Double public goods games and acid-mediated tumor invasion

Although I’ve spent more time thinking about pairwise games, I’ve recently expanded my horizons to more serious considerations of public-goods games. They crop up frequently when we are modeling agents at the cellular level, since interacts are often indirect through production of some sort of common extra-cellular signal. Unlike the trivial to characterize two strategy pairwise games, two strategy public-goods have a more sophisticated range of possible dynamics. However, through a nice trick using the properties of Bernstein polynomials, Archetti (2013,2014) and Peña et al. (2014a) have greatly increased our understanding of the public good, and I will be borrowing heavily from the toolbag and extending it slightly in this post. I will discuss the obvious continuation of this work by considering more than two strategies and several public goods together. Unfortunately, the use of public goods games here — and of evolutionary game theory (EGT) more generally — is not without controversy. This extension is not meant to address the controversy of spatial structure (although for progress on this, see Peña et al., 2014b), but the rigorous qualitative analysis that I’ll use (mostly in a the next post on this project) will allow me to side-step much of the parameter-fitting issues.

Of course, having two public goods games is only interesting if we couple them to each other. In this case, we will have one public good from which everyone benefits, but the second good is anti-correlated in the sense that only those that don’t contribute to the first can benefit from the second. A more general analysis of all possible ways to correlate two public-goods game might be a fun future direction, but at this point it is not clear what other correlations would be useful for modeling; at least in mathematical oncology.

By the way, if you are curious what mathematical oncology research looks like, it is often just scribbles like this emailed back and forth:

equations

I’ll use the rest of this post to guide you through the ideas behind the above sketch, and thus introduce you to the joint project that I am working on with Robert Vander Velde, David Basanta, and Jacob Scott. Treat this as a page from my open research notebook.


I’ll start with the second game. Let f(x) be the relative contribution of using a growth-factor — such as oxygen via aerobic respiration — to Alice’s fitness if x is the ratio of agents in Alice’s interaction neighborhood that are producing the growth-factor to those consuming it. The production of the growth-factor might be indirect — for instance producing VEGF to help vasculate nearby tissue and thus increase the abundance of oxygen — but it is not our goal to model all the potential complications, like time-delays, of this process. We will also assume that producing this growth-factor carries some constant cost c > 0.

Now moving onto the first game. Suppose that living without using the grown-factor produces some contaminant that is toxic to non-cancer cells, and maybe even to cancer cells in high enough quantities. In our aerobic vs. anaerobic respiration example, this might be the acid secreted by cells forfeiting aerobic respiration in favor of glycolysis. This is believed to help tumors achieve acid-mediated invasion (Gatenby & Gillies, 2004, 2007; Gatenby et al., 2006) and is one of the explanation for the Warburg effect in cancer progression. We’ll let m(x) be the relative contribution to Alice’s fitness of the contaminant if the proportion x of agents in Alice’s interaction is not using the growth-factor and thus producing the contaminant. There is no cost for producing the contaminant, except the implicit cost in not receiving the benefit f by functioning without using the growth-factor.

Based on the above considerations, we can write down the payoffs for the four strategies ‘\pm\pm‘ where the fist sign is ‘+‘ if the organism uses the growth-factor and the second sign is ‘+‘ if the organism produces — potentially indirectly — the growth factor. Note that in the case of the first sign, ‘‘ means to cooperate, since by not using the growth-factor, the agent cooperates in producing the contaminant; while for the second sign, ‘+‘ means to cooperate by producing the growth-factor. Thus the double defector strategy is ‘-+‘ and the universal cooperator is ‘+-‘. Let x_1 be the proportion of agents with strategy ‘‘, x_2 as strategy ‘++‘, x_3 as ‘+-‘, and x_4 as ‘-+‘.

In the above notation, the utilities (or finesses) for the four strategies are:

\begin{aligned}  U(--) & =  \sum_{n_1 + n_2 + n_3 + n_4 = n} {n \choose n_1, n_2, n_3, n_4} x_1^{n_1}x_2^{n_2}x_3^{n_3}x_4^{n_4} m(\frac{n_1 + n_4 + 1}{n + 1}) \\  U(++) & =  \sum_{n_1 + n_2 + n_3 + n_4 = n} {n \choose n_1, n_2, n_3, n_4} x_1^{n_1}x_2^{n_2}x_3^{n_3}x_4^{n_4} \Bigg( m(\frac{n_1 + n_4}{n + 1}) + f(\frac{n_2 + n_4 + 1}{n_2 + n_3 + 1})\Bigg) - c \\  U(+-) & =  \sum_{n_1 + n_2 + n_3 + n_4 = n} {n \choose n_1, n_2, n_3, n_4} x_1^{n_1}x_2^{n_2}x_3^{n_3}x_4^{n_4} \Bigg( m(\frac{n_1 + n_4}{n + 1}) + f(\frac{n_2 + n_4}{n_2 + n_3 + 1})\Bigg) \\  U(-+) & = \sum_{n_1 + n_2 + n_3 + n_4 = n} {n \choose n_1, n_2, n_3, n_4} x_1^{n_1}x_2^{n_2}x_3^{n_3}x_4^{n_4} m(\frac{n_1 + n_4 + 1}{n + 1}) - c,  \end{aligned}

