Is cancer really a game?

A couple of weeks ago a post here on TheEGG, which was about evolutionary game theory (EGT) and cancer, sparked a debate on Twitter between proponents and opponents of the idea of using EGT to study cancer. Mainly due to the limitations inherent to Twitter the dialogue fizzled. Instead, here we are expanding ideas in this guest blog post, and eagerly await comments from the others in the debate. The post is written by Philip Gerlee and Philipp Altrock, with some editing from Artem. We will situate the discussion by giving a brief summary of evolutionary game theory, and then offer commentary and two main critiques: how spatial structure is handled, and how to make game theoretic models correspond to reality.

When we speak of evolutionary game theory (EGT) we mean a dynamical system, typically the replicator equation. This system describes the changes over time in the relative sizes of sub-populations within one population of fixed size. Why would these sizes change at all? Well, the changes are driven by mutations and reproduction. Especially the reproduction part can be influenced by interactions between all, or some individuals. This concept is very general, as it encapsulates simple growth where the winner is always the fastest grower, as well as growth conditioned on interactions. For the latter, examples are predators and prey, or host and parasites, in which the growth of one sub-population clearly changes with the (relative) size of the other.

In summary, there are three classical assumptions in EGT. First we deal fixed population size; for every new offspring, an individual has to die [but see the ecological public goods game for a relaxation]. Second, one considers an a priori fixed number of sub-populations with all individuals identical within (often described as strategies or types). Typically no new types are allowed to spontaneously emerge [but see the discussion of best-response dynamics in section 3.3 of Hofbauer & Sigmund, 2003]. Third, interactions between types lead to reproductive advantage or disadvantage. In evolution, this currency is called fitness. It broadly defines a given type’s contribution to the pool of offspring in the future.

Can we ignore spatial structure?

Another classical assumption is that the population is well-mixed, or inviscid in Artem’s terminology. This concerns the individual interactions, which are linked to reproductive success (fitness). Often we assume that everyone interacts with everyone else. Then we do not have to care about where individuals are located in space, or if they managed to even interact at all before they reproduce and die. We only worry about the total number of individuals of a given type and say that a given individual interacts with all of them, somehow, and then reproduces. In addition the time-scale of interactions is taken to be infinitely faster than the scale of reproductive events.

There are of course many extensions of classical EGT that deal specifically with the assumption of well-mixed interactions. These extensions add spatial or social structure to the evolutionary game (see e.g. Santos & Pacheco 2005,Lieberman 2005, Szabó & Fáth 2007, and Ohtsuki 2009). Interestingly, this may change the way fitness depends on the number of individuals of each type. In these models it is assumed that offspring are always placed in the immediate vicinity of their parent [for an alternative, see Ohtsuki, Nowak, & Pacheco (2007)]. However, it crucially matters whether an individual dies to leave an empty spot free for invasion by another’s offspring from its neighborhood, or whether an offspring is produced first, which then replaces a randomly chosen individual (potentially its parent). These two processes can lead to very different outcomes (Maciejewski et al. 2014). The impact of the ratio of interactive and reproductive time scales was investigated by Roca & Cuesta (2006; for a survey, see Roca, Cuesta, & Sanchez, 2009), and the sample size of interactions by Hilbe (2010) — both were shown to influence the evolutionary outcome.

Central to EGT is the payoff table, which determines the fitness of each strategy in the presence of other strategies (including its own). This means that EGT focuses solely on frequency-dependent selection — the fact that fitness changes with the relative size of other sub-populations. It typically ignores density-dependent selection, where fitness can change with the absolute sizes of other sub-populations. In addition, one can easily imagine the importance of an external resource, which limits growth due to a carrying capacity. Of course a single mathematical or computational model that attempted to encompass all these things would be very complicated (consisting of many parts), as well as complex (the whole would be more than the sum of its parts). To model is to prioritize, and the practitioners of EGT have decided to focus on frequency-dependent selection.

In EGT-models of cancer the interactions, and in extension the payoffs, are often assumed to be mediated by some diffusible factors. These can be growth factors (Archetti, 2013), or detrimental waste products (Basanta et al., 2008). Since these factors also decay over time, and are removed by the vasculature, each cell producing factors will only affect cells in its spatial vicinity. This is, of course, a form of interaction that EGT would attempt to model. The question is how large this radius of interaction is. If it is on the order of the spatial extent of the whole population, then a well-mixed assumption might be reasonable. But if the radius of interaction is considerably smaller than the typical population radius, then the well-mixed assumption quickly becomes questionable. Recent experimental results by Marusyk et al. (2014) have shown that the presence of a specific growth factor producing subclone can alter the growth rate of the entire tumor. This suggests that at least in this case the radius of interaction can be considerable with respect to the population radius.

