# Abstracting evolutionary games in cancer

February 9, 2019 2 Comments

As you can tell from browsing the mathematical oncology posts on TheEGG, somatic evolution is now recognized as a central force in the initiation, progression, treatment, and management of cancer. This has opened a new front in the proverbial war on cancer: focusing on the ecology and evolutionary biology of cancer. On this new front, we are starting to deploy new kinds of mathematical machinery like fitness landscapes and evolutionary games.

Recently, together with Peter Jeavons, I wrote a couple of thousand words on this new machinery for Russell Rockne’s upcoming mathematical oncology roadmap. Our central argument being — to continue the war metaphor — that with new machinery, we need new tactics.

Biologist often aim for reductive explanations, and mathematical modelers have tended to encourage this tactic by searching for mechanistic models. This is important work. But we also need to consider other tactics. Most notable, we need to look at the role that abstraction — both theoretical and empirical abstraction — can play in modeling and thinking about cancer.

The easiest way to share my vision for how we should approach this new tactic would be to throw a preprint up on BioRxiv or to wait for Rockne’s road map to eventually see print. Unfortunately, BioRxiv has a policy against views-like articles — as I was surprised to discover. And I am too impatient to wait for the eventual giant roadmap article.

Hence, I want to share some central parts in this blog post. This is basically an edited and slightly more focused version of our roadmap. Since, so far, game theory models have had more direct impact in oncology than fitness landscapes, I’ve focused this post exclusively on games.

If we are going to talk about games in cancer then we might as well start with the biggest name for EGT in oncology — David Basanta. As a case study, let’s look at the progression of three models that he was involved with.

Basanta, Hatzikirou & Deutsch (2008a) wanted to understand the transition from benign to metastatic cancer. To become metastatic, it was believed that individual cells had to acquire motility so that they could move from their original site into say the circulatory system and then spread distantly across the body from there. But biologists believed that a cell faced a trade-off between growing and moving. So Basanta et al. (2008a) formalized this intuition as a two-strategy Go-vs-Grow game. They then analyzed this game, in part through simulation, and suggested how some hypothetical treatment would have to affect the game matrix so that it might hinder the emergence of motile cells.

Around that time, there was also a lot of interest in the role of Warburg effect in metastasis. It was believed that the acidity caused by the anaerobic glycolysis made tumours more likely to invade. So Basanta, Simon et al. (2008b), added a glycolytic strategy to their go vs grow game. They then analyzed the resulting three-strategy game and showed how it formalized the intuition that the presence of anaerobic glycolysis makes for a more invasive (for them: more motile cells) tumour.

A number of years later, I jumped in and in Kaznatcheev, Scott & Basanta (2015) we looked at a different avenue for extensions of the go-vs-grow game: spatial heterogeneity. After all, if we’re interested in motile cells for their role in metastasis then the intuition is that it only matters if we have motile cells near boundaries like blood vessels or basal membranes through which a tumour can enter a more global domain. Motile cells in the tumour body should matter less. But these two regions of the tumour have different local neighbourhood structures. We tried to approximate this edge effect and suggested that the more easily to sample (at least with liquid byopsis) tumour body can hide the more aggresive ratio of motile cells at the tumour edge.

Now, of course, one could easily imagine continuing this escalation of complexity and ‘realism’. One could ask about how glycolysis matters in the spatial setting (as I ask at the end of this post and as I think Patrick Ellsworth might have answered since).

This progression to more complicated and detailed models is a common pattern among EGT models in oncology. In fact, Marco Archetti even argued (or, at least, that’s how I understood it) at the Lorentz Center in Leiden that this is a healthy progression for the field. The fact that we are building more complicated models is a sign of a healthy, growing subfield. We should continue to confer models with more strategies and more complicated games (like non-linear ones).

I disagree.

The proliferation of progressively more complicated models is certainly a sign of a growing subfield. But as cancer researchers, we know that growth doesn’t need to imply health. I think we build more complicated and detailed theoretical models because we can and because it’s relatively straightforward; not because they’re needed and not because they are having a clear impact on cancer research (outside mathematical oncology).

This is due to another common aspect in our literature: our games rely almost exclusively on biological or clinical intuition. The exact game parameters are seldom measured. This EGT perspective has helped oncologists to express a number of interesting theoretical consequences of non-cell autonomous processes, but has only recently started to be translated into direct experimental work. And, of course, Marco realizes the importance of this experimental connection and has spearheaded the efforts in this direction (see Archetti, Ferraro & Christofori, 2015).

For integrating theory and experiment on this front of somatic evolution we have a new promising tactic – *abstraction*.

As I’ve discussed before, for a computer scientist, abstraction is a way to hide the complexity of a computer system. It is a way to make programs that can be used and re-used without having to re-write all the code for each new computer. In this sense an algorithm is an abstraction of the actual sequence of bit flips that carry out the physical process that is computation. To turn it around: the physical process carried out by your computer is then an implementation of some abstract algorithm. Abstraction and implementation are in some sense dual to each other.

Abstract objects or processes are multiply-realizable by a number of concrete objects or processes. The concrete objects might differ from each other in various ways, but if the implementations are ‘correct’ then the ways in which they differ are irrelevant to the abstraction. The abstraction is less detailed than the implementation, but captures essential features precisely.

It might seem like connecting more closely to experiment must always make a model less abstract. But this is not always the case: the act of measurement itself can be a way to abstract. This is achieved with phenomenological or effective (instead of reductive) theories. And I’ve discussed this before in the context of the game assay (see Kaznatcheev et al., 2019).

In the case of the game assay, we achieved abstraction by focusing on the fitness of types rather than tokens. The focus on abstract types lets us absorb all the details of spatial structure, interaction length-scales, reproductive strategies, etc. into the measurement of the type fitness. It is nature that figures out the particular computation that transforms token fitness into type fitness (see the figure above for 3 examples) and we do not need to know it once we are working at the level of the abstract effective game: the abstract measurement is enough to derive the predictions of the model. A downside, of course, is that we cannot actually describe the specific way token fitness is translated into type fitness in our system. But future work can push the abstraction down, so that more details of the implementation – such as the effects of spatial structure – can be extracted (see Kaznatcheev 2017,2018).

### References

Archetti, M., Ferraro, D.A., & Christofori, G. (2015). Heterogeneity for IGF-II production maintained by public goods dynamics in neuroendocrine pancreatic cancer. *Proceedings of the National Academy of Sciences of the United States of America*, 112(6): 1833-8

Basanta, D., Hatzikirou, H., & Deutsch, A. (2008a). Studying the emergence of invasiveness in tumours using game theory. *The European Physical Journal B*, 63(3): 393-397

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

Kaznatcheev, A., Scott, J. G., & Basanta, D. (2015). Edge effects in game-theoretic dynamics of spatially structured tumours. *Journal of The Royal Society Interface*, 12(108): 20150154.

Kaznatcheev, A. (2017). Two conceptions of evolutionary games: reductive vs effective. BioRxiv, 231993.

Kaznatcheev, A. (2018). Effective games and the confusion over spatial structure. *Proceedings of the National Academy of Sciences*, 201719031.

Kaznatcheev, A., Peacock, J., Basanta, D., Marusyk, A., & Scott, J. G. (2019). Fibroblasts and Alectinib switch the evolutionary games played by non-small cell lung cancer. *Nature Ecology & Evolution* bioRxiv, 179259.

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