports the idea that the emergence of phenotypic heterogeneity and resistance to cytotoxic drugs can be considered as a process of selection in tumor cell populations. In this framework, can we explain intra-tumor heterogeneity in terms of selection driven by the local cell environment? Can we overcome the emergence of resistance and favor the eradication of cancer cells by using combination therapies? Bearing these questions in mind, we develop a model describing cell dynamics inside a tumor spheroid under the effects of cytotoxic and cytostatic drugs. Cancer cells are assumed to be structured as a population by two real variables standing for space position and the expression level of a phenotype of resistance to cytotoxic drugs. The model takes explicitly into account the dynamics of resources and anticancer drugs as well as their interactions with the cell population under treatment. We analyze the effects of space structure and combination therapies on phenotypic heterogeneity and chemotherapeutic resistance. Furthermore, we study the efficacy of combined therapy protocols based on constant infusion and bang-bang delivery of cytotoxic and cytostatic drugs.

Reaction-diffusion systems for spatio-temporal intracellular protein networks: A beginner's guide with two examples.

Spatio-temporal dynamics of a variety of proteins is, among other things, regulated by post-translational modifications of these proteins. Such modifications can thus influence stability and biochemical activities of the proteins, activity and stability of their upstream targets within specific signalling pathways. Commonly used mathematical tools for such protein-protein (and/or protein-mRNA) interactions in single cells, namely, Michaelis-Menten and Hill kinetics, yielding a system of ordinary differential equations, are extended here into (non-linear) partial differential equations by taking into account a more realistic spatial representation of the environment where these reactions occur. In the modelling framework under consideration, all interactions occur in a cell divided into two compartments, the nucleus and the cytoplasm, connected by the semipermeable nuclear membrane and bounded by the impermeable cell membrane. Passive transport mechanism, modelled by the so-called Kedem-Katchalsky boundary conditions, is used here to represent migration of species throughout the nuclear membrane. Nonlinear systems of partial differential equations are solved by the semi-implicit Rothe method. Examples of two spatial oscillators are shown. Namely, these are the circadian rhythm for concentration of the FRQ protein in Neurospora crassa and oscillatory dynamics observed in the activation and regulation of the p53 protein following DNA damage in mammalian cells.

The dynamics of p53 in single cells: physiologically based ODE and reaction-diffusion PDE models.

The intracellular signalling network of the p53 protein plays important roles in genome protection and the control of cell cycle phase transitions. Recently observed oscillatory behaviour in single cells under stress conditions has inspired several research groups to simulate and study the dynamics of the protein with the aim of gaining a proper understanding of the physiological meanings of the oscillations. We propose compartmental ODE and PDE models of p53 activation and regulation in single cells following DNA damage and we show that the p53 oscillations can be retrieved by plainly involving p53-Mdm2 and ATM-p53-Wip1 negative feedbacks, which are sufficient for oscillations experimentally, with no further need to introduce any delays into the protein responses and without considering additional positive feedback.

The p53 protein and its molecular network: modelling a missing link between DNA damage and cell fate.

Various molecular pharmacokinetic-pharmacodynamic (PK-PD) models have been proposed in the last decades to represent and predict drug effects in anticancer chemotherapies. Most of these models are cell population based since clearly measurable effects of drugs can be seen much more easily on populations of cells, healthy and tumour, than in individual cells. The actual targets of drugs are, however, cells themselves. The drugs in use either disrupt genome integrity by causing DNA strand breaks, and consequently initiate programmed cell death, or block cell proliferation mainly by inhibiting factors that enable cells to proceed from one cell cycle phase to the next through checkpoints in the cell division cycle. DNA damage caused by cytotoxic drugs (and also cytostatic drugs at high concentrations) activates, among others, the p53 protein-modulated signalling pathways that directly or indirectly force the cell to make a decision between survival and death. The paper aims to become the first-step in a larger scale enterprise that should bridge the gap between intracellular and population PK-PD models, providing oncologists with a rationale to predict and optimise the effects of anticancer drugs in the clinic. So far, it only sticks at describing p53 activation and regulation in single cells following their exposure to DNA damaging stress agents. We show that p53 oscillations that have been observed in individual cells can be reconstructed and predicted by compartmentalising cellular events occurring after DNA damage, either in the nucleus or in the cytoplasm, and by describing network interactions, using ordinary differential equations (ODEs), between the ATM, p53, Mdm2 and Wip1 proteins, in each compartment, nucleus or cytoplasm, and between the two compartments. This article is part of a Special Issue entitled: Computational Proteomics, Systems Biology & Clinical Implications.