One might also argue that the population is well-mixed since cancer cells are highly motile and move around in the tumor. This would then yield interactions with a considerable number of cells and hence provide a rational basis for the well-mixed assumption. A back of the envelope calculation yields the following:

The cell cycle of a tumor cell is often taken to be 24 hours, but this is probably an underestimate for in vitro conditions. Let’s say 100 hours. The typical size of a tumor is roughly 10 cm. The mean displacement (i.e. square root of mean square displacement) after 100 hours is approximately 100 micrometer (Wu et al., 2014), i.e. 0.1% of the assumed linear size. Note that the migration data is taken from single cells moving at low density, and hence that the rate of migration in a tightly packed tumor is probably much smaller. This result suggests that migration contributes little towards “mixing” the population.

But do we really care how well-mixed the population is? Does it matter for the dynamics of the game? Yes, these things matter greatly. It has been established that local interactions can fundamentally shift the dynamics of a game. For example, in the “prisoner’s dilemma” game the well-mixed case always gives rise to defection as the dominant strategy, while in a spatial extended population cooperation can persist, and coexistence of strategies is observed (Roca, 2009). This effect is also seen in the Rock-Scissors-Paper game, and more importantly also in an actual realization of the game in a bacterial population where the interactions are mediated by a diffusible toxin (Kerr, 2002). For clarity, a game mediated by diffusible factors shows co-existence in a spatial setting and dominance by a single strategy in the well/mixed case. To model the spatial setting correctly we need a model that can account for space in some way or another (looking at pair correlations provides a powerful middle-ground, see e.g. van Baalen, 1998 and the extensive discussions of Ohtsuki & Nowak (2006) on TheEGG here and here).

This suggests an interesting direction of research, in which one develops and investigates agent-based models that can interpolate between completely local, and global interactions. Then, fitness would be determined by interactions with a certain neighborhood or with the entire population. At what range of interaction does the well-mixed assumption, or mean-field description, breakdown? At half the population size? Or at a couple of cell sizes? Or does it depend on the fitness landscape? We really need to know!

Fitting theory to reality

In cancer biology, the different sub-populations are tumor cells of different kind, cells of the tumor environment, immune cells, and some (or many) others. One can well imagine that there might be a multitude of cell-to-cell interactions, just like in a typical EGT model. But how are we to say whether frequency-dependent selection plays an important part in tumor evolution in the first place? In order to answer this question we need to know how well an EGT model fits actual data.

The follow-up question then becomes: what parameters do you have to measure in potentially growing populations that are experiencing frequency-dependent selection? For constant selection between two cell populations you need two growth rates. By normalization this leads to one free parameter, the ratio of these rates. If you want to know how frequency-dependent selection plays out, you need at least two more measurements: one to describe how type I interacts with type II, and one for the complementary ‘reaction’. Naively, adding frequency dependence we have hence moved from a 1-dimensional parameter space to a 3-dimensional one. Theoretically, this may and has been simplified to a 2-dimensional parameter space, since again, we might normalize the two new parameters and measure them relative to each other.

From an experimental population dynamics point of view, two things remain problematic. First, the normalization procedure is a theoretical concept that allows us to capture the qualitative behavior. If there is an equilibrium, it is not changed by this normalization. However, the time it might take the system to come close to the equilibrium is affected. Especially in cancer, when we want to examine a tumor cell populations dynamics, this time might be a highly relevant observable. Second, another tacit assumption is that the interactions between the two types is linear. Linearity means that adding a certain amount of type II cells will only affect the system directly proportional to that amount. It might turn out be problematic to assume linearity. Experimental verification or falsification can only be done measuring both interaction parameters. This is why one might truly consider the two-type of cell system with frequency dependent selection as a problem of at least three free parameters.

Now, how many more parameters do you need (to measure) if you want to add a third type of cell? Five: one for each of the first two types describing their interactions with the third, two parameters for type III describing its interactions with the first two, and the third type’s autonomous growth rate. For a connection of the frequency-dependent fitness values of three types of cell one may need eight parameters. And not only that; these eight parameters describe only the intrinsic fitness changes. Neither do they include spatial aspects of the interactions, nor do they directly capture external (physical) circumstances. You may ask yourself here how one would ever accurately measure and confirm these interaction parameters, or be able to falsify the underlying assumptions of linear frequency dependent selection. Clever experimental design under the awareness of the many potential pitfalls that the standard theory rarely talks about will be crucial.

