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Tumors as chaotic attractors Cite this: Mol. BioSyst., 2014, 10, 172
Svetoslav Nikolov,abc Olaf Wolkenhauerad and Julio Vera*e Malignant tumor growth, progression and evolution share several of the features associated with strange attractors, a special type of dynamical behavior of non-linear systems. In this framework, the genetic instability of tumor cells represents a local-scale driving force for instability that lets the tumor evolve with time, while selective pressures discarding inefficient cell clones play the role of a dissipative process, providing the tumor with global stability and robustness with respect to perturbations. This setup induces global stability combined with local instability, and is in our opinion the real source of tumor robustness and plasticity, and ultimately the origin of the remarkable resistance of many tumors to anti-cancer therapies. A scientific program taking into consideration the vision of tumors as strange attractors requires the development of experimental and computational tools to link the micro- and mesoscopic perspectives of tumor biology. We here develop the
Received 5th August 2013, Accepted 31st October 2013
idea of a tumor as a dynamical system, sharing characteristics with strange attractors and investigate several
DOI: 10.1039/c3mb70331b
the tremendous non-linearity of the biochemical regulatory circuits governing the individual fate and the
features of such systems, including: (a) local instability combined with global stability, a property inherent to interactions between tumor cells; (b) self-similarity at different levels; (c) and strong sensitivity to initial
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conditions in the composition and features of the tumor and its microenvironment.
1. Introduction: cancer as a multi-level dynamical system Cancer is a major health problem covering some 100 separate diseases with different risk factors and epidemiology, originating in different cell types and organs. If we focus on cancer manifested as solid tumors, they are characterized by accumulation of genetically abnormal cells featuring, individually or collectively, some or all of the following phenotypic traits: unlimited proliferation, evasion from endogenous and exogenous programmed cell death, independence from growth signals and/or insensitivity to antigrowth signals, metabolic reprogramming and invasiveness.1,2 The perspective that has driven oncologic research for decades is that of cancer as a genetic disease that arises from mutations in susceptible genes and subsequent somatic evolution.3 In the last few years this picture has been refined with the description of timedependent regulatory networks. These networks are rich in regulatory motifs that include positive or negative feedback as well as a
Department of Systems Biology and Bioinformatics, Institute for Informatics, University of Rostock, 18051 Rostock, Germany b Institute of Mechanics, Bulgarian Academy of Science, Acad. G. Bonchev Str., bl.4, 1113 Sofia, Bulgaria c University of Transport, Geo Milev str., 158, 1574 Sofia, Bulgaria d Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University, Stellenbosch, South Africa e Laboratory of Systems Tumor Immunology, Dept. of Dermatology, Faculty of Medicine, University of Erlangen-Nurnberg, Erlangen, Germany. E-mail: [email protected]
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coherent and incoherent feedforward loops, whose regulation is difficult to understand because their effect on the dynamics of the systems is often non-linear.4–7 Whether the origin of cancer is genetic and resides in the microscopic world of genes and mutations within single cells, or whether it is emerging from a failure of tissue organization, it actually manifests as a mesoscopic structure, the tumor. To make the problem more cumbersome, tumors are not simple aggregations of genetically abnormal cells, but they display an amazing genetic heterogeneity between the cells that compose them.8,9 Those cells are not isolated from the outer world, but rather establish complex interactions between them and with their microenvironment, which ultimately are expressed as selective pressures like hypoxia, lack of nutrients or apoptosis that govern the fate of the tumor as a whole.10,11 Thus, from a global perspective tumors are also complex dynamical systems and the natural language to analyze such systems is dynamical systems theory. This framework allows us to handle concepts like complexity, non-linearity, robustness, sensitivity and (in)stability of a system, which actually emerge also in tumors, the cancer cells and the regulatory networks governing them. In this paper, we develop the idea of a tumor as a nonlinear dynamical system that has some of the properties associated with strange attractors, a class of dynamical behavior displayed by some chaotic systems. These properties are: (a) local instability combined with global stability, a property inherent to the tremendous nonlinearity of the biochemical regulatory circuits governing the individual fate and the interactions between tumor cells; (b) self-similarity at different levels; and (c) strong sensitivity to initial conditions in the composition and features of the tumor and its microenvironment.
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From biological systems to strange attractors: a quick guide to dynamical systems theory Natural systems are ensembles of physical entities that interact with each other.12 Biology, rather than being an exception, is a realization of this idea: biological entities are organized in systems, from interacting proteins to the entire biosphere. When biological systems become structurally complex (e.g., too many proteins), their analysis evades direct reasoning and intuition, making mathematical modeling a necessary tool.5 There are many instances of mathematical models, but one very widespread class of models suitable to understand biochemical and cell population systems are ordinary differential equations. These models describe spatio-temporal changes of biological entities using equations with the following structure:
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X d C¼ Fi ðS; k; CÞ dt i In these equations, C are the time-dependent variables accounting for the biological entities whose evolution over time is analyzed (i.e., interacting proteins or tumor cell populations), Fi are the rate equations formalizing interactions between the biological entities, S are inputs (external stimulus affecting the system) and k are parameters (fixed numbers associated with given features like rate of proliferation or protein cleavage). With this kind of model, one can trace the evolution of the system properties over time, the so-called trajectories (Fig. 1A). Under some conditions, the trajectories evolve towards steady-state configurations of the system in which the properties of the system C stay constant (stability), while others keep the system changing constantly and generate instability (Fig. 1B). When biological systems are enriched with complex regulatory motifs like feedback and forward loops, changes in the value of critical biological parameters k induce sudden transitions between stable and unstable configurations of the system, the so called bifurcations (e.g. a mutation increasing the cell proliferation rate can transform a rather stable microscopic neoplastic lesion into a growing macroscopic tumor; Fig. 1C, see also ref. 4). Furthermore, systems containing multiple interconnected instances of these motifs can, under some conditions, display chaotic behavior. This behavior is characterized by a high sensitivity to initial conditions that makes the dynamics of the system appear irregular. This is the case of, for example, multi-looped negative feedback systems.13 Such systems must be described and analyzed through the use of chaos theory.14,15 A dynamical structure that was discovered using this theory are strange or chaotic attractors.16 An attractor is a configuration (a set of values for the variables) towards which the system evolves over time (Fig. 1D). An attractor can be a stable fixed point or steadystate if, in the absence of external stimuli, the system stays in this configuration once it achieves it. Attractors can also be stable limit cycles, when the values of the system variables are periodically repeated, leading to self-sustained oscillations. Finally, in a strange attractor the values of the system variables remain permanently confined in a delimited region of all the phase space. However, within that region the system will go through the same point more than once. This is a special kind of behavior detected in some nonlinear dynamical systems, characterized by two contradictory properties: from a global perspective they are stable structures, but locally they are unstable. Furthermore, the chaotic attractors display two other properties of chaotic systems: fractal structure or self-similarity (e.g., some properties of the system as a whole are similar to those displayed by a part of itself) and high sensitivity to initial conditions (e.g., the difference between two similar configurations of the system diverges exponentially over time). In strange attractors, global dissipative processes compensate destabilizing local driving forces and neutralize noise and perturbations to maintain the system in a global steady-state regime. In nature there are some cases of mesoscopic systems behaving as strange attractors. An example are Benard cells, which appear in the plane horizontal layer of fluids subjected to convection.17 Here, turbulent flow combines with the effect of gravity and thermal dissipation to create globally stable structures emanating from a microscopically unstable milieu.
while the selective pressures discarding inefficient cell clones play the role of a dissipative process, providing the tumor with global stability and robustness with respect to perturbations. 2.1. Local instability in tumors originated from genetic instability
Fig. 1
Tumors as dynamical systems.
