# Evolutionary dynamics of acid and VEGF production in tumours

July 14, 2016 2 Comments

Today was my presentation day at ECMTB/SMB 2016. I spoke in David Basanta’s mini-symposium on the games that cancer cells play and postered during the poster session. The mini-symposium started with a brief intro from David, and had 25 minute talks from Jacob Scott, myself, Alexander Anderson, and John Nagy. David, Jake, Sandy, and John are some of the top mathematical oncologists and really drew a crowd, so I felt privileged at the opportunity to address that crowd. It was also just fun to see lots of familiar faces in the same place.

My talk was focused on two projects. The first part was the advertised “Evolutionary dynamics of acid and VEGF production in tumours” that I’ve been working on with Robert Vander Velde, Jake, and David. The second part — and my poster later in the day — was the additional “(+ measuring games in non-small cell lung cancer)” based on work with Jeffrey Peacock, Andriy Marusyk, and Jake. You can download my slides here (also the poster), but they are probably hard to make sense of without a presentation. I had intended to have a preprint out on this prior to today, but it will follow next week instead. Since there are already many blog posts about the double goods project on TheEGG, in this post I will organize them into a single annotated linkdex.

The posts on acidity as a public good and vascularization as a club good, span more than a year and a half. During that time the project has drifted and the notations have changed. So the equations from different posts aren’t always directly comparable, and the early posts especially use different variable names than my presentation.

**Bernstein polynomials and non-linear public goods in tumours**

This project began with Robert and David’s excitement about some recent results by Marco Archetti on nonlinear public goods in cancer:

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.

Archetti, M. (2014). Evolutionary dynamics of the Warburg effect: glycolysis as a collective action problem among cancer cells. *Journal of Theoretical Biology*, 341: 1-8.

In this post, I outline the main technique that Archetti uses for the analysis of nonlinear public goods, and so of the reasons why we might care about public goods in cancer.

**Memes, compound strategies, and factoring the replicator equation**

I wanted to couple the goods that Archetti considered. Making sense of the resulting dynamics, required jumping back and forth between different representations of the replicator equations. It relied on exploiting a factor structure to the game payoffs. Since I was already using this sort of trick for thinking about symmetries in tag-based models, I thought it might be useful to discuss it explicitly in case others find uses for it, too.

**Double public goods games and acid-mediated tumor invasion**

With the factoring trick in hand, I was able to finally write down the two anti-correlated goods game. And using the same approximations as Archetti, to solve for the equations of the general non-linear case. Of course, writing down the equations is only half the battle. The bigger half is making sense of them. In this post, I analyzed only the approximate linear case.

**Pairwise games as a special case of public goods**

The non-linear case has many possible dynamic regimes depending on the particular forms of the benefit functions that are chosen. In this post, I considered the simplest case: a non-linear function defined piecewise by three points. This allowed the dynamics to be equivalent to a three-strategy matrix game, so it was easy to classify the possible dynamic regimes and start to make sense of them using Bomzian analysis:

Bomze, I.M. (1983). Lotka-Volterra equation and replicator dynamics: A two-dimensional classification. *Biological Cybernetics*, 48(3): 201-211.

Bomze, I.M. (1995). Lotka-Volterra equation and replicator dynamics: new issues in classification. *Biological Cybernetics*, 72(5): 447-453.

**From linear to nonlinear payoffs in the double public goods game**

Looking for larger classes of non-linear benefit functions to make sense of, I focused on a nice property of the approximate linear case. This allowed for an easy analysis of non-linear benefit functions where the benefit of oxygen is linear in the proportion of glycolytic cells, but arbitrary in the proportion of VEGF (over)producers among the aerobic cells.

**Don’t treat the player, treat the game: buffer therapy and bevacizumab**

Going back to the three-point non-linear case, or the small interaction group limit, I wanted to start making sense of treatment. In this post, I stressed one of the main focuses of evolutionary game theory in oncology: don’t target individual cell-types, but change the micro-environment so that the natural evolutionary dynamics can work in our favour. Based on suggestions from Jake, I started considering anti-acidity buffer therapy and “anti-benefit-of-vasculature” VEGF inhibitors.

**Cancer metabolism and voluntary public goods games** by Robert Vander Velde

As we continued to work on the linear case, Robert noticed an analogy to the optional public goods game developed by Hauert and colleagues:

Hauert, C., De Monte, S., Hofbauer, J., & Sigmund, K. (2002). Replicator dynamics for optional public good games. Journal of Theoretical Biology, 218 (2), 187-94 PMID: 12381291

After the presentation, this was one of the connections that interested Tatsuya Sasaki. He has been thinking about the effects of norms on optional public goods games, and was excited to see an oncological realization of the game. I wonder if violations of the norms that Sasaki studies could be combined with the anarchy in the organism view of cancer to gain extra insight into the disease.

**Acidity and vascularization as linear goods in cancer**

Robert didn’t get a chance to go into mathematical details in his post. So in this post, I showed how the techniques of Hauert et al. (2002) can be used to give us an exact solution for linear benefit functions. With a solution of the linear case, I could classify the dynamics into 3 qualitative distinct regimes and show how the order of transitions between those regimes (as we treat the game) can affect the transient levels of heterogeneity. In the process, I tried to make a case for neoadjuvant therapies that normalize the less common cell sub-types before the main therapy goes after the most common type.

**Multiple realizability of replicator dynamics**

After the presentation, Benjamin Werner asked about the limitation of the replicator dynamics that our work relies on. In particular, if our results only apply in the case of constant sized populations. His worry was similar to some of the concerns expressed previously on TheEGG by Philip Gerlee and Philipp Altrock. I wanted to address these issues head on, and discuss microdynamical realizations of the replicator equation that allow for non-constant population sizes. In this post, I discuss some such realizations and try to highlight the flexibility of replicator dynamics.

**Hamiltonian systems and closed orbits in replicator dynamics of cancer**

Finally, one of our main results for the linear benefit functions is the existence of an internal dynamics equilibrium that taking Archetti’s prior work in isolation would have ruled out. In this post, I wanted to use the techniques of Hauert et al. (2012) to show that the orbits around the internal fixed point are closed, and what consequences this can have for the timing of therapy.

Throughout the posts, and on my GitHub, you can find links to some Mathematica scripts to explore our model. These are useful for the numeric examples, generating figures, and as sanity checks. Feel free to play around with them.

I’ve also written a number of posts on the poster and the second part of my presentation concerned with measuring the games that experimental cancer populations play. However, I still need to write a number of new posts to make that story self-contained, and so I will save the second project’s linkdex for another time.

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