Team:UNAM-Genomics Mexico/Modeling/GT

Background

As self-respecting geeks, we are gamers. In this case, we wanted to corroborate a Worst-Case-Scenario using Game Theory. The Scenario in question is the following:

The transgenic turns out to be highly incompetent in N fixation, so you knock-out said pathway. If we generate two strains, one per gas, and mix them: What will happen?

Introduction

So we believe the system will behave as a Snowdrift game. This game is characterized by two antagonizing forces. Each force has two options, of which one is the favorite. However, if both select the favorite, everybody is penalized. It is a variation of the Chicken Game where the penalization is not death. You may consult the All-Knowing-Oracle on this topic [http://en.wikipedia.org/wiki/Chicken_(game) here].

The Setting

We believe our system will behave as said game because we have two players: WildType and TransGenic. Each has two choices: Stasis and Reproduce. This generates the following payoff table.

As is the case with the Hawk-Dove game, this setting has three [http://en.wikipedia.org/wiki/Nash_equilibria Nash Equilibria].

The Steady States

The 3 [http://en.wikipedia.org/wiki/Nash_equilibria Nash Equilibria] are the following: Two are pure contingent strategy profiles, in which each strain plays for the win while the other simply stays put until displaced; the third one is a mixed equilibrium, in which each strain randomly shifts from one strategy to the other.

In organic parlance, this means that the scenario where both reproduce is unfeasible because resources are limited. Moreover, fixation of either one is unlikely: WildType is the only one that fixates nitrogen, and since life requires nitrous compounds this generates a selective pressure for it; TransGenic is much more efficient in terms of resources/cell.division, and therefore is naturally selected for. The final alternative where both do nothing is also unlikely as life never stays put...

Interestingly, this Game Theory formalism doesn't discriminate between an heterogeneous population of choices, and a shifting population of homogeneous choices. Therefore, the same expected result arises from a population of free-for-all cells as well as for 2 massive populations of synchronized cells. In formal words, Strategy Polymorphism & Strategy Mixing yield the same result.

One particular property of these three Nash Equilibria, is that for the case when there exists an [http://en.wikipedia.org/wiki/Uncorrelated_asymmetry Uncorrelated Asymmetry] (i.e. row players know they are row players, as do column players), and continue to randomly choose again, the mixed equilibrium becomes an [http://en.wikipedia.org/wiki/Evolutionarily_stable_strategy Evolutionary Steady Strategy]. This means that said Steady State is resistant to random mutation shifting it from the equilibrium. In other words, Natural Selection keeps it in place. Therefore, since our strains are fixed as either WildType or TransGenic, and they can't shift from one to the other, the final steady state will be a resistant ratio of both strains present. Id est, they will coexist (and they won't destroy the world's biodiversity in the process).

We could refine our payoff matrix by calculating better values for each of the green strategies. This would enable us to calculate the expected final ratio of the populations. However, that would require extreme analysis of the chassis, as well as sophisticated experimental measurements, both actions that lie outside the current project's ambitions.

Conclusions

As is the case in the traditional Snowdrift game, the steady state mechanics involve an equilibrium of both green events, generating a fixed ratio of both populations at steady time.