The genetic instability of most or part of the cells constituting a tumor is one of the first proved hallmarks of the cancer18 and some authors indicate that genetic instability might be a prerequisite for tumors to develop.19,20 This process induces rapid changes in the microscopic landscape of the tumor, including fast clonal succession of tumor cells.21 This is a source of permanent instability, altering the tumor composition at the microscopic level and enriching with genetic variability the micro-ecosystem of the tumor. Thus, genetic instability of tumor cells is the biological translation ´lezof the local instability found in strange attractors (Fig. 2). Gonza Garcı´a and co-authors8 combined data from single-cell genotyping of tumor laser micro-dissections and mathematical modeling to show that genetic instability and mutator phenotypes provide a source for tumor heterogeneity, but also that the generated spatial heterogeneity in the tumor is maintained over time.
2. Hypothesis
2.2.
We support the idea that tumor dynamics display several of the features associated with strange attractors. The genetic instability of tumor cells is one source of instability, the local-scale driving force that keeps the tumor operating and evolving over time,
In our hypothesis, the selective pressures operating on the genetically heterogeneous tumor cell community play the role of dissipative processes. Hypoxia, lack of nutrients or growth factors, natural or therapy-related apoptotic signals, when operating
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Global stability in tumors induced by selective pressures
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Fig. 2 Tumors as systems with local instability and global stability. From a mesoscopic perspective, tumors are robust long-lasting structures that perform extraordinarily well with respect to environmental perturbations. However, genetic instability and fast clonal succession, hallmarks of cancer, make tumors microscopically unstable and prone to genetic variability in their composing tumor cells. In Vera et al.22 we used multi-level modelling to show how heterogeneous tumors, composed by tumor cells with different genetic signatures (red, blue and grey colored cells) evolve mechanisms, which make them resistant to chemotherapy. At the mesoscopic scale these tumors are robust structures resisting the therapy (left figure). At the microscopic level, the selective forces driven by the therapy change drastically the composition of the tumor, extinguishing populations of non-resistant clones (blue and grey lines) while offering an opportunity to proliferate to chemoresistance tumor cell populations (red line).
over the heterogenic tumor composition, discard cells holding inefficient genotypes, favor well-adapted clones and channel tumor microevolution. The consequence of this fast evolutionary process happening in the tumor is that from a mesoscopic, global perspective tumors perform extraordinarily well with respect to environmental perturbations, including therapies, and therefore are stable, robust long-lasting structures. This ability of tumors to persist despite internal (genetic) instability and external (immune survey, therapy, metabolic burden) perturbations is one of the key tumor features explaining the limited success of current cancer therapy approaches.2,23 From our perspective, it is actually a synergistic combination of local genetic instability and global tumor stability and persistence which characterizes intractable tumors (Fig. 2). The adaptability to changing microenvironments that has been observed in tumors is similar to the ability of systems behaving like strange attractors to persist despite strong perturbations. In this regard, our vision of tumor dynamics as strange attractors is similar, but not identical to that of cancer as an evolutionary and ecological process.3,10,11 2.3. Tumors as systems integrated by a plethora of interconnected and multi-level regulatory loops. A plausible origin for chaotic behavior The dynamics of tumors is governed by nested regulatory loops that play their role at different organization scales, from intracellular
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signaling to cell-to-cell communication and tumor–stroma interaction. Nonlinear regulatory motifs that are found often in tumors include positive and negative feedback loops, as well as coherent and incoherent feedforward loops.5,6 Just to mention one instance of these loops controlling tumor dynamics, positive feedback loops, non-linear regulatory structures that can induce signal amplification and multistability, appear in tumors at least at three different organizational levels: the intracellular signaling pathways, the cell-to-cell communication within the tumor, and the tumor–stroma interaction (Fig. 3). The literature contains evidence for many signaling regulatory systems in which a protein regulates transcriptionally or posttranscriptionally the activity and expression of an upstream protein in the fashion of a positive feedback loop (e.g., p53 regulating miR-34a in Yamakuchi et al.;24 E2F1 regulating EGFR in Alla et al.,25 NFkB regulating in COX-2 in Daniluk et al.26). For example, Alla and co-authors25 found that the transcription factor E2F1 activates the expression of EGFR, a plasma membrane receptor upstream of the signaling cascade promoting E2F1 stabilization and activity. In some tumors, E2F1 overexpression triggers its own signal amplification through this circuit. Tumor cells can secrete growth factors and other cytokines in an autocrine positive feedback loop manner, promoting their own carcinogenic activation and that of surrounding cancer cells.27–29 For example, Holterman and coauthors29 found that
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Fig. 3 Tumors as systems integrated by a plethora of interconnected and multi-level regulatory loops. The dynamics of tumors is governed by nested regulatory loops playing their role at different organization scales, from intracellular signaling to cell-to-cell communication and tumor–stroma interaction. This is the case of positive feedback loops, which appear in tumors at least at three different organizational levels: the tumor intracellular signaling networks, the cell-to-cell communication within the tumor (intra-tumor communication), and the interaction between the tumor and its microenvironment (tumor–stroma communication). The system integrated by EGFR and E2F1 is a case of positive feedback loop.25 These loops are regulatory structures in which the activation of a signalling event (here, activation of E2F1, E2F1*) positive regulates a signalling process upstream (activation of EGFR, EGFR*). Positive feedback loops can induce signal amplification and ultrasensitivity. In case of ultrasensitivity, the system is able to transform graded input signals (a smooth change in EGF concentration, EGF) into discrete all-or-none output (switch in concentration of E2F1*), creating actual input signal thresholds that determine the activation of the system.