Is the study of evolutionary games a purely intellectual enterprise or do we hope to learn something new about cancer? If we hope to achieve the former then simple mean-field models are acceptable, or even necessary for deeper theoretical understanding. In the latter case, if we want to describe the dynamics of cancer we must first make sure that EGT-models are the appropriate tool. Is it possible to fit a three species EGT-model with its at least seven parameters to in vitro or in vivo data? Or should these models rather be viewed as tools that sharpen our intuitive reasoning — heuristics in Artem’s terminology — about cellular interactions and their influence on tumor evolution, as a means to visualize and communicate the perceived importance of interactions in tumor biology. What if, at some point, our ideas about EGT and cancer make it into lab or clinic? If connected with data from cell lines or patients, are we to apply a simpler modeling paradigm, or stick with EGT? We might be better off revising our ideas about the applicability of EGT to cancer already at this premature stage.

References

Archetti, M. (2013). Evolutionary game theory of growth factor production: implications for tumour heterogeneity and resistance to therapies. British Journal of Cancer, 109(4): 1056-1062.

Basanta, D., Simon, M., Hatzikirou, H., & Deutsch, A. (2008). Evolutionary game theory elucidates the role of glycolysis in glioma progression and invasion. Cell Proliferation, 41(6): 980-987.

Beckman, R. A., Schemmann, G. S., & Yeang, C. H. (2012). Impact of genetic dynamics and single-cell heterogeneity on development of nonstandard personalized medicine strategies for cancer. Proceedings of the National Academy of Sciences, 109(36): 14586-14591.

Hilbe, C. (2011). Local replicator dynamics: a simple link between deterministic and stochastic models of evolutionary game theory. Bull Math Biol, 73: 2068-2097.

Hofbauer, J., & Sigmund, K. (2003). Evolutionary game dynamics. Bulletin of the American Mathematical Society, 40(4): 479-519.

Kerr, B., Riley, M. A., Feldman, M. W., & Bohannan, B. J. (2002). Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors. Nature, 418(6894): 171-174.

Lieberman, E., Hauert, C., & Nowak, M. A. (2005). Evolutionary dynamics on graphs. Nature, 433(7023): 312-316.

Maciejewski, W., Fu, F., & Hauert, C. (2014). Evolutionary game dynamics in populations with heterogenous structures. PLoS Computational Biology, 10(4): e1003567.

Marusyk, A., Tabassum, D.P., Altrock, P.M., Almendro, V., Michor, F., & Polyak, K. (2014). Non-cell-autonomous driving of tumour growth supports sub-clonal heterogeneity. Nature, 514 (7520), 54-8 PMID: 25079331

Ohtsuki H, & Nowak MA (2006). The replicator equation on graphs. Journal of Theoretical Biology, 243(1): 86-97

Ohtsuki, H., Nowak, M. A., & Pacheco, J. M. (2007). Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs. Physical Review Letters, 98(10): 108106.

Ohtsuki, H., Iwasa, Y., & Nowak, M. A. (2009). Indirect reciprocity provides only a narrow margin of efficiency for costly punishment. Nature, 457(7225): 79-82.

Roca, C. P., Cuesta, J. A., & Sánchez, A. (2006). Time scales in evolutionary dynamics. Physical Review Letters, 97(15): 158701.

Roca, C. P., Cuesta, J. A., & Sánchez, A. (2009a). Effect of spatial structure on the evolution of cooperation. Physical Review E, 80(4): 046106.

Roca, C. P., Cuesta, J. A., & Sánchez, A. (2009b). Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics. Physics of Life Reviews, 6(4): 208-249.

Romero, P. A., & Arnold, F. H. (2009). Exploring protein fitness landscapes by directed evolution. Nature Reviews Molecular Cell Biology, 10(12): 866-876.

Santos, F. C., & Pacheco, J. M. (2005). Scale-free networks provide a unifying framework for the emergence of cooperation. Physical Review Letters, 95(9): 098104.

Szabo, G., & Fath, G. (2007) Evolutionary games on graphs. Physics Reports, 446(4): 97–216.

Van Baalen, M., & Rand, D. A. (1998). The unit of selection in viscous populations and the evolution of altruism. Journal of Theoretical Biology, 193(4): 631-648.

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About Philip Gerlee
I do mathematics for biological systems. Mainly cancer, but also microbial evolution and ecosystem dynamics. Currently assistant professor at Chalmers University of Technology, Gothenburg, Sweden.

20 Responses to Is cancer really a game?

  1. Jon Awbrey says:

    Cancer is the body’s attempt to solve a problem by brute-force branch while failing to bound?

  2. Sinead says:

    Cancer is the result of a cell’s quest for immortality.