Physiologically based mathematical models to optimize therapies against metastatic colorectal cancer: a mini-review.

Understanding and improving the effects of combined drug treatments in metastatic colorectal Cancer (mCRC) is a multidisciplinary and multiscale problem, that can benefit from a systems biology approach. Although a quite limited number of active drugs have been approved for clinical applications, a variety of combined delivery regimen options are actually used in the clinic, so that choosing between them, or designing new ones, is not an obvious task, which calls for some rationalization based on physiological principles. We propose some physiologically based molecular pharmacokinetics-pharmacodynamics models for the main cytotoxic drugs used in the clinic and call for others describing more recently used agents, such as associated with monoclonal antibodies. We also advocate simultaneously designing models of the proliferating cell populations under therapeutic control, as cancer is primarily a disruption of physiological control on tissue proliferation. These two types of models are based on differential equations to continuously describe both the fate of drugs in the organism, from infusion until pharmacological effects, and their impact on the proliferation of cell populations, healthy and tumor. The multiscale nature of colorectal cancer, from the disruption of intracellular pathways to tumor growth observed at the macroscopic level, together with its frequent multilocal extension by simultaneous metastases in various healthy tissues of the organism at the time of diagnosis, and later, call for multiscale mathematical models. We thus propose a multi-level vision of cytotoxic drug use in the clinic, in which the weapon in the hands of clinicians, a drug combination regimen, the targets -wanted and unwanted -on which it exerts its effects, molecular pathways in proliferating cell populations, and the environment of the latter in a whole organism, are all considered in order to design a rationale for appropriate shooting, i.e., treatment optimization under patient-tailored constraints.

Applying ecological and evolutionary theory to cancer: a long and winding road.

Since the mid 1970s, cancer has been described as a process of Darwinian evolution, with somatic cellular selection and evolution being the fundamental processes leading to malignancy and its many manifestations (neoangiogenesis, evasion of the immune system, metastasis, and resistance to therapies). Historically, little attention has been placed on applications of evolutionary biology to understanding and controlling neoplastic progression and to prevent therapeutic failures. This is now beginning to change, and there is a growing international interest in the interface between cancer and evolutionary biology. The objective of this introduction is first to describe the basic ideas and concepts linking evolutionary biology to cancer. We then present four major fronts where the evolutionary perspective is most developed, namely laboratory and clinical models, mathematical models, databases, and techniques and assays. Finally, we discuss several of the most promising challenges and future prospects in this interdisciplinary research direction in the war against cancer.

Age-structured cell population model to study the influence of growth factors on cell cycle dynamics.

Cell proliferation is controlled by many complex regulatory networks. Our purpose is to analyse, through mathematical modeling, the effects of growth factors on the dynamics of the division cycle in cell populations. Our work is based on an age-structured PDE model of the cell division cycle within a population of cells in a common tissue. Cell proliferation is at its first stages exponential and is thus characterised by its growth exponent, the first eigenvalue of the linear system we consider here, a growth exponent that we will explicitly evaluate from biological data. Moreover, this study relies on recent and innovative imaging data (fluorescence microscopy) that make us able to experimentally determine the parameters of the model and to validate numerical results. This model has allowed us to study the degree of simultaneity of phase transitions within a proliferating cell population and to analyse the role of an increased growth factor concentration in this process. This study thus aims at helping biologists to elicit the impact of growth factor concentration on cell cycle regulation, at making more precise the dynamics of key mechanisms controlling the division cycle in proliferating cell populations, and eventually at establishing theoretical bases for optimised combined anticancer treatments.

A spatial physiological model for p53 intracellular dynamics.

In this paper we design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of ATM by DNA damage. The p53 protein is known to have a central role in the response of the cell to cytotoxic or radiotoxic insults resulting in DNA damage. A reasonable requirement for a model describing intracellular signalling pathways is taking into account the basic feature of eukaryotic cells: the distinction between nucleus and cytoplasm. Our aim is to show, on a simple reaction network describing p53 dynamics, how this basic distinction provides a framework which is able to yield expected oscillatory dynamics without introducing either positive feedbacks or delays in the reactions. Furthermore we prove that oscillations appear only if some spatial constraints are respected, e.g. if the diffusion coefficients correspond to known biological values. Finally we analyse how the spatial features of a cell influence the dynamic response of the p53 network to DNA damage, pointing out that the protein oscillatory dynamics is indeed a response that is robust towards changes with respect to cellular environments. Even if we change the cell shape or its volume or better its ribosomal distribution, we observe that DNA damage yields sustained oscillations of p53.