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Cancers of the breast and other tissues arise from aberrant decision-making by cells regarding their survival or death, proliferation or quiescence, damage repair or bypass. These decisions are made by molecular signaling networks that process information from outside and from within the breast cancer cell and initiate responses that determine its survival and reproduction. Because the molecular logic of these circuits is difficult to comprehend by intuitive reasoning alone, we present some preliminary mathematical models of the basic decision circuits in breast cancer cells, with an eye to understanding better their susceptibility or resistance to endocrine therapy.
Cancer is a collection of diseases characterized by misregulation of the biomolecular pathways that control cellular processes of metabolism and growth, DNA replication and repair, mitosis and cell division, autophagy and apoptosis (programmed cell death), de-differentiation, motility and angiogenesis1. Molecular cell biologists have amassed a large body of information about the genes and proteins involved in these pathways and have some good ideas about how they go awry in certain types of cancers. However, most of our understanding of the molecular basis of cancer relies on intuitive reasoning about highly complex networks of biochemical interactions2–4. Intuition is clearly not the most reliable tool for querying the behavior of complex regulatory networks. Wouldn’t it be better if we could frame a reaction network in precise mathematical terms and use computer simulations to work out the implications of how the network functions in normal cells and malfunctions in cancer cells?
Of primary interest to cancer biologists is how cancer cells differ from normal cells in their responses to endogenous signals (such as growth and death factors, cell-cell and cell-matrix contacts) and to exogenous treatments (including cytotoxins, radiation, endocrine therapy). Cell responses—signal transduction, cell-fate decisions, adaptation—are intrinsically dynamic phenomena, so it is essential to understand the temporal evolution of biochemical signaling networks in response to particular stimuli. Ordinary differential equations, based on biochemical reaction kinetics, are an appropriate tool for addressing these questions. In principle, ODE models can provide a comprehensive, unified account of many experimental results, and a reliable tool for predicting novel cell behaviors. ODE models of yeast cell growth and division have lived up to these expectations5–8. But is it possible to build useful models of the considerably more complex regulatory networks in mammalian cells? We intend, in this article, to provide a roadmap for a detailed mathematical model of the estrogen signaling network in breast epithelial cells.
Our roadmap is built on the idea that a cell is an information processing system, receiving signals from its environment and its own internal state, interpreting these signals, and making appropriate cell-fate decisions, such as growth and division, movement, differentiation, self-replication, or cell death9. In plants and animals, these cell-level decisions are crucial to the growth, development, survival and reproduction of the organism. A hallmark of cancer cells is faulty decision-making: they proliferate when they should be quiescent, they survive when they should die, they move around when they should stay put1. To understand the origin, pathology and vulnerabilities of cancer cells, we must understand how normal cells make decisions that promote the survival of the organism as a whole, and how cancer cells make decisions that promote their own survival and reproduction with fatal results for the organism they inhabit10.
Viewing the living cell as an information processing system, we can (conceptually, at least) distinguish an input level, a processing core, and output devices (FIG. 1). As input, a cell receives information from its surroundings (such as extracellular ligands that bind to cell-surface receptors or to nuclear hormone receptors) and from its internal state (such as DNA damage, misfolded proteins, low energy level and oxidative stress). These signals are processed by chemical reaction networks that integrate information from many sources and compute a response. A response could take the form of the activation or inactivation of key integrator or effector proteins that drive the cell’s functional output devices. Of most interest to cancer biologists are the functional modules that control cell growth and division, motility and invasion, stress responses and apoptosis.
Although there may be many ways to subdivide the information processing system of a cell, there is clearly a need to divide and conquer the staggering complexity of the system11–13. Fortunately, it is not necessary to model the protein reaction networks in all their complexity, because it is usually possible to identify a set of key ‘integrator’ and ‘decision-making’ proteins that determine the cell’s response to input signals. Unfortunately, living cells are not like human-engineered systems, where modules are designed not to interfere much with one another14. Cellular modules have significant crosstalk and shared components. So although we must divide the system into modules to reduce the initial modeling complexity, we must also put the modules back together into a complete system that properly captures the information processing capabilities of living cells.