  3. It is great to have a forum in which to discuss topics like this so thanks to Philip & Philip for expressing their views. As someone that has been working on game theoretical models of cancer for a few years, one could expect me to disagree with them. I don’t. At least I agree with a big part of their assesment. Somewhat less with the conclusions.

    I still remember one of my first presentations on game theoreitcal models of cancer where I mentioned three aspects that are hard to study with game theory:

    1. Spatial issues
    2.Population issues
    3.Quantitative isses

    I would go on detail over these three but P&P have done a good work in describing not only how they constitute a limitation of many game theoretical models.
    I would like only to qualify that many of these limitations can be overcome. Even if at the expense of making these models more complicated. One example is the work of Tom Vincent, Joel Brown and Bob Gatenby where they consider continuum ranges of phenotypes in their games.

    I do not think I need to work too hard to defend that mathematical models based on game theory can be of use in cancer research. The reason is the growing body of literature in the field. Because of researchers like Tomlinson, Bach, Pacheco, Dingli, Brown, Gatenby, Csikasz-Nagy, Archetti, Austin, etc, we know of the power of game theoretical models to tease apart key aspects of cancer evolutionary dynamics. Their work has showed us that we can learn new things about cancer with simple models.

    It is true that most of these examples are heuristic. Cancer Research is a field with too many “theories” and too few people making sure these theories are rigorous. Heuristic approaches are necessary and will be for a long time. This leads me to the title of the post. Titles can be curious things. Like Darwin’s “On the origin of species”, P&P’s post’s title has little to do with the content of the post. This is a shame because whether cancer is a game or not is a relevant element of this conversation. It is my belief, and the assumption my models make, that cancer IS a game. In the sense that cancer follows Darwinian evolution where the interactions between different cells drive the all important selection, I believe that cancer is indeed a game. Understanding these interactions is thus, for me, of fundamental importance and heuristic models are very useful for these endevours.

    Does this mean that evolutionary game theory is the only mathematical tool we need? Certainly not: cancer is a game but many tools can be used to understand how the interactions between different cell types and the physical microenvironment drive the way it grows and evolves.
    Simple games are analytically tractable and what we learn from them can be used to produce more complex computational models where, for instance, the role of space can be better understood.

    Furthermore, it might be worth thinking about the way we think about models. We can spend a lot of time thinking about all the limitations that a modelling approach might have (space, population and so on) but modelling is aboutexplaining as much as we can with the simplest model. Oftentimes an experimental collborator will ask me why I did not include something they consider important in a model. I do not think I am alone as a mathematical modeller in struggling to balance the complexity of what we study with the simplicity we hope our models should have. Almost any feature you can think of will be dear to some researchers but surely a mathematical model that includes explicitly every aspect of a cancer is neither practical nor desirable.
    It is difficult to develop predictive models of evolution but that is the challenge if we want to understand cancer growth and treatment resistance. If we do not get our qualitative understanding right first then there is little reason to jump into more complex and labour intensive models.

    • Philip Gerlee says:

      Thanks for your reply. I agree that we need many different mathematical tools to study cancer. The fact that many articles have been written about EGT applied to cancer does, however, not make it an appropriate tool. A lot has been written about e.g. string theory, yet is has no empirical connection to observed fact.

      What I’m asking for is empirical data that backs up the assertion that frequency based selection (in a fixed, infinite population) dominates the evolutionary process inside a tumour.

      • I cannot produce this data on my own Philip, I just do modelling. It will take time since most experimentalist do not think experiments lile those would be exciting to them. As theoreticians though, we can try to test the plausability of these ideas using existing results (like we did with the double bind model) and suggest exciting hypotheses that might lead to interesting experiments that might lead to data that could motivate more quantitative games. That’s what I am trying to do.

  4. Robert Vander Velde says:

    “Recent experimental results by Marusyk et al. (2014) have shown that the presence of a specific growth factor producing subclone can alter the growth rate of the entire tumor. This suggests that at least in this case the radius of interaction can be considerable with respect to the population radius.” It seems that non-spatial models are unusually suited to tumors in this case. In the case of the prisoner’s dilemma cooperators cannot survive in well mixed populations because defectors get the same benefits as cooperators, and if Marusyk et al. are correct this is true for tumors also (at least when it comes to public goods).