deregulation of the transcription factor Ets-1 promotes secretion of transforming growth factor a (TGFa) in a variety of cancer cell lines. TGFa is a diffusible growth factor that promotes tumor cell proliferation in vivo and autonomous growth in culture and, when secreted by the tumor cells, elicit an autocrine positive feedback loop. Scaling up one level in the tumor structure, tumors cells can emit chemical signals aiming to reconfigure their microenvironment and recruit non-cancer cells, which in turn secrete growth factors and cytokines promoting growth, survival and/or aggressiveness of the cancer cells that ‘‘call’’ them.30–32 This sort of cytokine mediated tumor–stroma interaction fulfills the requirements to be considered a paracrine positive feedback loop. Tsuyada and colleagues31 found that cancer-associated fibroblasts produce and secrete higher levels of CCL2, a chemokine which stimulates cancer cells and cancer stem cells self-renewal. While CCL2 expression requires the prior activation of the STAT3 signaling in fibroblasts by cancer-secreted cytokines, the system operates like a chemokine-mediated positive feedback loop. Taken together, this multiplicity of nonlinear, multi-level regulatory motifs may be the actual origin of the chaotic behavior hypothesized here: it is known that complex systems containing multiple interconnected nonlinear regulatory loops can, under some conditions, display chaotic behavior.33 This is the case of multi-looped nonlinear negative feedback systems, in which perturbation of specific parameters of the system may transform periodic oscillations into chaotic behavior.13 2.4. Strong sensitivity to the initial conditions or why long-term tumor evolution is rather unpredictable Another key property of strange attractors held by tumors is the strong sensitivity to initial conditions: small differences in the
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features of two almost identical tumors (e.g., genetic traits, tumor composition or microenvironment) may lead to significantly different future properties and behavior (Fig. 4). This divergence between similar tumors actually increases with time, which indicates the longer the time, the more unpredictable the behavior of a tumor with respect to a similar one. This would be the case when comparing a patient’ tumor with a similar tumor model. This behavior is a direct consequence of the existence of the many nested non-linear regulatory systems operating in the tumor at different organization scales. Multistability in regulatory networks governing tumor cells sum up with the non-linear nature of the tumor cell-to-cell interactions and the tumor–stroma communication to create a dense forest of divergent trajectories for tumors that initially were genotypically and phenotypically similar. At the root of this sensitivity we find again the synergy between local genetic instability and global selective pressures, which make small initial differences in the genotype and composition of tumors able to provoke fast divergent tumor evolution, poor accuracy in the prognosis and inefficacy of proposed anti-cancer therapies in the long-term. We recently showed that small changes in the composition of a heterogeneous tumor, substantiated in terms of the genotype of critical signaling circuits, can change drastically the fate of the tumor when genotoxic drug treatment is applied.22
3. Consequences of the hypothesis for tumor understanding and therapy The hypothesis we here propose has interesting consequences in the way we envision tumor progression and how anti-cancer therapies could be designed and applied.
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Fig. 4 Strong sensitivity of tumor trajectories to initial conditions. Small differences in the features of two almost identical tumors may lead to significantly different future properties and behavior. This divergence between similar tumors actually increases with time: the longer the time, the more unpredictable the behavior of a tumor with respect to a similar one. Here, we plot the dynamics of two tumors (T1 and T2) with an almost identical composition (two subpopulations of tumor cells, C1 and C2 with different properties concerning chemosensitivity). When the T1 and T2 are subjected to a selective pressure (genotoxic stress), the initially negligible difference in composition between them gets quickly amplified over time (in tumor T2 the initially negligible population of C2 chemoresistant cells expands). Thus, the initially almost identical tumors get totally different fates (genotoxic stress abolishes T1, while T2 survives and further expands).
3.1. Tumor understanding requires integration of spatio-temporal scales The identification of tumors as strange attractors involves microscopic single-cell phenomena as the source of local instability and mesoscopic selective pressures as a source of global stability. Under these conditions, the true understanding of tumor biology requires methodological frameworks and experimental techniques able to link individual tumor cell genetics and tumor composition: individual cell genetics to account for the microscopic driving force of tumor evolution, while tumor composition to account for the dissipative processes that confer global stability and robustness to the tumor. 3.2. Investigation of tumor heterogeneity is essential in any successful approach towards cancer understanding Any analysis based on experimental data techniques ignoring the genetic heterogeneity of the tumors neglect the source of tumor instability, the driven force of tumor evolution and robustness. Thus, experimental techniques ignoring tumor heterogeneity are prone to make incomplete and misleading interpretations on tumor etiology, as well as wrong predictions on tumor therapy. Experimental techniques based on cell-sorting and single cell microscopy are therefore a must, and agent based models a good modeling approach to handle this feature. 3.3. Time-limited potential for predictions on tumor evolution and therapy As a consequence of the strong sensitivity of strange attractors to initial conditions, any prediction based on data about the current genetic composition and structure of a tumor has a very limited time-interval of validity. As we have explained, the fast divergence in the evolution of strange attractor tumors will change quickly the properties of the tumor analyzed and therefore the validity of any therapeutic assessment based on them.
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3.4.
Cancer therapies as double-edged swords
The sensitivity of tumor evolution with respect to initial conditions has necessarily an effect on the design and effectiveness of anti-cancer therapies, especially in therapies that are designed in consecutive steps. According to our hypothesis, the tumor properties will diverge fast under new selective pressures as that of therapeutic interventions. Thus, in combined multi-step therapies the application of the step-one therapy will alter the internal composition and dynamics of the tumor, in a manner that can undermine the efficacy of the step-two therapy. This effect is dramatic in case of therapies based on the use of genotoxic drugs. In surviving cancer cells, these therapies can increase genetic instability and DNA errors, which in turns may enrich tumor heterogeneity. This can confer the tumor with additional fuel to boost microevolution, overcome selective pressures and ultimately adapt to chemotherapy.34 It is not a minor question to think whether genotoxic drugs could be in some cases a bad first-choice treatment due to their ability to generate a wider genetic pool of cancer cells and promote potential chemoresistance.22,35
4. A scientific program to investigate tumors as strange attractors In this paper we hypothesized that tumor dynamics behaves as strange attractors from the perspective of their organization and dynamics. In this sense, genetic instability plays the role of the local-scale driving force providing the source of local instability, while the selective pressures operating over the tumor cell populations act as a mesoscopic dissipative process. When combined, both counteracting forces provide the tumor with global stability, robustness and plasticity, the ultimate origin of the remarkable ability of tumors to develop strategies to evade anti-cancer therapies.
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The dynamical feature of strong sensitivity to initial conditions makes the microscopic and mesoscopic properties of very similar tumors diverge over short time spans, reducing the accuracy of any long term predictions about the best long term therapeutic strategies made through data-based tumor analysis. Therapies inducing genotoxic stress, when not successful in providing fast and complete remission of the tumor, increase the genetic pool of the tumor, and provide additional fuel to boost microevolution, chemoresistance and aggressiveness. A scientific program taking into consideration this vision of tumors as strange attractors is possible. It requires the development of experimental and computational tools to link the microand mesoscopic perspective of tumor biology. From the data acquisition perspective, we will need more and better experimental techniques providing just-in-time multi-level quantitative data of the tumor. Seminal examples of these techniques are available.36–39 From a computational perspective, the aim is to develop models able to generate a dynamical genotype–phenotype mapping of tumors. Hybrid, multi-level mathematical models, integrating information from different spatio-temporal scales, seem the right tool to further develop in the coming years.8,22,40–42 In our opinion, this conceptual approach constitutes an advanced setup of the Cancer Systems Biology paradigm.6,43,44 Traditional contradicting hypothesis have supported that the origin of cancer is either genetic and resides in the microscopic world of genes and mutations within single cells, or it is emerging from a failure of tissue organization.45 However, in the light on new experimental results concerning the tumor– microenvironment interaction we know now that both hypotheses are not mutually exclusive.46 When the tumor–microenvironment interaction is considered, given cancer genotypes may have the ability to induce tissue remodeling, whereas certain environmental changes may as well generate a microenvironment favorable to increase the usually low predisposition for mutations.47 Under this condition, tumors become definitively complex, multiscale, and highly nonlinear systems, prone to be described under the dynamical systems theory as here proposed.
