Is cancer really a game?
December 1, 2014 20 Comments
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.
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