where the n_i \geq 0 and {n \choose n_1, n_2, n_3, n_4} are the multinomial coefficients. Note that c being positive means that we always have U(–) > U(-+) so the strategy ‘-+‘ will not be present in any stable equilibrium. For now, I am only interested in the equilibrium analysis of this system, so we can set x_4 = 0 and then factor our replicator equation with x_1 = p and x_2 = (1 - p)q. One of the first components we’ll need is the gradient for q:

\begin{aligned}  U(++) - U(+-) &= \sum_{n_1 + n_2 + n_3 = n} {n \choose n_1, n_2, n_3} x_1^{n_1}x_2^{n_2}x_3^{n_3} \Bigg( f(\frac{n_2 +  1}{n_2 + n_3 + 1}) - f(\frac{n_2}{n_2 + n_3 + 1})\Bigg) - c \\  & = \Bigg( f(q + \frac{1}{(1 - p)(n + 1)}) - f(q)\Bigg) - c \\  & = \frac{1}{(1 - p)(n + 1)}\frac{\partial}{\partial q}f(q) - c,  \end{aligned}

where we go from the first to second line by applying the multinomial generalization of Bernstein’s approximation theorem (for example, see Duchoň, 2011; in a future post, I will provide a slightly cleaner probabilistic proof of this by generalizing Levasseur, 1984). Since the dynamic equation for q is given by \dot{q} = q(1 - q)(U(++) - U(+-)), we know that — apart from the two trivial equilibria at q = 0 and q = 1 — we have an internal equilibrium when there exists a q^* such that \frac{\partial}{\partial q}f(q^*) = (1 - p)(n + 1)c. Now, we can look for the gradient for p:

\begin{aligned}  U(--) - (qU(++) + (1 - q)U(+-)) = &\; U(--) - U(+-) - q(U(++) - U(+-)) \\   = &\; \sum_{k = 0}^n {n \choose k} p^{k}(1 - p)^{n - k} \Bigg( m(\frac{k}{n + 1} + \frac{1}{n + 1}) - m(\frac{k}{n + 1}) \Bigg) \\ & - \sum_{n_1 + n_2 + n_3 = n} {n \choose n_1, n_2, n_3} x_1^{n_1}x_2^{n_2}x_3^{n_3} f(\frac{n_2}{n_2 + n_3 + 1}) \\  & - q(U(++) - U(+-)) \\  = &\; m(p + \frac{1}{n + 1}) - m(p) - \overbrace{(f(q) + q(U(++) - U(+-)))}^{Q(q)} \\  = &\; \frac{1}{n+1}\frac{\partial}{\partial p}m(p) - Q(q).  \end{aligned}

From the above we can characterize the dynamics of p. We know that — apart from the the two trivial quilibria at p = 0 and p = 1 — we have an internal equilibrium when there exists a p^* such that \frac{\partial}{\partial p}m(p^*) = (n + 1)Q(q). Note that if we are looking at q = 0 or q = q*, we have Q(q) simplifying to just be f(q).

Also, just to be pedantic I want to write down the simplified dynamic system of equations explicitly:

\begin{aligned}  \dot{p} & = p(1 - p)\Bigg(\frac{1}{n+1}\frac{\partial}{\partial p}m(p) - Q(q)\Bigg) \\  \dot{q} & = q(1 - q)\Bigg(\frac{1}{(1 - p)(n + 1)}\frac{\partial}{\partial q}f(q) - c\Bigg)  \end{aligned}

Note that for p = 0 we recover Archetti (2013) and for q = 0 or q = 1 we get Archetti (2014). In the other cases, if we want the equilibrium points (p*,q*) then we to solve the following system of equations:

\begin{aligned}   \frac{\partial}{\partial p}m(p^*) & = (n + 1)f(q^*) \\  \frac{\partial}{\partial q}f(q^*) & = (1 - p)(n + 1)c  \end{aligned}

At this point, it is important to do a sanity check by considering linear public goods f(q) = b_fq and m(p) = b_m p. This results in a stable internal equilibrium with q^* = \frac{1}{n+1}\frac{b_m}{b_f} and (1 - p^*) = \frac{1}{n + 1}\frac{b_f}{c} or in the simplex coordinates (x^*_1, x^*_2, x^*_3) = (1 - \frac{b_f}{c(n + 1)}, \frac{b_f}{c(n + 1)^2}, \frac{b_f}{c(n + 1)} - \frac{b_f}{c(n + 1)^2}). From this, we can conclude that if \frac{b_f}{c} > n + 1 then we will have all cooperators in the growth-factor public good. In our running example, this means that among the aerobic cells everybody will invest in VEGF. If \frac{b_m}{b_f} > n + 1 the we will have all cooperators in the contamination public good. In our running example, this means that all the cells will revert to glycolosis. These cases are in agreement with more traditional approaches like Santos et al. (2008) hitchhikes.