5. A concluding remark on other complementary (and not competing) hypothesis In this opinion paper we hypothesized that malignant tumor growth, progression and evolution share several of the features associated with strange attractors, a special regime for chaotic dynamical systems. In the last decades, a number of papers have associated features of chaotic systems to cancer.48–51 Some of these papers are devoted to the idea of assigning fractal dimension, a property often linked to chaotic behavior, to the ´ spatial features of tumor growth.48 In a series of papers, Bru and co-authors, hypothesized and confirmed experimentally that the spatial properties and the growth of some cell colonies and tumors are characterized by a fractal dimension.49,50 The spatial fractality of tumors is not considered in our hypothesis, although it could be complementary. Dinicola and coauthors,51
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Molecular BioSystems
made use of the terminology associated to non-linear dynamical systems to hypothesize that cell phenotypes are emergent properties arising from the integration of intracellular and microenvironmental components, interaction in which spatial features play an important role. Dalgleish52 envisioned the interaction between the tumor and the immune system from the perspective of the chaos theory and discuss how immunotherapy could modify the dynamical features of this interaction. In line with this, Letellier and coworkers53 analyzed a simple but highly nonlinear mathematical model of cell populations, which accounted for the interaction between host, immune and tumor cells. This model was able to generate for some model parameterizations several properties associated to chaotic systems, which is in agreement with our hypothesis. However, in our opinion the chaotic behavior can emerge as well in the dynamics of the complex intracellular regulatory networks regulating the interaction between host, immune and tumor cells. Going down to the gene networks involved in cancer emergence, Kauffman and co-authors54 proposed the existence of the so-called ‘‘cancer attractors’’. These cancer attractors would be abnormal configurations for the gene regulatory networks, phenotypically viable and proliferative, but not feasible for normal stem or differentiated cell types. In their opinion, cancer cells reach and get trapped in these abnormal attractors via mutations or epigenetic modifications. They think that the cancer attractor hypothesis can account for a number of experimental observations that seem to evade mechanistic explanation. This is the case, for example, of memory effects that can transform transient expression of a tumor promoting gene into permanent malignant phenotypes, which is explained in their hypothesis as the transition between two cancer attractors. This hypothesis is coincident in several aspects with the features associated to the existence of chaotic attractors. For example, they suggest that many of the properties of the tumors derive from the enormous complexity of networks involved in tumor initiation and progression. This complexity multiplies in an unpredictable manner the number possible phenotypic scenarios found in malignant tumor progression. From a completely different perspective, one could say that some of the features associated to tumor progression here linked to tumor attractors could be justified by assuming stochasticity at the single cell or at the level of tumor cell populations. A large amount of papers have been published in this direction in the last years. Oudenaarden and coworkers55 support the idea that critical steps in the gene expression process hold randomness, making the entire process fundamentally stochastic. According to them, this feature has important consequences in the regulation of intracellular networks, the triggering of phenotypic responses and ultimately can account for the cell-to-cell variability observed with single cell experimental techniques. However, some others have found that in many cases gene networks are rather robust to this noise source.56 From a totally different perspective, others have made extensive use of stochastic models to predict the fate of populations of tumor cells. They have analyzed the consequences of single cell random mutations in the context of tumor initiation and progression,3 and derived probabilistic
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models that predict the success of anticancer therapies and may assist in redesigning them.57 Even Kolmogorov,58 one of the founding fathers of the probability theory indicated that: ‘‘the possibility of using, in the treatment of a real process, schemes of well-determined or of only stochastically definite processes stands in no relation to the question whether the real process is itself determined or random’’. From a pragmatic point of view, in most of the cases, like the models to predict the output of medical treatments, the use of stochastic (or deterministic) models does not rely in fundamental mechanistic properties of the system modeled, but on the necessity to develop applicable phenomenological and problem-oriented models, which are actually valid under some assumptions. Thus, one does not have to think which model is the correct one, but rather which model is the appropriate one under the investigated scenario. In line with this, in Liao et al.33 it is suggested that in some cases stochastic representations can describe well chaotic systems even though the system is not fundamentally random, but deterministic. The underlying motivation for this apparent contradiction is that some properties, which can be associated both to stochasticity and deterministic chaos, can emerge in deterministic systems when they are composed by complex, large enough, interconnected and insufficiently determined biological interaction networks. That is, exactly the type of networks that we know is governing cancer progression! To conclude, the stochastic and the non-linear systems perspectives on the tumor dynamics can be perfectly integrated when one makes use of more complex theoretical frameworks and tools able to analyze the effect of stochastic noise in chaotic systems.59,60
Acknowledgements This work was supported by the German Federal Ministry of Education and Research (BMBF) as part of the projects eBio:miRSys [0316175A] and eBio:SysMet [0316171]. The authors thank the critical comments of Florian Wendland during the revision of the manuscript.
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34 C. C. Maley, B. J. Reid and S. Forrest, Cancer Epidemiol. Biomarkers Prev., 2004, 13, 1375–1384. 35 R. A. Gatenby, Nature, 2009, 459, 508–509. 36 W. Schubert, B. Bonnekoh, A. J. Pommer, L. Philipsen, ¨ckelmann, Y. Malykh, H. Gollnick, M. Friedenberger, R. Bo M. Bode and A. W. M. Dress, Nat. Biotechnol., 2006, 24, 1270–1278. ¨usser, M. Gossen, G. Dittmar and M. Selbach, 37 B. Schwanha Proteomics, 2009, 9, 205–209. 38 M. R. Schweiger, M. Kerick, B. Timmermann and M. Isau, Cancer Metastasis Rev., 2011, 30, 199–210. 39 M. Hidalgo, E. Bruckheimer, N. V. Rajeshkumar, I. GarridoLaguna, E. D. Oliveira, B. Rubio-Viqueira, S. Strawn, M. J. Wick, J. Martell and D. Sidransky, Mol. Cancer. Ther., 2011, 10, 1311–1316. 40 S. L. Spencer, R. A. Gerety, K. J. Pienta and S. Forrest, PLoS Comput. Biol., 2006, 2, e108. ¨ger and N. Grabe, ¨tterlin, C. Kolb, H. Dickhaus, D. Ja 41 T. Su Bioinformatics, 2013, 29, 223–229. 42 D. V. Guebel, U. Schmitz, O. Wolkenhauer and J. Vera, Mol. Biosyst., 2012, 8, 1230–1242. ¨thgen, 43 O. Wolkenhauer, C. Auffray, S. Baltrusch, N. Blu H. Byrne, M. Cascante, A. Ciliberto, T. Dale, D. Drasdo, ¨nther, D. Fell, J. E. Ferrell, D. Gallahan, R. Gatenby, U. Gu B. D. Harms, H. Herzel, C. Junghanss, M. Kunz, I. van Leeuwen, P. Lenormand, F. Levi, M. Linnebacher, J. Lowengrub, ¨fer, P. K. Maini, A. Malik, K. Rateitschak, O. Sansom, R. Scha ¨rrle, C. Sers, S. Schnell, D. Shibata, J. Tyson, J. Vera, K. Schu
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M. White, B. Zhivotovsky and R. Jaster, Cancer Res., 2010, 70, 12–13. O. Wolkenhauer, C. Auffray, R. Jaster, G. Steinhoff and O. Dammann, Pediatr. Res., 2013, 73, 502–507. C. Sonnenschein and A. M. Soto, Semin. Cancer Biol., 2008, 18, 372–377. L. A. Gilbert and M. T. Hemann, Cell, 2010, 143, 355–366. I. P. Witz, Adv. Cancer Res., 2008, 100, 203–229. E. Ahmed, Int. J. Theor. Phys., 1993, 32, 353–355. A. Bru, S. Albertos, J. Luis Subiza, J. L. Garcia-Asenjo and I. Bru, Biophys. J., 2003, 85, 2948–2961. ´, D. Casero, S. de Franciscis and M. A. Herrero, Math. A. Bru Comput. Modell., 2008, 47, 546–559. S. Dinicola, F. D’Anselmi, A. Pasqualato, S. Proietti, E. Lisi, A. Cucina and M. Bizzarri, OMICS, 2011, 15, 93–104. A. Dalgleish, QJM, 1999, 92, 347–359. C. Letellier, F. Denis and L. A. Aguirre, J. Theor. Biol., 2013, 322, 7–16. S. Huang, I. Ernberg and S. Kauffman, Semin. Cell Dev. Biol., 2009, 20, 869–876. A. Raj and A. van Oudenaarden, Cell, 2008, 135, 216–226. ´, J. Timmer and M. Kollmann, L. Løvdok, K. Bartholome V. Sourjik, Nature, 2005, 438, 504–507. H. Haeno, M. Gonen, M. B. Davis, J. M. Herman, C. A. IacobuzioDonahue and F. Michor, Cell, 2012, 148, 362–375. A. Kolmogorov, Math. Ann., 1931, 104, 415–458. T. He and S. Habib, Chaos, 2013, 23, 033123. T. Kapitaniak, Chaos in Systems with Noise, World Scientific, 1990.