A comprehensive model of the information processing system of mammalian cells is not yet available, but we can provide a roadmap of how a modeler might capture, in mathematical form, the molecular events controlling cell growth, proliferation, damage responses and programmed death. Our approach is illustrated by simple mathematical models of the mechanisms involved in the initial susceptibility of breast cancer cells to anti-estrogen therapy and their subsequent development of anti-estrogen resistance. The value of this enterprise will be measured ultimately by new insights provided by the model into the logic and functionality of estrogen-receptor signaling pathways and by the effectiveness of the model as a tool for experimental prediction and design.
The growth and proliferation of breast tissue is normally responsive to estrogen, a steroid hormone that binds to and activates the estrogen receptors (ERα and ERβ), nuclear transcription factors that regulate the expression of genes that orchestrate survival and proliferation. In many neoplastic breast cells, the ER signaling network contributes to controlling the relative rates of cell proliferation and programmed cell death, with pro-survival and proliferation signals overwhelming pro-death and quiescence signals.
Of the 180,000 cases of invasive breast cancer newly diagnosed each year in the USA, more than 70% express ERα (ER+ cells)15. Many of these tumors are initially responsive to endocrine therapy alone, and many also respond to a combination of cytotoxic chemotherapies16,17. Endocrine therapy can consist of anti-estrogens (such as Tamoxifen or Fulvestrant), which bind to and neutralize ER, and/or aromatase inhibitors (such as Letrozole or Exemestane), which block the synthesis of estrogen. Unfortunately, many ER+ tumors recur as incurable, endocrine-resistant cancer cells18.
The advantages and limitations of endocrine therapies have been known for over 30 years. To make significant new advances in the treatment of advanced breast cancer, we need a better understanding of the ER signaling network19. For example, how does ER signaling function in normal breast cells? How does it malfunction in ER+ breast cancer cells that respond to endocrine therapy? How is it further misregulated in antiestrogen-resistant and aromatase inhibitor-resistant cancer cells? And how are cell survival and proliferation maintained in ER cancer cells?
FIGURE 1 provides an overview of the ER signaling network and its major output devices (cell growth and division, apoptosis and autophagy). From a combination of classical molecular biology studies and high-throughput transcriptomic analyses, we identified an initial set of transcription factors that are intimately connected with ER signaling in breast cancer cell lines20. Subsequently, we and others have established the functional relevance of several of these factors, including NFκB, a pro-survival transcription factor that is highly expressed in hormone-resistant cells compared to hormone-sensitive cells21–23; IRF1, a pro-death transcription factor that is down-regulated in endocrine-resistant cells24–27; and XBP1, a transcription factor involved in the unfolded protein response and induction of autophagy, which is highly expressed in its active (spliced) variant in endocrine-resistant cells27,28. Given that FIG. 1 correctly captures some of the key regulatory components and their interactions, interpreting it at a mathematical level should provide novel and useful insights into the decision-making processes in normal and transformed breast cells.
As useful as FIG. 1 is for providing a guide to intuitive reasoning about the probable effects of perturbations to this network, a molecular interaction graph can deliver much more information about the potential dynamic behavior of the control system if it is translated into reasonable mathematical terms suitable for computer simulation. In that case, the computer can keep track of the dynamic consequences of multiple and often conflicting interactions29,30.
In keeping with our roadmap perspective, we will begin by modeling the separate modules in FIG. 1: the ‘decision modules’ (cell cycle and apoptosis), the ‘stress modules’ (autophagy and unfolded protein response), and the ‘signal processing modules’ (ER and growth factor signal transduction networks). As we go along, we will describe how the ‘integrator and effector proteins’ mediate communication between these modules.