    As to your argument in general, I’d like to respond to your three “assumptions” in terms of the definition of evolutionary game theory. This is how I’d define evolutionary game theory: “evolutionary game theory is a model of evolution using frequency dependent selection”. Of your assumptions only the last one is necessary by definition and models have been created which violate the first two (Wakano et al., 2009) (Hofbauer, 1985). Furthermore these two assumptions constitute a problem with most models, since most models assume constant population size and the possible phenotypes in most models are limited by the need for simplicity and/or the researcher’s imagination. It would be great to see more models with varying population size since the assumption that one “individual” is instantly replaced by another is pretty ridiculous in any situation and especially in cancer, but I don’t think the problem is unique to evolutionary game theory. It would also be nice to see models whose phenotypes are limited by the laws of physics and time (if such a model is even possible), but again this problem is not unique to evolutionary game theory. Therefore, if one wants to criticize evolutionary game theory’s applicability to cancer these assumptions shouldn’t be the focus. Instead I ask this question: Does the frequency of a phenotype affect the fitness of a cell in the tumor? I think the answer is yes, although absolute numbers might matter more, in which case a modified evolutionary game theory whose definition depends on absolute numbers is more appropriate.

    One problematic aspect of evolutionary game theory which you seem to touch on is that small changes in the model lead to radical changes in the outcome (eg. Spatial structure allows cooperation). This makes it difficult to approach the “truth” as you increase the realism of the model. This makes evolutionary game theory a problematic tool, but again I think considering frequency dependent selection is important and you wouldn’t reject a uniquely representative function that doesn’t have a limit as x->infinity just because it’s inconvenient.

    You might get some agreement with David and Artem about how problematic testing evolutionary game theory in tumors is. But I think a rough idea can be garnered by looking at tumor tissue with only two initial strategies/phenotypes (by taking two subclones from a tumor and then growing them together but apart from the other subclones). Unfortunately biology is messy and, with mutations in DNA proofreading pathways, cancer is even messier, other strategies will eventually arise due to mutation. But just like mathematical models, biological experiments are simplifications. In the end I think they’re worth doing. In any case we need create the models before we can test there applicability to the lab.

    Wakano, J. Y., Martin A. Nowak & Hauert, Ch. (2009) “Spatial Dynamics of Ecological Public Goods”,Proc. Natl. Acad. Sci.

    J. Hofbauer (1985) “The Selection Mutation Equation”, Journal of Mathematical Biology

  5. Philip Gerlee says:

    Great to see more people joining the discussion. What I’m worried about is building an understanding of tumour evolution (based on frequency dependent selection), that we communicate to cancer biologists, that is possibly fundamentally flawed. I also think that frequency dependent selection might be an important factor in tumour evolution, but before I see empirical evidence that this is so, I’m reluctant to communicate these ideas as if they were true.

    Don’t get me wrong, I definitely think there is room for speculative modelling in the spirit of: “let’s assume this mechanism and see what follows logically from it”, but I also think that caution is needed.

    As to your second point about adding more complexity, I find it very problematic that the prediction of a model does not converge as it becomes more realistic. How are we to know when the prediction is to be trusted?

    • I think with this we are getting to the crux of the matter. It’s frequency-dependent selection not just something that happens (we probably agree on that) but a KEY driver?
      This is the hypothesis that drives my work and I believe (and from his reply, so does Robert) that is very consistent with what we know about tumours, their heterogeneity, the role of the factors they produce as public goods, etc.
      But you are right, mathematical models should be used to compare evolutionary outcomes of frequency-dependent and non-frequency dependent selection, see what the two difference assumptions lead to and validate.

  6. Marc Harper says:

    I don’t have a horse in this race but I’ve made some observations from your debate thus far that may be useful.

    There seems to be significant disagreement here on what constitues “Evolutionary Game Theory”, as some see it as requiring an infinite or fixed population size, a fixed or non-innovative set of phenotypes, frequency dependent fitness, etc. There are many contexts in which games are used in population models — in well-mixed finite populations, via the Moran or Fermi processes, on finite graphs — sometimes static, sometimes with active linking and active node addition / deletion (so a non-fixed population size), with continuous trait spaces (which goes back to Cressman’s work from at least 2004), with non-fixed phenotypes [1], and many other variations. Perhaps “Evolutionary Dynamics” is a better name for the field now.

    Nevertheless, in all of the above contexts, game theory is often used just a convenient way to define a fitness landscape, and of course landscapes can be non-linear or not based on a game matrix. I’m not sure if I understand what “Is cancer really a game?” is even asking. Regardless, in each context there is typically a region of parameter space in which cooperation is stable for the prisoner’s dilemma, so it’s not generally true that e.g. merely adding spatial structure means that cooperators cannot survive. (I am also not sure that the claims of Marusyk et al necessarily requires a spatial model, and it seems similar to [2].) Whether cooperation is viable depends heavily on the specific spatial structure, and for a given landscape, it’s often not hard to cook up a weird graph that yields results differing from the well-mixed population behavior. But it’s also often the case that the outcomes are quite similar qualitatively to the outcomes of the vanilla replicator equation or the Moran process.