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OPINION
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Tumors as chaotic attractors Cite this: Mol. BioSyst., 2014, 10, 172
Svetoslav Nikolov,abc Olaf Wolkenhauerad and Julio Vera*e Malignant tumor growth, progression and evolution share several of the features associated with strange attractors, a special type of dynamical behavior of non-linear systems. In this framework, the genetic instability of tumor cells represents a local-scale driving force for instability that lets the tumor evolve with time, while selective pressures discarding inefficient cell clones play the role of a dissipative process, providing the tumor with global stability and robustness with respect to perturbations. This setup induces global stability combined with local instability, and is in our opinion the real source of tumor robustness and plasticity, and ultimately the origin of the remarkable resistance of many tumors to anti-cancer therapies. A scientific program taking into consideration the vision of tumors as strange attractors requires the development of experimental and computational tools to link the micro- and mesoscopic perspectives of tumor biology. We here develop the
Received 5th August 2013, Accepted 31st October 2013
idea of a tumor as a dynamical system, sharing characteristics with strange attractors and investigate several
DOI: 10.1039/c3mb70331b
the tremendous non-linearity of the biochemical regulatory circuits governing the individual fate and the
features of such systems, including: (a) local instability combined with global stability, a property inherent to interactions between tumor cells; (b) self-similarity at different levels; (c) and strong sensitivity to initial
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conditions in the composition and features of the tumor and its microenvironment.
1. Introduction: cancer as a multi-level dynamical system Cancer is a major health problem covering some 100 separate diseases with different risk factors and epidemiology, originating in different cell types and organs. If we focus on cancer manifested as solid tumors, they are characterized by accumulation of genetically abnormal cells featuring, individually or collectively, some or all of the following phenotypic traits: unlimited proliferation, evasion from endogenous and exogenous programmed cell death, independence from growth signals and/or insensitivity to antigrowth signals, metabolic reprogramming and invasiveness.1,2 The perspective that has driven oncologic research for decades is that of cancer as a genetic disease that arises from mutations in susceptible genes and subsequent somatic evolution.3 In the last few years this picture has been refined with the description of timedependent regulatory networks. These networks are rich in regulatory motifs that include positive or negative feedback as well as a
Department of Systems Biology and Bioinformatics, Institute for Informatics, University of Rostock, 18051 Rostock, Germany b Institute of Mechanics, Bulgarian Academy of Science, Acad. G. Bonchev Str., bl.4, 1113 Sofia, Bulgaria c University of Transport, Geo Milev str., 158, 1574 Sofia, Bulgaria d Stellenbosch Institute for Advanced Study (STIAS), Wallenberg Research Centre at Stellenbosch University, Stellenbosch, South Africa e Laboratory of Systems Tumor Immunology, Dept. of Dermatology, Faculty of Medicine, University of Erlangen-Nurnberg, Erlangen, Germany. E-mail: [email protected]
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coherent and incoherent feedforward loops, whose regulation is difficult to understand because their effect on the dynamics of the systems is often non-linear.4–7 Whether the origin of cancer is genetic and resides in the microscopic world of genes and mutations within single cells, or whether it is emerging from a failure of tissue organization, it actually manifests as a mesoscopic structure, the tumor. To make the problem more cumbersome, tumors are not simple aggregations of genetically abnormal cells, but they display an amazing genetic heterogeneity between the cells that compose them.8,9 Those cells are not isolated from the outer world, but rather establish complex interactions between them and with their microenvironment, which ultimately are expressed as selective pressures like hypoxia, lack of nutrients or apoptosis that govern the fate of the tumor as a whole.10,11 Thus, from a global perspective tumors are also complex dynamical systems and the natural language to analyze such systems is dynamical systems theory. This framework allows us to handle concepts like complexity, non-linearity, robustness, sensitivity and (in)stability of a system, which actually emerge also in tumors, the cancer cells and the regulatory networks governing them. In this paper, we develop the idea of a tumor as a nonlinear dynamical system that has some of the properties associated with strange attractors, a class of dynamical behavior displayed by some chaotic systems. These properties are: (a) local instability combined with global stability, a property inherent to the tremendous nonlinearity of the biochemical regulatory circuits governing the individual fate and the interactions between tumor cells; (b) self-similarity at different levels; and (c) strong sensitivity to initial conditions in the composition and features of the tumor and its microenvironment.