The interesting case is that if both conditions are violated then unlike the single public-good (which would go toward all defectors), this anti-correlated pair of public goods will result in a coexistence of all 3 types of strategies. In our running example, we will have a tumor with some cells glycolotic, while others continue aerobic respiration with some of them producing VEGF. I am not sure if this co-existence is frequently observed in clinical cases, so I welcome any references. Note that this result would have been impossible to achieve with two decoupled linear public goods games, so the anti-correlated coupling of games is essential.

Of course, linear public goods are of limited appeal. In the next post on this project I will use the sign pattern technique of Gokhale & Traulsen (2010) and Peña et al. (2014a) to look at the general case of two anti-correlated non-linear public goods games. As far as I can tell, this post offers one of the first discussions of the EGT of multiple interacting public goods. Multiple public goods games have been briefly looked at in economics by Sable & Kling (2001), Ehlers (2001,2004), and Umezawa (2012) but as far as I can tell their interests in multiple public goods game is in the context of matching markets (which is fundamentally an equilibrium selection problem) while I am looking at them as social dilemmas (which is fundamentally playing with the tension of Nash Eq. that is not Pareto optimal). However, dear reader, if you know of existing work on the dynamics of multiple interacting public goods then please mention it in the comments!

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.

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

Duchoň, M. (2011). A generalized Bernstein approximation theorem. Tatra Mountains Mathematical Publications, 49(1): 99-109.

Ehlers, L. (2001). Independence axioms for the provision of multiple public goods as options. Mathematical Social Sciences, 41(2): 239-250.

Ehlers, L. (2002). Multiple public goods and lexicographic preferences: replacement principle. Journal of Mathematical Economics, 37(1): 1-15.

Gatenby, R. A., & Gillies, R. J. (2004). Why do cancers have high aerobic glycolysis? Nature Reviews Cancer, 4(11): 891-899.

Gatenby, R.A., Gawlinski, E.T., Gmitro, A.F., Kaylor, B., & Gillies, R.J. (2006). Acid-mediated tumor invasion: a multidisciplinary study. Cancer Research, 66(10): 5216-5223.

Gillies, R.J., & Gatenby, R.A. (2007). Adaptive landscapes and emergent phenotypes: why do cancers have high glycolysis?. Journal of Bioenergetics and Biomembranes, 39(3): 251-257.

Gokhale, C. S., & Traulsen, A. (2010). Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences, 107(12): 5500-5504.

Levasseur, K.M. (1984). A probabilistic proof of the Weierstrass approximation theorem. Amer. Math. Monthly, 91: 249-250.

Peña, J., Lehmann, L., & Nöldeke, G. (2014a). Gains from switching and evolutionary stability in multi-player matrix games. Journal of Theoretical Biology, 346, 23-33 PMID: 24380778

Peña, J., Nöldeke, G., & Lehmann, L. (2014b) Relatedness and synergies of kind and scale in the evolution of helping. arXiv: 1412.1729

Sable, K. A., & Kling, R. W. (2001). The Double Public Good: A Conceptual Framework for“Shared Experience”Values Associated with Heritage Conservation. Journal of Cultural Economics, 25(2): 77-89.

Santos, F. C., Santos, M. D., & Pacheco, J. M. (2008). Social diversity promotes the emergence of cooperation in public goods games. Nature, 454(7201): 213-216.

Umezawa, M. (2012). The replacement principle for the provision of multiple public goods on tree networks. Social Choice and Welfare, 38(2): 211-235.

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About Artem Kaznatcheev
From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

18 Responses to Double public goods games and acid-mediated tumor invasion

  1. Robert Vander Velde says:

    I think this paper may be interesting and somewhat related (but not exactly what you’re looking for): http://www.nature.com/nbt/journal/v31/n6/full/nbt.2530.html

    However it seems that even though they refer to the “same tumor” I think this only means the same tumor “type”, a confusion that might be interesting on this blog for more philosophical reasons. Also it reminds me of explaining the difference between doing a “single channel recording” and recording from a cell with a single ion channel “type” in electrophysiology.