We start with the module controlling DNA replication and division, events that are triggered by cyclins and cyclin-dependent kinases (CDKs)31,32, the retinoblastoma protein (RB), which regulates members of the E2F-family of transcription factors, and late-G1 and early-S phase cyclins (type A and E cyclins)33–35. RB also down-regulates the expression of ribosomal RNA genes, thereby inhibiting the production of new ribosomes and the cell’s capacity for increased protein synthesis34,36–39. Hence, we can think of RB as a major ‘brake’ on cell growth and division, which must be released before a cell can grow and divide. This release is the job of the cyclin D-dependent kinases (CYCD1–3 in combination with CDK4 or CDK6), which phosphorylate RB and reduce its inhibitory effect on E2Fs33,40. CYCD is an unstable protein, and it is not present in quiescent cells because its transcription regulators, including MYC, AP1 and β-catenin, are inactive. These transcription regulators are activated by proliferative signals, such as growth factors, cytokines, nuclear hormone receptors and integrins, causing the concentration of CYCD to rise. The increasing concentration of CYCD must be converted into a digital decision: shall the cell undergo a new round of DNA replication and division or remain in G1 phase?
This decision is apparently made by a bistable switch, created by the interaction between RB, E2F and cyclin E (CYCE)41,42. The molecular interactions among these three proteins (FIG. 2a) are characterized by a positive feedback loop (E2F up-regulates CYCE, CYCE–CDK2 inactivates RB, RB inactivates E2F) and an auto-activation loop (E2F family members can activate their own transcription). According to mathematical models (Ref. 41 and supplementary information S1 (Box)), these sorts of positive feedback loops create a signal-response curve (FIG. 2b) with alternative stable steady states: an OFF state (RB active, E2F low, CYCE low), and an ON state (RB inactive, E2F high, CYCE high). The OFF state corresponds to quiescent cells (arrested in G1 phase of the cell cycle) and the ON state to proliferating cells (progression through S-G2-M phases)43. Careful measurements of the expression of CYCD and E2F in fibroblast cells responding to changes in serum concentration confirm the predictions of the model (FIG. 2c and d)41. Entry into the mammalian cell cycle in these non-cancerous cells is controlled by a bistable switch that is biased to the OFF state by signals that down-regulate CYCD and E2F (and possibly by signals that up-regulate RB), and that is switched ON by signals that up-regulate CYCD and E2F (see supplementary information S1 (Box) for further information). In cancer cells, this crucial decision point may still be intact44, but it is likely that in many cancers the bistable switch is disrupted by mutations that break the underlying feedback circuits45.
CYCD is a classic integrator and effector protein: its level integrates the pro- and anti-proliferative signals being received by the cell, and the activity of CYCD-dependent kinases effects the commitment of the cell to a new round of DNA synthesis and cell division. Pro-proliferative signals, such as estrogen acting through ERα, increase CYCD expression by activating its transcription factors. By contrast, cell-cell contacts result in cytoplasmic sequestration of β-catenin and down-regulation of CYCD expression. One of the hallmarks of many cancers is the loss of contact inhibition. A different mode of action is exemplified by the anti-proliferation factor, transforming growth factor β (TGF-β, which up-regulates synthesis of p27Kip1, an inhibitor of CYCD-dependent kinases. In breast cancer cells TGF-β is a key regulator of the anti-proliferative effects of anti-estrogens46,47, and CYCD expression is associated with poor response to Tamoxifen48. In summary, we might think of CYCD level as a rheostat that varies up and down continuously in response to pro- and anti-proliferative signals, respectively41, 43. When CYCD level exceeds a certain threshold, the RB-E2F-CYCE switch converts the CYCD signal into a discrete decision to begin a new round of DNA synthesis and cell division. Triggering this switch is therefore dependent on many factors affecting the level of active CYCD, such as estrogen, β-catenin, p27Kip1 and TGF-β49,50.
Once a cell has committed to the G1/S transition, it will proceed through S, G2 and M phases, even if the pro-proliferative signals are removed and CYCD disappears. However, when this cell divides and the other classes of cyclins (A, B and E) are degraded, RB will return and arrest the cell in a quiescent state.