    So I see some possible strawmen being argued:
    (1) That EGT is unsuitable because the replicator equation or makes some unrealistic or unverified assumptions in regards to cancer
    (2) Relaxing some of these assumptions leads to models that cannot model cancer because of

    … and I think that cancer is not well-defined enough in this discussion (there are many types and modes of cancer). It seems like you all may be converging on “frequency depenence” of the fitness landscape as the key feature of evolutionary game theory, but perhaps “population state dependent” or “population distrbution dependent” are better terms (since the population frequencies may cause information loss in a given model, e.g. for a non-constant finite population size).

    So perhaps the question should be “Are the fitness landscapes of population dynamics of [insert type] cancer/tumor dependent on the population state/distribution?” It seems like there is a lot of agreement that the answer to this is yes, since the biology suggests so (e.g. Marusyk et al) and if not then the possible dynamical behaviors are rather limited. If we include the rest of the body outside of the tumor, then it is very much the case that the population dynamics are dependent on the other cells in the body (since the tumor needs blood, oxygen, etc. and will also die if the host dies).

    The other charge levied against the utility of EGT for studying cancer — and the best IMO of all those on this page so far — is that there are few studies applying or fitting EGT models to real data. Part of the issue is the difficultly of obtaining suitable population data, i.e. many snapshots of the population distribution of real tumors over time. Population time series can be hard to obtain for plants and animals (though see [4]) even without including spatial structure variations. It’s hard to imagine how to gather such data without damaging or destroying a tumor (you would need e.g. a very accurate non-invasive scan done regularly).

    Many authors simply try to generate results that match the qualitative features of cancer or the distribution of some trait or feature, and often there is very good agreement. New such studies arrive all the time [2] [3]. Only recently with the advent of high-throughput genomics are data sets capable of validating models really starting to be created. Human population genetics is a good example where large data sets are starting to allow certain models to be tested (e.g. neutral evolution and exponential growth models of genomic variation). Such results partially mitigate the lack of data issue in my opinion, but at some point one has to consult reality for the validity of one’s models.

    Lastly, this debate has partially occurred in the literature already, see e.g.
    Dingli et al, “Reply: Evolutionary game theory: lessons and limitations, a cancer perspective” (2009) http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2795434/
    Also, there is a recent special issue on the theme ‘Game Theory and Cancer’ that you all may find of interest [5], which includes such titles as:
    “The ecology of cancer from an evolutionary game theory perspective” and “Evolutionary game theory for physical and biological scientists. Training and validating population dynamics equations”.

    [1] Huang et al, “Emergence of stable polymorphisms driven by evolutionary games between mutants” http://www.nature.com/ncomms/journal/v3/n6/full/ncomms1930.html

    [2] Bickel et al, “Evolution of Proliferation and the Angiogenic Switch in Tumors with High Clonal Diversity” http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0091992

    [3] Kianercy et al, “Critical transitions in a game theoretic model of tumour metabolism”, http://rsfs.royalsocietypublishing.org/content/4/4/20140014.full

    [4] “The Global Population Dynamics Database” http://www3.imperial.ac.uk/cpb/databases/gpdd

    [5] http://rsfs.royalsocietypublishing.org/content/4/4.toc

    • Philip Gerlee says:

      The unifying thought, it seems to me at least, in EGT is the reliance on frequency dependent selection to explain evolutionary outcomes. Of course there are exceptions, such as Novak et al. [1] recent contribution, where they assume that interactions affect carrying capacity, but a vast majority of EGT applied to cancer relies on frequency-dependence. In the light of this, what we are asking the following:

      1. Is frequency dependence a driving force in tumour evolution?

      2. If so, are models that assume infinite and well-mixed populations, i.e. the replicator equation suitable?

      3. How do we fit such models to data?

      As to (1) I would say that there is some data pointing in this direction, but not enough to motivate most EGT-models of cancer. As suggested in the original post (2) could even be investigated on a purely computational basis. In relation to (3) you highlight the experimental difficulties of measuring subclone abundance over time in a growing tumour. Given the possibilities of labelling cells and the advanced imaging techniques that exist I don’t see any real obstacles. For example cells expressing different phenotypes could be fluorescently labelled and grown inside a collagen matrix or even subcutaneously on a mouse.

      You also bring up two examples of papers that match qualitative features of the system at hand. The paper by Bickel et al. is well outside the scope of this discussion. It is really a multi-scale model (sub-cellular, tissue and evolutionary components) onto which adaptive dynamics is applied. The paper by Pienta et al. on the other hand is exactly the kind of study we are critical of. The “game” between the cells is taken for granted, and when the simple game is shown to disagree with data the authors invoke a speculative reinforcement learning model, which then recapitulates the observation that hypoxic cells are rarely completely glycolytic. This to me does not constitute “very good agreement”.

