Team:Tokyo Tech/Projects/RPS-game/index.htm
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Rock-Paper-Scissors game
1. The Hands
2. The Judge
2.1 Using AND-Gate promoters to create Judges
2.2 Creating Parts that responded correctly to our set of Signaling Molecules
2.3 Improving PlsrA3>
2.4 Improving PlasI
3. The Randomizers
3.1 Single Colony Isolation
3.2 Conditional Knockout by Recombination
3.3 The Randomizers
3.4 Survival of one strain
In 1996 Durret and Levin described a system of three types of bacteria that competed for survival in dynamic that resembled a Rock-Paper-Scissors (RPS) game. The bacteria used two main evolutionary stable strategies (ESS) to outcompete their rivals: the production of a toxin (a bacteriocin called colicin) that was toxic to other strains and a higher birth rate than their rival strains. The three types of bacteria described in the model by Durret and Levin were colicin-producing E. coli (R), colicin-resistant E. coli (P) and colicin-sensitive E. coli (S). The colicin producer outcompeted the colicin sensitive by producing the colicin, the colicin sensitive bacteria outcompeted the colicin resistant because it's birth rate was higher than that of the colicin resistant, and the colicin resistant outcompeted the colicin producer because it' birth rate was higher than that of the colicin producer. The colicin resistant bacteria were also able to produce colicin, but at a lower energetic cost, which allowed them to have a higher birth rate. The system was described by the following general differential equations ∑_(i=1)^n?u_i < 1
(du_i)/dt=β_i u_i u_0- u_i ( δ_i+∑_(j=1)^(n-1)??γ_j u_j ?)
u_0=1-∑_(i=1)^n?u_i
Where
u_i i's concentration in arbitrary units (a.u.) β_i i's birth rate δ_i i's death rate γ_j i's death rate due to j's bacteriocin u_0 carrying capacity In the model described by Durret and Levin's paper the equations were as follows: Producer (du_1)/dt=β_1 u_1 u_0- u_1 δ_1 Resistant (du_2)/dt=β_2 u_2 u_0- u_2 δ_2 Sensitive (du_3)/dt=β_3 u_3 u_0- u_i ( δ_3+γ_1 u_1+ γ_2 u_2 ) Setting the parameters as follows, the following graph was created by Durret and Levin. β_1=3, β_2=3.2, β_3=4 γ_1 = 3, γ_2 = 0.5 δ_1 = δ_2 = δ_3 =1 Note that these parameters satisfy the following relations: β_3>β_2>β_1, γ_1> γ_2, and δ_1=δ_2= δ_3
In this model, however, there is no case where the colicin-producer can survive for infinitively long periods of time if the colicin-resistant's initial concentration is greater than zero. This can be seen in the graph drawn by Durret and Levin, where the lines along the axis for u_1 (in the rightmost lower corner) always converge either to u_3 or u_2. We highlighted this fact by circling the point in the graph below.
The limited-resistance strain survives if u_1= 0.29, u_2= 0.32, u_3= 0.29 (a.u.).
The sensitive strain survives if u_1= 0.29, u_2= 0.29, u_3= 0.32 (a.u.).
In practice, this minimal variation of concentrations will cause the outcome of Rock, Paper or Scissors signaling molecule to be random. Consequently, we can conclude that this randomizer is not only feasible but also practical and effective (let alone interesting).