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From biological systems to strange attractors: a quick guide to dynamical systems theory Natural systems are ensembles of physical entities that interact with each other.12 Biology, rather than being an exception, is a realization of this idea: biological entities are organized in systems, from interacting proteins to the entire biosphere. When biological systems become structurally complex (e.g., too many proteins), their analysis evades direct reasoning and intuition, making mathematical modeling a necessary tool.5 There are many instances of mathematical models, but one very widespread class of models suitable to understand biochemical and cell population systems are ordinary differential equations. These models describe spatio-temporal changes of biological entities using equations with the following structure:
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X d C¼ Fi ðS; k; CÞ dt i In these equations, C are the time-dependent variables accounting for the biological entities whose evolution over time is analyzed (i.e., interacting proteins or tumor cell populations), Fi are the rate equations formalizing interactions between the biological entities, S are inputs (external stimulus affecting the system) and k are parameters (fixed numbers associated with given features like rate of proliferation or protein cleavage). With this kind of model, one can trace the evolution of the system properties over time, the so-called trajectories (Fig. 1A). Under some conditions, the trajectories evolve towards steady-state configurations of the system in which the properties of the system C stay constant (stability), while others keep the system changing constantly and generate instability (Fig. 1B). When biological systems are enriched with complex regulatory motifs like feedback and forward loops, changes in the value of critical biological parameters k induce sudden transitions between stable and unstable configurations of the system, the so called bifurcations (e.g. a mutation increasing the cell proliferation rate can transform a rather stable microscopic neoplastic lesion into a growing macroscopic tumor; Fig. 1C, see also ref. 4). Furthermore, systems containing multiple interconnected instances of these motifs can, under some conditions, display chaotic behavior. This behavior is characterized by a high sensitivity to initial conditions that makes the dynamics of the system appear irregular. This is the case of, for example, multi-looped negative feedback systems.13 Such systems must be described and analyzed through the use of chaos theory.14,15 A dynamical structure that was discovered using this theory are strange or chaotic attractors.16 An attractor is a configuration (a set of values for the variables) towards which the system evolves over time (Fig. 1D). An attractor can be a stable fixed point or steadystate if, in the absence of external stimuli, the system stays in this configuration once it achieves it. Attractors can also be stable limit cycles, when the values of the system variables are periodically repeated, leading to self-sustained oscillations. Finally, in a strange attractor the values of the system variables remain permanently confined in a delimited region of all the phase space. However, within that region the system will go through the same point more than once. This is a special kind of behavior detected in some nonlinear dynamical systems, characterized by two contradictory properties: from a global perspective they are stable structures, but locally they are unstable. Furthermore, the chaotic attractors display two other properties of chaotic systems: fractal structure or self-similarity (e.g., some properties of the system as a whole are similar to those displayed by a part of itself) and high sensitivity to initial conditions (e.g., the difference between two similar configurations of the system diverges exponentially over time). In strange attractors, global dissipative processes compensate destabilizing local driving forces and neutralize noise and perturbations to maintain the system in a global steady-state regime. In nature there are some cases of mesoscopic systems behaving as strange attractors. An example are Benard cells, which appear in the plane horizontal layer of fluids subjected to convection.17 Here, turbulent flow combines with the effect of gravity and thermal dissipation to create globally stable structures emanating from a microscopically unstable milieu.
while the selective pressures discarding inefficient cell clones play the role of a dissipative process, providing the tumor with global stability and robustness with respect to perturbations. 2.1. Local instability in tumors originated from genetic instability
Fig. 1
Tumors as dynamical systems.
The genetic instability of most or part of the cells constituting a tumor is one of the first proved hallmarks of the cancer18 and some authors indicate that genetic instability might be a prerequisite for tumors to develop.19,20 This process induces rapid changes in the microscopic landscape of the tumor, including fast clonal succession of tumor cells.21 This is a source of permanent instability, altering the tumor composition at the microscopic level and enriching with genetic variability the micro-ecosystem of the tumor. Thus, genetic instability of tumor cells is the biological translation ´lezof the local instability found in strange attractors (Fig. 2). Gonza Garcı´a and co-authors8 combined data from single-cell genotyping of tumor laser micro-dissections and mathematical modeling to show that genetic instability and mutator phenotypes provide a source for tumor heterogeneity, but also that the generated spatial heterogeneity in the tumor is maintained over time.
2. Hypothesis
2.2.
We support the idea that tumor dynamics display several of the features associated with strange attractors. The genetic instability of tumor cells is one source of instability, the local-scale driving force that keeps the tumor operating and evolving over time,
In our hypothesis, the selective pressures operating on the genetically heterogeneous tumor cell community play the role of dissipative processes. Hypoxia, lack of nutrients or growth factors, natural or therapy-related apoptotic signals, when operating
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Global stability in tumors induced by selective pressures
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Fig. 2 Tumors as systems with local instability and global stability. From a mesoscopic perspective, tumors are robust long-lasting structures that perform extraordinarily well with respect to environmental perturbations. However, genetic instability and fast clonal succession, hallmarks of cancer, make tumors microscopically unstable and prone to genetic variability in their composing tumor cells. In Vera et al.22 we used multi-level modelling to show how heterogeneous tumors, composed by tumor cells with different genetic signatures (red, blue and grey colored cells) evolve mechanisms, which make them resistant to chemotherapy. At the mesoscopic scale these tumors are robust structures resisting the therapy (left figure). At the microscopic level, the selective forces driven by the therapy change drastically the composition of the tumor, extinguishing populations of non-resistant clones (blue and grey lines) while offering an opportunity to proliferate to chemoresistance tumor cell populations (red line).
over the heterogenic tumor composition, discard cells holding inefficient genotypes, favor well-adapted clones and channel tumor microevolution. The consequence of this fast evolutionary process happening in the tumor is that from a mesoscopic, global perspective tumors perform extraordinarily well with respect to environmental perturbations, including therapies, and therefore are stable, robust long-lasting structures. This ability of tumors to persist despite internal (genetic) instability and external (immune survey, therapy, metabolic burden) perturbations is one of the key tumor features explaining the limited success of current cancer therapy approaches.2,23 From our perspective, it is actually a synergistic combination of local genetic instability and global tumor stability and persistence which characterizes intractable tumors (Fig. 2). The adaptability to changing microenvironments that has been observed in tumors is similar to the ability of systems behaving like strange attractors to persist despite strong perturbations. In this regard, our vision of tumor dynamics as strange attractors is similar, but not identical to that of cancer as an evolutionary and ecological process.3,10,11 2.3. Tumors as systems integrated by a plethora of interconnected and multi-level regulatory loops. A plausible origin for chaotic behavior The dynamics of tumors is governed by nested regulatory loops that play their role at different organization scales, from intracellular
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signaling to cell-to-cell communication and tumor–stroma interaction. Nonlinear regulatory motifs that are found often in tumors include positive and negative feedback loops, as well as coherent and incoherent feedforward loops.5,6 Just to mention one instance of these loops controlling tumor dynamics, positive feedback loops, non-linear regulatory structures that can induce signal amplification and multistability, appear in tumors at least at three different organizational levels: the intracellular signaling pathways, the cell-to-cell communication within the tumor, and the tumor–stroma interaction (Fig. 3). The literature contains evidence for many signaling regulatory systems in which a protein regulates transcriptionally or posttranscriptionally the activity and expression of an upstream protein in the fashion of a positive feedback loop (e.g., p53 regulating miR-34a in Yamakuchi et al.;24 E2F1 regulating EGFR in Alla et al.,25 NFkB regulating in COX-2 in Daniluk et al.26). For example, Alla and co-authors25 found that the transcription factor E2F1 activates the expression of EGFR, a plasma membrane receptor upstream of the signaling cascade promoting E2F1 stabilization and activity. In some tumors, E2F1 overexpression triggers its own signal amplification through this circuit. Tumor cells can secrete growth factors and other cytokines in an autocrine positive feedback loop manner, promoting their own carcinogenic activation and that of surrounding cancer cells.27–29 For example, Holterman and coauthors29 found that
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Fig. 3 Tumors as systems integrated by a plethora of interconnected and multi-level regulatory loops. The dynamics of tumors is governed by nested regulatory loops playing their role at different organization scales, from intracellular signaling to cell-to-cell communication and tumor–stroma interaction. This is the case of positive feedback loops, which appear in tumors at least at three different organizational levels: the tumor intracellular signaling networks, the cell-to-cell communication within the tumor (intra-tumor communication), and the interaction between the tumor and its microenvironment (tumor–stroma communication). The system integrated by EGFR and E2F1 is a case of positive feedback loop.25 These loops are regulatory structures in which the activation of a signalling event (here, activation of E2F1, E2F1*) positive regulates a signalling process upstream (activation of EGFR, EGFR*). Positive feedback loops can induce signal amplification and ultrasensitivity. In case of ultrasensitivity, the system is able to transform graded input signals (a smooth change in EGF concentration, EGF) into discrete all-or-none output (switch in concentration of E2F1*), creating actual input signal thresholds that determine the activation of the system.
