Bernstein polynomials and non-linear public goods in tumours
November 7, 2014 13 Comments
By analogy, or maybe homage, to standard game theory, when we discuss the payoffs of an evolutionary game, we usually tell the story of two prototype agents representing their respective strategies meeting at random and interacting. For my stories of yarn, knitting needles, and clandestine meetings in the dark of night, I even give these players names like Alice and Bob. However, it is important to remember that these are merely stories, and plenty of other scenarios could take their place. In the case of replicator dynamics there is so much averaging going on that it is often just better to talk about the payoffs as feedback between same-strategy sub-populations of agents. The benefit of this abstraction — or vagueness, if you prefer — is that you don’t get overwhelmed by details — that you probably don’t have justification for, anyway — and focus on the essential differences between different types of dynamics. For example, the prisoner’s dilemma (PD) and public goods (PG) games tell very different stories, but in many cases the PD and linear PG are equivalent. Of course, ‘many’ is not ‘all’ and my inclusion of ‘linear’ should prompt you to ask about non-linear public goods. So, in this post I want to provide a general analysis of replicator dynamics for non-linear public goods games following the method of Bernstein polynomials recently used by Archetti (2013, 2014). At the end, I will quickly touch on the two applications to mathematical oncology that Archetti considers. SInce I am providing a more general analysis, I will use notation inspired by Archetti, but defined more precisely and at times slightly differently — some symbols will be the same in name but not in value, so if you’re following along with the paper then pay close attention.
Public goods games
Let’s start with a story. In most biological contexts, we think of a public good as some sort of substance that is released into the ambient space that the cells inhabit and that diffuses out to some distance. This substance can be consumed by the agent that released it but also other agents in order to gain some sort of fitness effect, usually positive — hence public “good” and not “bad”. The effects need not be direct. For instance, when modeling oxygination the public good would be the oxygen that cells use for respiration while the relevant molecule that cells release is vascular endothelial growth factor (VEGF) that causes blood vessels to grow and thus deliver oxygen. In this example, the specific molecule released is only indirectly related to the public good that it causes. This level of indirect-ness is not atypical, because the physical molecules that cells release usually undergo at least some sort of transformation in the ambient environment before they are consumed.
For simplicity, we will assume that the strategy that causes an increase in the public-good — such as the cells making VEGF; we will call this strategy cooperators — pay some fixed cost c to create the good. This cost is usually not the exact cost of producing the mediating molecule, but some amortization over the whole population of producers that takes into account things like probability of producing the molecule. Since we want to keep the model inviscid, we incorporate the diffusion distance not as a physical distance but as a count n of agents other than the focal agent that partake in the good — if you want to get this n from some physical distance then just take the volume that the distance corresponds to and multiply by the density of cells. Now, we suppose that we have cooperators and defectors — defectors are agents that don’t cause an increase in the public-good — with . To get these numbers of cooperators and defectors at each interaction, we sample from the overall strategy distribution. Hence, if the proportion of cooperators is x then our probability of sampling is . The resulting benefit of the public good is then if the focal agent is a cooperator (+1 because the focal agent contributes to the public good) and if the focal agent is a defector. As an example, if you want a linear public good with multiplicative factor k then you would have .
The above results in the fitness functions:
As always, this gives us the replicator dynamics . We will be interested in knowing if the resulting equation has any non-trivial equilibria (the trivial ones are x = 0 and x = 1), and if so where they are and their stability.
Even wihout the non-linearily of b, our equations look scary, with the ODE having degree n + 2. Thankfully, a little bit of mathematical analysis will help us greatly simplify our lives — at least in the limit of large n. This bit of analysis is Bernstein’s approximation theorem.
In 1885, Karl Weierstrass — the father of modern analysis — proved a now standard approximation theorem: for every continuous function f defined on a closed interval [0 1] and every there exists a polynomial p such that for all (Pinkus, 2000). Weierstrass’s proof was not constructive, which motivated Sergei N. Bernstein to find a constructive proof. His proof was not the first constructive one, Borel had arrived at a much less elegant construction earlier (for the history, see Pinkus, 2000), but it is the one that will help our public goods game. In particular, Bernstein showed that:
where the convergence is uniform — just like for Weierstrass. The proof itself relies heavily on the Chebyshev inequality — see Levasseur (1984) for a modern version of this probabilistic treatment — and a careful unpacking of the proof can give us a good bound on the rate of convergence.
We can now use this theorem to simplify our public goods game. Let and notice that by the definition of derivative, we have that where the derivative is taken with respect to x. Now, we have in the limit of large n:
At this point, it is good to pause for a sanity check. If you plug in the linear public goods game then the equation simplifies to . From this, we expect the game to have two regimes: selection towards all defectors when k < n, and toward all cooperators when k > n. This is in agreement with more traditional approaches. For example, it is the main mechanism on which the cooperation in Santos et al. (2008) hitchhikes.
Non-linear public goods in tumours
Of course, we didn’t build all this mathematical machinery just to analyze the linear public goods game, we really care about more complicated functional forms for that are found in biology. For Archetti (2013) this is the sigmoid function associated with production of growth factors, and for the follow up paper (Archetti, 2014) it is the double-sigmoid of acid production due to glycolysis. Depending on how you set the parameters of these benefit functions — as always with EGT, there is no perscription given for how we would go about measuring such parameters in vivo — you get qualitatively different dynamics. Archetti (2013,2014) finds all the typical dynamics of linear games, like pressure toward all cooperation, all defection, and bifurcation around or attraction to a internal fixed point. However, he also finds surprising dynamics like dynamic C of figure 1 (2013; similar dynamics are seen in type 2 of figure 1 in 2014), where there are two internal fixed points (impossible in linear games) with one being a bifurcation and one being an attractor — depending on the initial conditions, the population will go either to all defectors or a coexistence of cooperators and defectors.
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 PMID: 24075895
Levasseur, K.M. (1984). A probabilistic proof of the Weierstrass approximation theorem. Amer. Math. Monthly, 91: 249-250.
Pinkus, A. (2000). Weierstrass and approximation theory. Journal of Approximation Theory, 107(1): 1-66.
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.