      Lastly, with regards to the study by Marusyk et al. they did not use a spatial model. But neither did they take a “game” for granted. Instead they developed a sequence of non-spatial models of growing complexity.

      [1] Novak S, K Chatterjee, MA Nowak (2013). Density games. J theor Biol 334: 26-34

      • Marc Harper says:

        You seem to be making the following assumptions:
        a) That evolutionary game theory encompasses just frequency dependent models, and
        b) that frequency dependent models must have infinitely large and well-mixed populations.

        Then the implication seems to be that we are restricted to a small class of models — the replicator equation, adaptive dynamics, and possibly (?) some others. I will not argue that the replicator equation is a good model of the intracellular dynamics of a growing tumor. Surely one can get decent qualitative results in some circumstances (incorrect/incomplete models can yield insights), but without mutation and assuming an infinitely large population, I do not expect the replicator equation being a quantitatively accurate model for early stage or rapidly growing tumors.

        So I will assume that we are discussing internal tumor dynamics and that we are not restricted to the replicator equation. As I pointed out before, assumption (a) is a definitional issue which I will assume for now. I do not think that most researchers in evolutionary dynamics would agree with (b). There are many papers that discuss “evolutionary games” in finite populations going back to at least 2004 [1] and these models are described as having frequency dependent fitness landscapes. Such finite population models are often called “population games”. Evolutionary games have been studied on graphs, and selection may be local (death-birth) or global (birth-death), and the population is neither infinitely large nor well-mixed, but games are still involved. Both have limiting cases that yield the replicator equation. Why would either not be considered evolutionary game theory?

        So I disagree with the premise of (2),

        > 1. Is frequency dependence a driving force in tumour evolution?
        > 2. If so, are models that assume infinite and well-mixed populations, i.e. the replicator equation suitable?

        as there are relevant models that include frequency dependence and population structure dependence. The Bickel paper certainly seems to invoke multiple game-theoretic ideas (public goods interactions, cooperators and defectors, ESS, adaptive dynamics, etc.) but you apparently do not consider this to be EGT. Their model also has frequency dependence (masses of the clonal types). So I request a more precise definition of evolutionary game theory — specifically what models you consider to be included in EGT? What assumptions are necessary for a model to be considered EGT?

        (Re: the Bickel paper, at a recent conference there was a presentation of data and model outcomes but I do not see that specific diagram in the paper; perhaps they are planning another manuscript. So I’ll retract that as an example of matching real outcomes for now.)

        > Lastly, with regards to the study by Marusyk et al. they did not use a spatial model. But neither did they take a “game” for granted.

        Again, I still don’t know what a “game” is supposed to refer to in this context. Often fitness landscapes are specified by a game matrix A via f(x) = Ax, but the replicator equation and many other models allow for arbitrary fitness landscapes. Many authors consider arbitrary fitness landscapes as games, e.g. [2]. The landscape is still frequency dependent as long as it actually depends on the population variable(s), just not linearly so. Is that still a “game” in your terminology? What if we take f_i(x) = [(Ax)_i]^2, so that a game matrix is involved, but the landscape is nonlinear? There are many results on how transformations of the fitness landscape still yield the same trajectories (with a velocity change) and/or equilibria for the replicator equation [3]. Many authors study non-linear functions of game matrices and payoffs. For example, Traulsen et al use exponentiated payoffs but still describe many results in terms of evolutionary games. So it would help to know when you consider a model to be dependent on a game and whether that is equivalent to assuming any fitness landscape or just particular forms of the fitness landscape.

        There are many other “evolutionary dynamics”, such as the best reply dynamic, imitation dynamics, learning dynamics, etc. other than the replicator equation that are considered to be part of EGT — many are discussed in Hofbauer and Sigmund’s text “Evolutionary Games and Population Dynamics” [3]. Given that, the methods used in Kianercy et al would seem to still be both frequency dependent and within EGT. (By the way, many models are equivalent to the replicator equation, and the replicator equation is essentially the general time-independent ordinary differential equation on several variables [4]. So the breadth of models that could be reasonably consider to be EGT is potentially large. See also [5].)

        If you could clarify these two points — what is your definition of EGT (what models) and what is a “game” — then we can have a more fruitful discussion.