Like the decision to enter a new round of mitotic cell division, the commitment to apoptosis must reach an all-or-none decision point that is biased one way or the other by the summation of pro-death and pro-survival signals. Although the evidence is not conclusive, we believe that the irrevocable commitment to apoptosis is normally made in the activation of BAX and amplified by mitochondrial outer-membrane permeabilization (MOMP)51. In our mathematical models MOMP is governed by a bistable switch involving three families of proteins: BCL2-like, BH3-only and BAX-like proteins (FIG. 3a)52–57. In the OFF state, BAX is inactivated by binding to BCL2. Accumulation of BH3 proteins can displace BCL2 from BAX, leading to the self-amplifying activation of BAX (the ON state). Active BAX proteins create pores in the mitochondrial outer membrane, thereby releasing cytochrome C and SMAC to the cytoplasm, where cytochrome C promotes activation of ‘executioner’ caspases, and SMAC neutralizes the inhibitor of apoptosis (IAP) proteins that inhibit caspases55.
Based on our models (FIG. 3 and supplementary information S2 (Box)), the apoptosis-switch is in the OFF or ON position depending on the balance between BCL2-like proteins (the ‘brakes’) and BH3-only proteins (the ‘accelerators’). When the ratio of accelerators to brakes exceeds a certain critical value (the point in FIG. 3b where the OFF state disappears), then BAX is abruptly activated and MOMP-induced activation of executioner caspases ensues. The ‘snap-action’ kinetics of MOMP are consistent with this view of a bistable switch activating BAX proteins (FIG. 3c).
Whether or not apoptosis is controlled by a bistable ‘decision’ module and (if so) what is its biochemical basis are questions of considerable discussion in the literature51,53–56,58–62. This debate illustrates that fundamentally different mathematical models may be equally consistent with limited experimental data. Fortunately, the different models can be used to design additional experiments that will distinguish between alternative mechanisms. We suppose that apoptosis is governed by a one-way (irreversible) bistable switch, because apoptosis in normal cells is an all-or-nothing affair. We interpret the evidence to suggest that the decision is made upstream of MOMP and that the BH3-BCL-BAX module is the most likely locus for the bistable switch. Although the apoptotic switch may be disabled in some cancer cells, it is likely still functional in most cancers but more difficult to engage. For instance, in breast cancer cell lines, ER-mediated signaling upregulates antiapoptotic proteins, including BCL223,63,64, BCL-W64 and BCL365, making it harder to trigger apoptosis. Endocrine-therapy, by inactivating ER, moves these levels in the opposite direction, making it easier to trigger apoptosis.
Intracellular damage-processing modules have crucial roles in maintaining the viability of cells and organisms. For example, DNA damage activates kinases that phosphorylate and stabilize the transcription factor p53 (Ref.66). p53 up-regulates genes encoding repair enzymes and p21Cip1, which binds to and inhibits the activity of CDKs, thereby preventing the damaged cell from beginning a new round of DNA replication. DNA damage also prevents S- or G2-phase cells from entering mitosis by pathways involving inhibitory phosphorylation of CDKs and the production of stoichiometric CDK inhibitors. If the damage cannot be repaired in a timely fashion, then p53 upregulates production of BH3 proteins in an attempt to activate the apoptosis module. Whether apoptosis occurs or not depends on BH3 level relative to BCL proteins, thereby integrating the influences of pro- and anti-apoptotic agents, including ER-mediated signals. Effective mathematical models of these DNA damage-processing pathways, based on the cell-proliferation and cell-death networks described in FIG. 2 and and3,3, have been published52,55,66,67.
Other common inducers of stress in normal and cancer cells include hypoxia and oxidative stress68,69. These types of stress cause problems in intermediary metabolism, electron transport in mitochondria and protein folding in the endoplasmic reticulum, and these problems induce characteristic responses by the cell. The first response to low-level stress is a pro-survival mechanism, autophagy, which is thought to provide a steady supply of energy and raw materials by degrading the cell’s own proteins and lipids70. Unremitting stress can lead to cell death, either by excessive autophagy or by activation of apoptosis71.
The autophagosome is a subcellular organelle containing a selection of cellular proteins and other macromolecules that are destined for destruction. When the autophagosome fuses with a lysosome, its contents are hydrolyzed to amino acids and other small metabolites that can be used by the cell as sources of energy and raw materials for the biosynthesis of essential substances. Autophagy is controlled in large part by Beclin-1 (BECN1), a myosin-like, BCL2-interacting protein. When not bound to BCL2, BECN1 participates in a multi-protein complex that initiates the earliest stages of autophagosome assembly70–72.