deregulation of the transcription factor Ets-1 promotes secretion of transforming growth factor a (TGFa) in a variety of cancer cell lines. TGFa is a diffusible growth factor that promotes tumor cell proliferation in vivo and autonomous growth in culture and, when secreted by the tumor cells, elicit an autocrine positive feedback loop. Scaling up one level in the tumor structure, tumors cells can emit chemical signals aiming to reconfigure their microenvironment and recruit non-cancer cells, which in turn secrete growth factors and cytokines promoting growth, survival and/or aggressiveness of the cancer cells that ‘‘call’’ them.30–32 This sort of cytokine mediated tumor–stroma interaction fulfills the requirements to be considered a paracrine positive feedback loop. Tsuyada and colleagues31 found that cancer-associated fibroblasts produce and secrete higher levels of CCL2, a chemokine which stimulates cancer cells and cancer stem cells self-renewal. While CCL2 expression requires the prior activation of the STAT3 signaling in fibroblasts by cancer-secreted cytokines, the system operates like a chemokine-mediated positive feedback loop. Taken together, this multiplicity of nonlinear, multi-level regulatory motifs may be the actual origin of the chaotic behavior hypothesized here: it is known that complex systems containing multiple interconnected nonlinear regulatory loops can, under some conditions, display chaotic behavior.33 This is the case of multi-looped nonlinear negative feedback systems, in which perturbation of specific parameters of the system may transform periodic oscillations into chaotic behavior.13 2.4. Strong sensitivity to the initial conditions or why long-term tumor evolution is rather unpredictable Another key property of strange attractors held by tumors is the strong sensitivity to initial conditions: small differences in the
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features of two almost identical tumors (e.g., genetic traits, tumor composition or microenvironment) may lead to significantly different future properties and behavior (Fig. 4). This divergence between similar tumors actually increases with time, which indicates the longer the time, the more unpredictable the behavior of a tumor with respect to a similar one. This would be the case when comparing a patient’ tumor with a similar tumor model. This behavior is a direct consequence of the existence of the many nested non-linear regulatory systems operating in the tumor at different organization scales. Multistability in regulatory networks governing tumor cells sum up with the non-linear nature of the tumor cell-to-cell interactions and the tumor–stroma communication to create a dense forest of divergent trajectories for tumors that initially were genotypically and phenotypically similar. At the root of this sensitivity we find again the synergy between local genetic instability and global selective pressures, which make small initial differences in the genotype and composition of tumors able to provoke fast divergent tumor evolution, poor accuracy in the prognosis and inefficacy of proposed anti-cancer therapies in the long-term. We recently showed that small changes in the composition of a heterogeneous tumor, substantiated in terms of the genotype of critical signaling circuits, can change drastically the fate of the tumor when genotoxic drug treatment is applied.22
3. Consequences of the hypothesis for tumor understanding and therapy The hypothesis we here propose has interesting consequences in the way we envision tumor progression and how anti-cancer therapies could be designed and applied.
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Fig. 4 Strong sensitivity of tumor trajectories to initial conditions. Small differences in the features of two almost identical tumors may lead to significantly different future properties and behavior. This divergence between similar tumors actually increases with time: the longer the time, the more unpredictable the behavior of a tumor with respect to a similar one. Here, we plot the dynamics of two tumors (T1 and T2) with an almost identical composition (two subpopulations of tumor cells, C1 and C2 with different properties concerning chemosensitivity). When the T1 and T2 are subjected to a selective pressure (genotoxic stress), the initially negligible difference in composition between them gets quickly amplified over time (in tumor T2 the initially negligible population of C2 chemoresistant cells expands). Thus, the initially almost identical tumors get totally different fates (genotoxic stress abolishes T1, while T2 survives and further expands).
3.1. Tumor understanding requires integration of spatio-temporal scales The identification of tumors as strange attractors involves microscopic single-cell phenomena as the source of local instability and mesoscopic selective pressures as a source of global stability. Under these conditions, the true understanding of tumor biology requires methodological frameworks and experimental techniques able to link individual tumor cell genetics and tumor composition: individual cell genetics to account for the microscopic driving force of tumor evolution, while tumor composition to account for the dissipative processes that confer global stability and robustness to the tumor. 3.2. Investigation of tumor heterogeneity is essential in any successful approach towards cancer understanding Any analysis based on experimental data techniques ignoring the genetic heterogeneity of the tumors neglect the source of tumor instability, the driven force of tumor evolution and robustness. Thus, experimental techniques ignoring tumor heterogeneity are prone to make incomplete and misleading interpretations on tumor etiology, as well as wrong predictions on tumor therapy. Experimental techniques based on cell-sorting and single cell microscopy are therefore a must, and agent based models a good modeling approach to handle this feature. 3.3. Time-limited potential for predictions on tumor evolution and therapy As a consequence of the strong sensitivity of strange attractors to initial conditions, any prediction based on data about the current genetic composition and structure of a tumor has a very limited time-interval of validity. As we have explained, the fast divergence in the evolution of strange attractor tumors will change quickly the properties of the tumor analyzed and therefore the validity of any therapeutic assessment based on them.
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3.4.
Cancer therapies as double-edged swords
The sensitivity of tumor evolution with respect to initial conditions has necessarily an effect on the design and effectiveness of anti-cancer therapies, especially in therapies that are designed in consecutive steps. According to our hypothesis, the tumor properties will diverge fast under new selective pressures as that of therapeutic interventions. Thus, in combined multi-step therapies the application of the step-one therapy will alter the internal composition and dynamics of the tumor, in a manner that can undermine the efficacy of the step-two therapy. This effect is dramatic in case of therapies based on the use of genotoxic drugs. In surviving cancer cells, these therapies can increase genetic instability and DNA errors, which in turns may enrich tumor heterogeneity. This can confer the tumor with additional fuel to boost microevolution, overcome selective pressures and ultimately adapt to chemotherapy.34 It is not a minor question to think whether genotoxic drugs could be in some cases a bad first-choice treatment due to their ability to generate a wider genetic pool of cancer cells and promote potential chemoresistance.22,35
4. A scientific program to investigate tumors as strange attractors In this paper we hypothesized that tumor dynamics behaves as strange attractors from the perspective of their organization and dynamics. In this sense, genetic instability plays the role of the local-scale driving force providing the source of local instability, while the selective pressures operating over the tumor cell populations act as a mesoscopic dissipative process. When combined, both counteracting forces provide the tumor with global stability, robustness and plasticity, the ultimate origin of the remarkable ability of tumors to develop strategies to evade anti-cancer therapies.