        Finally, let’s address (1), whether cancer is frequency dependent. From Marusyk et al:

        “We found that tumour growth can be driven by a minor cell subpopulation, which enhances the proliferation of all cells within a tumour by overcoming environmental constraints and yet can be outcompeted by faster proliferating competitors, resulting in tumour collapse.”

        and from Bickel et al:

        “Angiogenesis clearly benefits tumors. In addition to nutrient delivery and waste removal, tumor microvessels provide routes for metastasis. However, all tumor cells receive the benefits of angiogenesis whether or not they participate in producing the signal. Therefore, the signal is a public good. As is well known from decades of research into the “free-rider” problem in economics and evolutionary biology, public goods are susceptible to exploitation by free-riders. In this context, free-riders would be clones that, by mutation or epigenetic alteration, decrease or stop their own production of proangiogenic signals. Since metabolic energy is required to produce the angiogenic signal, free-riders eliminate one drain on internal energy reserves with no immediate detriment. However, they gain an immediate advantage—saved energy reserves can be committed to proliferation and maintenance metabolism. Free-rider clones would therefore be expected to expand more rapidly than angiogenic clones due to their inherited advantage. The obvious fact that the tumor, and the free-riders themselves, would suffer hypoxia once free-riding becomes dominant is irrelevant. Natural selection does not act to benefit the tumor. Selection simply favors clones with the highest growth and survival potential once the chains of kin selection and other evolutionary forces compelling cooperation have been broken. In any environment, even severely hypoxic ones, free-rider clones will always have an advantage over angiogenic clones, all else being equal, because they have less demand for energy to produce a public good. Any angiogenic clone will certainly benefit from being angiogenic. But the free-rider benefits equally. The fact that cancer cells tend to disperse from unfavorable environments does not eliminate the problem. It simply spreads it. If the hallmarks of cancer are consequences of evolution, it is not immediately clear why the angiogenic switch persists in malignant tumors.”

        Both cases the population dynamics seems to have important frequency dependence where interactions (in the game theoretic sense) among cells affects fitness (and the dependence does not seem to depend heavily on the spatial distribution of the cooperators and defectors).

        [1] Taylor et al, “Evolutionary Game Dynamics in Finite Populations”, Bulletin of Mathematical Biology (2004)

        [2] Hofbauer, Josef, and William H. Sandholm. “Stable games and their dynamics.” Journal of Economic Theory 144.4 (2009): 1665-1693.

        [3] Hofbauer, Josef, and Karl Sigmund. Evolutionary games and population dynamics. Cambridge University Press, 1998.

        [4] http://johncarlosbaez.wordpress.com/2012/06/07/information-geometry-part-11/, see the comment labeled “8 June, 2012 at 4:45 pm”

        [5] Page and Nowak, “Unifying Evolutionary Dynamics” JTB 2002

        • Philip Gerlee says:

          Thanks for the reply and for asking me to be more precise. You asked me to clarify two things:

          (i) I prefer a narrow definition of EGT in line with what Maynard Smith uses in his book “Evolution and the theory of games”:

          “Evolutionary game theory is a way of thinking about evolution at the phenotypic level when the fitness of particular phenotypes depend on their frequencies in the population”.

          This definition might seem dated, but at least separates the topic from evolutionary dynamics in general. So, to me an EGT-model is any model that builds on this premise. As I’ve said before there is currently sparse evidence that frequency dependence is a major force in tumour evolution (the paper by Marusyk et al. is one notable exception). For an opposing view see e.g. Trevor Graham’s work on neutral evolution in cancer [1].

          (ii) As to the definition of a “game” I realised I’ve compressed a lot of ideas into one word. Let me expand. My main contention here is that many researchers formulate matrix games of evolving tumour cell populations based on vague empirical notions. The paper by Kiarnecy et al. is an example of this. Based on the number of ATP molecules produced by different metabolic pathways they construct a matrix which is meant to predict the evolutionary outcome of the system. Bickel et al. do the opposite. They formulate a detailed metabolic model, connect it to tissue scale dynamics (density of vessels etc.) and then apply adaptive dynamics to this multi-scale model. This to me is sensible. By taking the “game” for granted I mean the latter approach, where pay-offs are deduced from semi-quantitative data and plugged into a matrix.

          [1] Ann-Marie Baker, Biancastella Cereser, Samuel Melton, Alexander G. Fletcher, Manuel Rodriguez-Justo, Paul J. Tadrous, Adam Humphries, George Elia, Stuart A.C. McDonald, Nicholas A. Wright, Benjamin D. Simons, Marnix Jansen, Trevor A. Graham, Quantification of Crypt and Stem Cell Evolution in the Normal and Neoplastic Human Colon, Cell Reports, Volume 8, Issue 4, 21 August 2014, Pages 940-947

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