In FIG. 4 we propose a simple model for the initiation of autophagy (for details, see supplementary information S3 (Box). In this model, autophagy is regulated not as a toggle switch (as in FIG. 2 and and3)3) but as a rheostat, ramping up smoothly to higher levels as stress increases. As autophagy ramps up, BCL2 is released from its association with BECN1 and with the inositol trisphosphate receptor (IP3R). The results can be quite variable, including survival (moderate autophagy and inhibition of apoptosis), or apoptotic cell death, or autophagic cell death. Whether the autophagic response is functioning normally or abnormally in breast cancer cell lines is a matter of current investigation.
The accumulation of unfolded proteins in the endoplasmic reticulum causes a characteristic response73 that is intended to relieve the immediate problem (by re-folding or degrading the non-functional proteins and reducing the rate of protein synthesis) and to deal with the underlying stress (by inducing autophagy). The molecular basis of the UPR is well understood, and useful mathematical models have been presented in the literature73–76. In FIG. 5 and supplementary information S4 (Box) we present a simplified model of UPR, to illustrate the basic principles of this damage-response module. Both autophagy and the UPR are strongly implicated in the responsiveness of breast cancer cells to anti-estrogens19, 77.
To impose some order on the tangled web of macromolecular interactions within a living cell it is necessary to think in terms of functional modules. Nonetheless, we must take into account that there is significant crosstalk between modules; for example, the apparent mutual inhibition between autophagy and apoptosis71. Crosstalk between signal transduction pathways is well known; for example, overlapping cell survival pathways are implicated in the notorious plasticity of cells in response to cancer chemotherapeutics, including endocrine therapies49,78–80. Understanding the mechanisms and roles of crosstalk is a crucial concern as we try to assemble modules into more complex networks that can account for the complex responses of cells under realistic conditions, including the development of drug resistance in breast cancer cells.
As an example, consider the epidermal growth factor (EGF) family of signaling pathways. A growing body of evidence demonstrates that endocrine therapy, which is often effective in regression of early-stage ER+ breast cancer, may provoke cellular adaptation processes including activation of a spectrum of estrogen-regulated survival and proliferation genes, such as those involved in EGF signaling81–86. Interestingly, Pratt et al.65 observed that a population of MCF-7 cells could be divided into two sub-groups following withdrawal of estrogen: most cells retain an absolute dependency on estrogen and die as a result of the treatment, but some cells become estrogen-independent by switching to alternative survival and proliferation signals. If endocrine treatment is discontinued within a short period of time, before the resistant cells have established their phenotype by genetic or epigenetic modifications, then the acquired resistance can be reversed. For example, a population of MCF-7 cells that over-express EGFR or HER2 exhibit a bimodal distribution of receptors (FIG. 6c), and this distribution pattern can be reversibly controlled by manipulating the exposure of the cells to estrogen87–89. We take these observations as evidence for a bistable survival-switch working through crosstalk between ER and EGF signaling pathways. Although little is known about how it works, mutual inhibition between the two pathways is a likely source of bistability. In FIG. 6 and supplementary information S5 (Box) we present a simple model that could account for the effects of estrogen withdrawal on MCF-7 cells.
Crosstalk in cell signaling networks generates a large selection of discrete, stable, self-organized states; creating a degree of cell-fate plasticity that is necessary for a cell to switch adaptively and robustly among these different states. Although this plasticity is essential for normal cells to survive in noisy environments and to differentiate properly in response to various developmental cues, it may lead to robust development of resistance to cytotoxic drugs. Hence, understanding how crosstalk controls these phenotypic switches is of first importance for designing more effective cancer treatment strategies.