  2. Robert Vander Velde says:

    I also wanted to talk about a concern I had which might create problems for this project (or opportunities). I wanted to discuss the accuracy of what I wrote below (slightly different from what I wrote in an email):

    First some biochemistry terminology: R132 is a way of writing an amino acid, basically the arginine (R) at the 132nd spot in the protein. R132C is a way of writing a mutation, the arginine at position 132 is replaced with a cysteine (C). R132C in the protein isocitrate dehydrogenase (IDH1) is a common mutation in cancer that creates a glycolytic phenotype by breaking the citric acid cycle. UniProt is a good way to learn more about these mutations (under “Natural Variant”): http://www.uniprot.org/uniprot/O75874

    Now our model is based primarily on evolution and only takes into account biochemistry as a way to determine fitness. However, in normal (except for the mutation in IDH1) cells, glycolysis (as brought on by mutations in IDH1) induces the expression of HIF-1alpha which in turn increases the production of VEGF (Cairns and Mak, 2013). Hypoxia Inducible Factor (HIF-1alpha) is, as the name suggests induced by hypoxia (lack of oxygen) or similar conditions (like those brought on by glycolytic cells). It therefore triggers the up-regulation of VEGF so that blood vessels can grow and bring more oxygen. But remember, the cell can’t use oxygen (because of the IDH1 mutation). But the biochemical “premises” do in fact make sense (I’m using glycolysis, I must not have enough oxygen, I must bring oxygen) but produces a contradiction that makes no evolutionary sense (I’m using glycolysis, I must bring oxygen, but I can’t use oxygen, therefore I’m producing a public good I can’t use).

    This relationship ([glycolysis] -> [HIF-1alpha] -> [VEGF up-regulation]) should decouple over time, since as both our model and common sense show “VEGF producing + lactic acid producing (eg. Glycolytic)” is a nonviable strategy. It incurs the cost of producing VEGF (‘c’ in our model) and doesn’t benefit from the corresponding public good.

    I think this means we should be clear about our model only applying to the late stages of tumor evolution (since during early evolution, if I’m right, glycolysis and VEGF are biochemically linked and we don’t want to couple them right now), which Artem insisted on all along (for different reasons). However we have to consider if the timespans involved in tumor evolution are long enough for a complete decoupling to occur (and that might also depend on the cost of VEGF production).

    Cairns and Mak, (2013) Oncogenic Isocitrate Dehydrogenase Mutations: Mechanisms, Models, and Clinical Opportunities

  3. Artem, just so you know, Philip Gerlee on twitter has some serious concerns about the use of Bestrein Polynomials in our particular problem. The conversation is (unfortunately) only on twitter at the moment. The gist is that you cannot average over all the different groups invoved in the game as it imples that each cell enters into a unique game with n other cells.

    • Thanks for the heads up. That is the same critique he offered on Archetti’s work, and I think it is not a particularly strong one. I explained why in this comment, but I guess I didn’t do a clear enough job. The fundamental issue in question, has nothing to do with Bernstein polynomials (although there is a subtle point where the BernPoly gives us a slightly better analysis, but I’ve actually not touched in any of the posts on this point, because that is just icing on the cake), but with how we wrote down the payoffs. In particular, Philip seems to be opposed to population averages.

      Although there are some times when we should really have to worry about the stochastic to deterministic transition (in mathematical oncology, the area where we worry about this is when we discuss branching processes) and the discrete to continuous transition (note that the two are interrelated but not the same). This worry is relatively familiar to evolutionary game theorists, and here are my two favourite papers that discuss it carefully:

      Durrett, R., & Levin, S. (1994) The Importance of Being Discrete (and Spatial). Theoretical Population Biology, 46: 363-394.

      Shnerb, N.M., Louzoun, Y., Bettelheim, E., and Solomon, S. (2000) The importance of being discrete: Life always wins on the surface. PNAS, 97(19): 10322-10324.

      However, there is no real clear case for why stochasticity (and discreteness; although the subtle point about BernPoly is that it in a careful analysis it lets us ignore discreteness less than simple averaging) should matter in this case. Further, even if they are the most important aspects in this case, we won’t know it without having a deterministic model to compare to and say: look how different the results are. So, either way our model is useful, either just as good approximation of the ‘final story’ as the fully stochastic (which I wouldn’t be that surprised by), or as a first step to then figure out what exactly stochasticity would give you in these cases.

      I’ll try to expand on some of this later, but you are welcome to tweet this comment at Philip.

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  9. I think x_2* should be equal to b_m/(c*(n+1)^2), you have b_f/(c*(n+1)^2), this also affected your calculation of x_3*, not sure if that’s something you can change. Pretty important since b_m doesn’t affect the equilibria otherwise.

    • You are correct, it is a typo while doing all the copy and pasting and underscores. Each fraction in the three-strat expression where (n + 1) is squared, there should be a b_m instead of a b_f on top.

  10. Jorge Peña says:

    Just to let you know, a new version of Peña et al. (2014b) has been published in JTB. Here’s the link: http://www.sciencedirect.com/science/article/pii/S0022519315003185

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