Evolutionary game theory without interactions

When I am working on evolutionary game theory, I usually treat the models I build as heuristics to guide intuitions and push the imagination. But working on something as practical as cancer, and being in a department with many physics-trained colleagues puts pressure on me to think of moving more towards insilications or abductions. Now, Philip Gerlee and Philipp Altrock are even pushing me in that direction with their post on TheEGG. So this entry might seem a bit uncharacteristic, I will describe an experiment — at least as a theorist like me imagines them.

Consider the following idealized protocol that is loosely inspired by Archetti et al. (2015) and the E. coli Long-term evolution experiment (Lenski et al., 1991; Wiser et al., 2013; Ribeck & Lenski, 2014). We will (E1) take a new petri dish or plate; (E2) fill it with a fixed mix of nutritional medium like fetal bovine serum; (E3) put a known number N of two different cell types A and B on the medium (on the first plate we will also know the proportion of A and B in the mixture); (E4) let them grow for a fixed amount of time T which will be on the order of a cell cycle (or two); (E5) scrape the cells off the medium; and (E6) return to step (E1) while selecting N cells at random from the ones we got in step (E5) to seed step (E3). Usually, you would use this procedure to see how A-cells and B-cells compete with each other, as Archetti et al. (2015). However, what would it look like if the cells don’t compete with each other? What if they produce no signalling molecules — in fact, if they excrete nothing into the environment, to avoid cross-feeding interactions — and don’t touch each other? What if they just sit there independently eating their very plentiful nutrient broth?[1]

Would you expect to see evolutionary game dynamics between A and B? Obviously, since I am asking, I expect some people to answer ‘no’ and then be surprised when I derive some math to show that the answer can be ‘yes’. So, dear reader, humour me by being surprised.

Since I don’t have any cell cultures to play with, let’s fool around with a few equations. Let x be the proportion of A-cells in the population, and y be the nutritional content of the medium — normalized so that the most nutrient rich mix possible has y = 1 and distilled water has y = 0. For each cell we will have some (analytic) feeding function $f_A(y)$ and $f_B(y)$ which translates between the nutritional content of the medium and the organism’s fitness, such that:

$\dot{x} = x(1 - x)(f_A(y) - f_B(y))$[2]

It is important to note that $f_A$ and $f_B$ are functions of y and completely independent from x. At this point, we might be tempted to stop by saying that since experimental step (E2) uses a fixed mix of nutritional medium, we can just treat $f_A(y) - f_B(y)$ as a constant and thus (excluding the neutral case) we will always have the population converge to all-A or all-B depending on the sign. We will see no (non-trivial) evolutionary game dynamics.

But stopping here would be a bit disingenuous. The reason that we have to renew the medium on each cycle of the experiment is because it gets consumed between the replating. Further, the rate of consumption might differ between the two cell types. Suppose that each cell type consume the nutrients at rate $2k_A$ and $2k_B$, such that if $y_\text{in}$ was our initial level of nutrients in step (E2) then our final level it step (E5) is:

$y_\text{out} = y_\text{in}(1 - 2k_Ax - 2k_B(1 - x))$

We can assume that the cell cycle is significantly slower than the metabolic cycle, and start working with our average consumption (calling $y_\text{in}$ just y to save on LaTeX commands — I have a quota):

$\langle y \rangle_x = y(1 - k_B + x(k_B - k_A))$[3]

Now, our dynamics become:

$\dot{x} = x(1 - x)(f_A(\langle y \rangle_x) - f_B(\langle y \rangle_x))$[4]

and suddenly our gain function is no longer independent of x. That is the main slight-of-hand, but let’s take the trick to its conclusion.

We will expand the gradient function, noting that we picked analytic feeding functions, so $f_A(y) = \sum_{n = 0}^\infty A_n y^n$ for some sequence $\{ A_n \}$ and similar for $f_B(y)$ but with $\{ B_n \}$:

\begin{aligned} f_A(\langle y \rangle_x) - f_B(\langle y \rangle_x) & = \sum_{n = 0}^\infty (A_n - B_n)\langle y \rangle^n \\ & = \sum_{n = 0}^\infty (A_n - B_n) y^n (1 - k_B + x(k_B - k_A))^n \\ & = \sum_{n = 0}^\infty \frac{(A_n - B_n) y^n}{(k_A - k_B)^n} (x - \frac{1 - k_B}{k_A - k_B})^n \\ \end{aligned}

But that last line is just the power series of some analytic function g(x) with coefficients $\{ G_n = \frac{(A_n - B_n) y^n}{(k_A - k_B)^n} \}$ around the point $x_0 = \frac{1 - k_B}{k_A - k_B}$. In particular, given any desired gain function $g(x)$, there is some choice of feeding functions $f_A$ and $f_B$ (thus, their corresponding coefficients $\{ A_n \}$ and $\{ B_n \}$) and $k_A$ and $k_B$ such that the population follows identical dynamics. In other words, in this experimental set up, we can recreate any evolutionary game dynamics without having the cells interacting but just based on how they turn nutrition into reproduction.