Mathematical modeling of intracellular molecular regulatory networks is an essential part of a systems-approach to cancer biology90. Intuitive reasoning must be complemented by mathematical models when the molecular regulatory network under consideration is large, complex and interconnected, and when we are dealing with quantitative aspects of signaling and control91. A well crafted mathematical model allows us to integrate crucial information about the genetics, molecular biology and physiology of cancer cells into a quantitative hypothesis, amenable to computer simulation, mathematical analysis, and detailed comparison to experimental data. By computing the behavior of the model under a variety of experimental conditions and comparing these simulations to the observed behavior of cells, we can determine whether our hypothetical molecular mechanism is sufficient to account for the known behavior of cells. If and when our model passes this first test (‘post-diction’), we can use it to predict the behavior of cells under novel experimental conditions, and use these quantitative predictions to test the efficacy of the model. Even when are models are not in full agreement with experiments, we can be confident that the problem is in some part of the model rather than in faulty reasoning about its consequences. Indeed, the model can help us track down the origin of the problem(s) and consider alternative hypotheses.
Mathematical modeling of intercellular control systems related to breast cancer development, though still in its infancy, is beginning to provide some useful insights. For example, a sophisticated model of p53 signaling in MCF-7 cells successfully predicted a novel role for Wip1 in a negative feedback loop from p53 to an upstream kinase in the DNA damage-signaling pathway92. A recent model of the HER2-ER signaling network identified novel drug targets for trastuzumab-resistant cells93. A dynamic model of combinatorial cancer therapy suggested promising treatment strategies that were subsequently verified experimentally94.
In this review, we have presented a roadmap for the mathematical modeling component of an integrative, systems biology of endocrine responsiveness in ER+ breast cancer. The hard work is yet to be done: formulating and verifying models, estimating kinetic parameters, making non-obvious predictions and testing them by quantitative experimental measurements. Is it just a matter of time before an effective, integrated model of regulatory networks in breast cancer cells is informing the next wave of experiments and therapies? Successful ODE models of cell cycle regulation, growth factor signaling, programmed cell death and the unfolded protein response suggest that there are no fundamental barriers to accurate, predictive models of complex control systems in mammalian cells. On the other hand, effective modeling is hampered by many significant differences—genetically and phenotypically—among different types of mammalian cells. Extending models to cancer cells, which are notoriously unstable genetically, will be even more difficult. High-throughput data collection and analysis will be helpful in identifying important differences among cell types and between normal cells and their cancerous derivatives95–97.
Despite the seeming wealth of data on molecular mechanisms controlling mammalian cell proliferation and stress responses, there is often a distinct lack of reliable, quantitative measurements of these mechanisms under conditions that are relevant to model formulation and testing. Another impediment to modeling intracellular control systems stems from the fact that the behavior of populations of cells (e.g., graded response to drug treatment) may not reflect the behavior of single cells (all-or-none decision in response to the drug). At present, modelers are still struggling with how best to cope with all these competing issues.
In addition, there are other relevant theoretical considerations that we have not described in this review. For instance, realistic models of molecular regulation must take into account the compartmentalization of eukaryotic cells. For another, the restricted number of genes, mRNAs and protein molecules in a single cell generate unavoidable stochastic fluctuations in molecular control networks. Intracellular information-processing systems must be robust to these fluctuations in most circumstances, although in some circumstances these fluctuations may be exploited to generate a range of possible outcomes (‘bet-hedging’). Thirdly, our models only bridge the scales from molecular networks to cell physiology. Breast tumors exist in a complex microenvironment that affects the dynamic signaling within and among cancer cells. Modeling these effects adds new layers of complexity. Other kinds of mathematical models are needed to describe how cells are organized into multi-cellular tissues, interacting with extracellular matrix, recruiting vasculature, and eventually metastasizing to distant parts of the body98–100. Models at these higher scales are beginning to be integrated with molecular-level descriptions of intracellular control systems (e.g., the cell cycle) and of intercellular communication (e.g., Wnt signaling)101,102.
We expect that all these modeling challenges will soon be overcome and that a new generation of mathematical models will soon be providing new insights into the molecular foundations of endocrine responsiveness in breast cancer.
This work was supported in part by NIH Grants U54-CA149147 (to RC), R01-GM078989 (to JJT and WB), by NSF Grants DMS-0342283 (to JJT and Paul Brazhnik), DBI-0904340 (to AV), and by fellowships to CC and IT provided by the Virginia Tech graduate program in Genetics, Bioinformatics and Computational Biology.