# Cancer metabolism and voluntary public goods games

When I first came to Tampa to do my Masters[1], my focus turned to explanations of the Warburg effect — especially a recent paper by Archetti (2014) — and the acid-mediated tumor invasion hypothesis (Gatenby, 1995; Basanta et al., 2008). In the course of our discussions about Archetti (2013,2014), Artem proposed the idea of combining two public goods, such as acid and growth factors. In an earlier post, Artem described the model that came out of these discussions. This model uses two “anti-correlated” public goods in tumors: oxygen (from vasculature) and acid (from glycolytic metabolism).

The dynamics of our model has some interesting properties such as an internal equilibrium and (as we showed later) cycles. When I saw these cycles I started to think about “games” with similar dynamics to see if they held any insights. One such model was Hauert et al.’s (2002) voluntary public goods game.[2] As I looked closer at our model and their model I realized that the properties and logic of these two models are much more similar than we initially thought. In this post, I will briefly explain Hauert et al.’s (2002) model and then discuss its potential application to cancer, and to our model.

### Voluntary public goods game

This game involves a typical linear public goods game where the payoff for defectors is:

$U_{D}=\frac{rn_c}{S}$

and the payoff for cooperators is:

$U_{C}=\frac{r(n_c + 1)}{S}-c$,

where r > 1 is the interest, c > 0 is the cost of cooperation, $n_c$ is the number of other cooperators in an interaction group, and S is the total number of cooperators and defectors in the interaction group.

The voluntary public goods game also adds a third strategy, the loners, with a constant payoff $U_{L}=\sigma c$, where $0 < \sigma < r-1$. In the absence of cooperators $U_{L} = \sigma c > 0 = U_{D}$, this allows loners to take over the population. When loners increase S decreases, and since the good is distributed only among non-loners, this allows cooperators to begin taking over the population (since the condition r > S, required in public goods games for cooperation, can be satisfied).[3] S then increases as the number of cooperators increases and defectors gradually become fitter and begin to take over the population.

We therefore get the cyclical dynamics shown in the figure on the left.

In this model cooperation can evolve even if r is less than the size of the interaction group (N), this would not occur in the absence of loners. This model is therefore used as an explanation for cooperation.

### Applications to cancer

Back to cancer metabolism. Let’s look at the Warburg effect: when tumor cells use anaerobic metabolism in the presence of oxygen (I will call cells which do this “Warburg cells”) they do not benefit from the oxygen brought in by VEGF producing cells. In a sense, Warburg cells are loners in a voluntary public goods game where oxygen (brought by VEGF) is the public good. If the Warburg effect is inherently beneficial to tumor cells (perhaps by retaining nutrients necessary for increased anabolism in cancer as suggested by Vander Heiden et al. (2009)) then we can assign a constant fitness to glycolytic cells as Hauert et al. (2002) do for loners in the voluntary public good.

Hauert et al.’s (2002) model is still applicable, however, if Warburg cells participate in a linear public goods game where they are the cooperators and the other cells (both cooperators and defectors in the oxygen public goods game) are defectors. In that case, Warburg cells would have the fitness:

$U_W = \frac{b_a (n_W + 1)}{N}$

and the non-Warburg cells would have $U_N = \frac{b_a n_W}{N}$ added to their fitness.

From the above, notice that $U_W = U_N + \frac{b_a}{N}$. Since a common factor to all fitnesses does not matter, we could then subtract $U_N$ from the fitness of all cells without changing the dynamics. In this case Warburg cells are left with a constant fitness gain (over the mean) of:

$U_L=\sigma c=\frac{1}{N}b_a$,

Thus, this means that the linear public goods for acidity and linear club good for vascularization have the same mathematical dynamics as the voluntary public goods game.

The section above is equivalent to the case of $a(x_L)=b_a x_L$ in Artem’s previous post. Here $a(x_L)$ is the contribution of the Warburg cell’s public good to the fitness of tumor cells and $x_L$ is the proportion of Warburg cells/loners. We consider this public good to be acid. This uses an alternative explanation of the Warburg effect in which Warburg cells produce acid which promotes tumor invasion (and helps tumors cells survive) by destroying the extracellular matrix.

In conclusion, cancer metabolism presents a possible example of a real life voluntary public goods game (and a subset of rock-paper-scissors games). It also demonstrates that Warburg cells can promote costly VEGF production in other cells even if the benefit is linear. This is in contrast to Archetti (2013) which requires the benefit of growth factors to be non-linear. This model therefore shows that the presence of Warburg cells causes the evolutionary dynamics of VEGF production to be distinct from those of other growth factors in ways beyond simple changes in parameters.

Archetti (2014) points out that acid as a public good probably does not have a linear benefit, and may even over-saturate. Furthermore, the relationship (via oxygen) between fitness and the production of vasculature inducing molecules is likely non-linear. We’ve seen some interesting differences between the linear and non-linear cases but that will have to wait until a future post. Therefore, while the voluntary public goods games gives some interesting insights there is still work that can be done to make this model more applicable to cancer.

### Notes and References

1. I came to the University of South Florida with the goal of working with the Integrated Mathematical Oncology (IMO) department at Moffitt. I learned about the IMO from a Nature Outlook, at the time I was doing undergraduate research in electrophysiology and merely found the contents of the articles interesting. However, by the time I graduated with a B.Sc. in biophysics and started looking for graduate programs I had decided that my interests lie in mathematical oncology. After arriving in Tampa I started working with Dr. David Basanta and Artem, since I had also developed an interest in evolutionary game theory. I am currently working on my PhD under Dr. Bob Gatenby and doing more wet lab work than previously (while still working closely with the IMO).
2. I learned about the voluntary public goods game by playing with the simulations on Dr. Chrisoph Hauert’s website. I later had the opportunity to learn some game theory from Dr. Hauert in person while working under Dr. Sarah Otto (during the summer just prior to my arrival in Tampa). This was my first taste of bio-mathematics research. I was fortunate enough to grow up in the Vancouver area and to attend UBC which has many great faculty members involved in bio-mathematics and evolution.
3. Notice that the good is not distributed to all members of the population, but only to the non-loners. It is therefore not an actual public good, but a club good. This is one of the reasons for Artem’s recent change in terminology from double public goods to a club good coupled to a public good. Also notice that the previously discussed ecological public goods game of Hauert et al. (2006) has some similar properties. The mechanism of cooperation is the same: the good is shared by a group of varying size and that size oscillates around the critical value of S = r.

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

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.

Gatenby, R. A. (1995). The potential role of transformation-induced metabolic changes in tumor-host interaction. Cancer Research, 55(18): 4151-4156.

Hauert, C., De Monte, S., Hofbauer, J., & Sigmund, K. (2002). Replicator dynamics for optional public good games. Journal of Theoretical Biology, 218 (2), 187-194 PMID: 12381291

Hauert, C., Holmes, M., & Doebeli, M. (2006). Evolutionary games and population dynamics: maintenance of cooperation in public goods games. Proceedings of the Royal Society B: Biological Sciences, 273(1600): 2565-2571.

Vander Heiden, M., Cantley, L., Thompson, C. (2009) Understanding the Warburg effect: The metabolic requirements of cell proliferation. Science, 324: 1029-1033.