As an example, suppose you want to recreate an arbitrary cooperate-defect game. In that case, you need to create the gain function of g(x) = U + x(U + V – 1) where U, V are the game parameters. To achieve this, just pick any $k_A$ and $k_B$ such that $1 - k_B = k_B - k_A$[5]; $A_0$ and $B_0$ such that $A_0 - B_0 = U$; and $A_1$, $B_1$, and your initial nutrient concentration y such that $\frac{A_1 - B_1}{1 - k_B}y = U + V - 1$; for all higher $A_n$, $B_n$ just have them equal to each other (by example setting them all to zero, giving you linear feeding functions).

I encourage you, dear reader, to plug the above examples into the derivation to check that you in fact recreate any linear game. I won’t go into more detail in this post, but I will show a more detailed application for quadratic games when I discuss Archetti et al. (2015), since we need at least a quadratic gain function to recreate the qualitative aspects of their results.

Notes and References

1. For those that attended the last evolutionary game theory reading group where we discussed Archetti et al. (2015), this was the alternative hypothesis that I called ‘hungry hungry hippos’. For those that weren’t there, this post is a starting sketch of one of my critiques or Archetti et al. (2015), although I will probably write a full post on this later. To give you the basic gist: I suggested that the authors did not eliminate the possibility of the hungry hungry hippos hypothesis, and so didn’t demonstrate that the interaction is in fact a public goods game. Of course, demonstrating things beyond a shadow of a doubt it a silly criterion; they definitely convinced me that they observed a public-goods game in action, but I want their result to be true so I try to compensate by being extra skeptical — at least for the sake of argument.
2. Since there is a natural time discretization here at every experimental cycle and replating, it would be more fitting to write the discrete-time replicator dynamics. This would result in the same exact argument, but require me to write a lot more fractions, which makes the math slightly less readable. You could also write the Lotka-Volterra equations, if you are bothered by the fact that the population size will be changing before the replating (some people seem to believe that replicator dynamics absolutely require a static population size), but then you can just return to my equations through the standard transform.
3. Here you could argue that due to the population growth, this is not the right way to average, since I should weigh more heavily the later time given the larger population compared to the start of the cycle. However, you should be careful with this argument, because the natural response becomes: how will you measure $k_A$ and $k_B$ then, except by running a full cycle? More importantly, it is irrelevant how exactly the average is gotten or if $k_A$ and $k_B$ are constants or some complicated functions on y. For the main slight-of-hand in this article, all that matters is that $\langle y \rangle_x$ is a function of x (which I said in words above). If it is a more complicated function then linear then it just makes the math harrier without changing any of the conclusions (obviously, the mapping between the feeding function coefficients and the desired gain function coefficients will get more complicated, but will still allow an arbitrary gain function).
4. If you don’t like the separation of metabolic and reproductive time-scales that I did here then you can replace this system by writing down a second ODE that explicitly models the decay of y and reset it to its initial value every T time-units. You could even use the Lotka-Volterra equations with renormalizing being pulsed every T time-units (if you want to properly calculate $\langle y \rangle_x$). This is fun for numeric solutions, but makes a simple analytic demonstration intractable. However, you can hopefully see how it only gives you more knobs to play with and thus even easier to recreate (an approximation of) arbitrary game dynamics. You might object on reductionists grounds, saying that this complicated systems of ODES ‘better describes reality’, but then some of the comments from footnote [3] apply. How do you plan to measure $f_A$, $f_B$, $k_A$, and $k_B$ without the discretizations and approximations that the platting is forcing on us. Our understanding of biology isn’t sufficient to calculate these functions from the cell genome or biochemical network, so you’ll have to rely on phenomenology which will return you approximations like the ones I am making.
5. In fact, you have more freedom with $k_A$ and $k_B$ but then it makes the conditions on $A_1$ and $B_1$ a little more complicated, so I picked this constraint to make it easier for anybody recreating my algebra.

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, 112 (6), 1833-8 PMID: 25624490

Lenski, R. E., Rose, M. R., Simpson, S. C., & Tadler, S. C. (1991). Long-term experimental evolution in Escherichia coli. I. Adaptation and divergence during 2,000 generations. American Naturalist, 1315-1341.

Ribeck, N., & Lenski, R. E. (2014). Modeling and quantifying frequency-dependent fitness in microbial populations with cross-feeding interactions. bioRxiv, 012807.

Wiser, M. J., Ribeck, N., & Lenski, R. E. (2013). Long-term dynamics of adaptation in asexual populations. Science, 342(6164): 1364-1367.