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The dynamical feature of strong sensitivity to initial conditions makes the microscopic and mesoscopic properties of very similar tumors diverge over short time spans, reducing the accuracy of any long term predictions about the best long term therapeutic strategies made through data-based tumor analysis. Therapies inducing genotoxic stress, when not successful in providing fast and complete remission of the tumor, increase the genetic pool of the tumor, and provide additional fuel to boost microevolution, chemoresistance and aggressiveness. A scientific program taking into consideration this vision of tumors as strange attractors is possible. It requires the development of experimental and computational tools to link the microand mesoscopic perspective of tumor biology. From the data acquisition perspective, we will need more and better experimental techniques providing just-in-time multi-level quantitative data of the tumor. Seminal examples of these techniques are available.36–39 From a computational perspective, the aim is to develop models able to generate a dynamical genotype–phenotype mapping of tumors. Hybrid, multi-level mathematical models, integrating information from different spatio-temporal scales, seem the right tool to further develop in the coming years.8,22,40–42 In our opinion, this conceptual approach constitutes an advanced setup of the Cancer Systems Biology paradigm.6,43,44 Traditional contradicting hypothesis have supported that the origin of cancer is either genetic and resides in the microscopic world of genes and mutations within single cells, or it is emerging from a failure of tissue organization.45 However, in the light on new experimental results concerning the tumor– microenvironment interaction we know now that both hypotheses are not mutually exclusive.46 When the tumor–microenvironment interaction is considered, given cancer genotypes may have the ability to induce tissue remodeling, whereas certain environmental changes may as well generate a microenvironment favorable to increase the usually low predisposition for mutations.47 Under this condition, tumors become definitively complex, multiscale, and highly nonlinear systems, prone to be described under the dynamical systems theory as here proposed.
5. A concluding remark on other complementary (and not competing) hypothesis In this opinion paper we hypothesized that malignant tumor growth, progression and evolution share several of the features associated with strange attractors, a special regime for chaotic dynamical systems. In the last decades, a number of papers have associated features of chaotic systems to cancer.48–51 Some of these papers are devoted to the idea of assigning fractal dimension, a property often linked to chaotic behavior, to the ´ spatial features of tumor growth.48 In a series of papers, Bru and co-authors, hypothesized and confirmed experimentally that the spatial properties and the growth of some cell colonies and tumors are characterized by a fractal dimension.49,50 The spatial fractality of tumors is not considered in our hypothesis, although it could be complementary. Dinicola and coauthors,51
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made use of the terminology associated to non-linear dynamical systems to hypothesize that cell phenotypes are emergent properties arising from the integration of intracellular and microenvironmental components, interaction in which spatial features play an important role. Dalgleish52 envisioned the interaction between the tumor and the immune system from the perspective of the chaos theory and discuss how immunotherapy could modify the dynamical features of this interaction. In line with this, Letellier and coworkers53 analyzed a simple but highly nonlinear mathematical model of cell populations, which accounted for the interaction between host, immune and tumor cells. This model was able to generate for some model parameterizations several properties associated to chaotic systems, which is in agreement with our hypothesis. However, in our opinion the chaotic behavior can emerge as well in the dynamics of the complex intracellular regulatory networks regulating the interaction between host, immune and tumor cells. Going down to the gene networks involved in cancer emergence, Kauffman and co-authors54 proposed the existence of the so-called ‘‘cancer attractors’’. These cancer attractors would be abnormal configurations for the gene regulatory networks, phenotypically viable and proliferative, but not feasible for normal stem or differentiated cell types. In their opinion, cancer cells reach and get trapped in these abnormal attractors via mutations or epigenetic modifications. They think that the cancer attractor hypothesis can account for a number of experimental observations that seem to evade mechanistic explanation. This is the case, for example, of memory effects that can transform transient expression of a tumor promoting gene into permanent malignant phenotypes, which is explained in their hypothesis as the transition between two cancer attractors. This hypothesis is coincident in several aspects with the features associated to the existence of chaotic attractors. For example, they suggest that many of the properties of the tumors derive from the enormous complexity of networks involved in tumor initiation and progression. This complexity multiplies in an unpredictable manner the number possible phenotypic scenarios found in malignant tumor progression. From a completely different perspective, one could say that some of the features associated to tumor progression here linked to tumor attractors could be justified by assuming stochasticity at the single cell or at the level of tumor cell populations. A large amount of papers have been published in this direction in the last years. Oudenaarden and coworkers55 support the idea that critical steps in the gene expression process hold randomness, making the entire process fundamentally stochastic. According to them, this feature has important consequences in the regulation of intracellular networks, the triggering of phenotypic responses and ultimately can account for the cell-to-cell variability observed with single cell experimental techniques. However, some others have found that in many cases gene networks are rather robust to this noise source.56 From a totally different perspective, others have made extensive use of stochastic models to predict the fate of populations of tumor cells. They have analyzed the consequences of single cell random mutations in the context of tumor initiation and progression,3 and derived probabilistic
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models that predict the success of anticancer therapies and may assist in redesigning them.57 Even Kolmogorov,58 one of the founding fathers of the probability theory indicated that: ‘‘the possibility of using, in the treatment of a real process, schemes of well-determined or of only stochastically definite processes stands in no relation to the question whether the real process is itself determined or random’’. From a pragmatic point of view, in most of the cases, like the models to predict the output of medical treatments, the use of stochastic (or deterministic) models does not rely in fundamental mechanistic properties of the system modeled, but on the necessity to develop applicable phenomenological and problem-oriented models, which are actually valid under some assumptions. Thus, one does not have to think which model is the correct one, but rather which model is the appropriate one under the investigated scenario. In line with this, in Liao et al.33 it is suggested that in some cases stochastic representations can describe well chaotic systems even though the system is not fundamentally random, but deterministic. The underlying motivation for this apparent contradiction is that some properties, which can be associated both to stochasticity and deterministic chaos, can emerge in deterministic systems when they are composed by complex, large enough, interconnected and insufficiently determined biological interaction networks. That is, exactly the type of networks that we know is governing cancer progression! To conclude, the stochastic and the non-linear systems perspectives on the tumor dynamics can be perfectly integrated when one makes use of more complex theoretical frameworks and tools able to analyze the effect of stochastic noise in chaotic systems.59,60
Acknowledgements This work was supported by the German Federal Ministry of Education and Research (BMBF) as part of the projects eBio:miRSys [0316175A] and eBio:SysMet [0316171]. The authors thank the critical comments of Florian Wendland during the revision of the manuscript.
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