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 PlsrA

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 Durret & Levin (1996)
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 existence of this instability along the u_1 axis does not allow us to construct a randomizer that can function based on minimal differences in the initial concentrations of the three different populations of bacteria. For that reason, we modified the differential equations of the model so that any of the three types of bacteria could win for infinitely long periods of time (could win definitely). More specifically, we limited the resistance of the colicin-resistant bacteria in the sense that it would produce a type of bacteriocin that is only toxic to itself and to the sensitive strain, and additionally the resistant strain would also be vulnerable to the colicin produced by the colicin producer. Producer (du_1)/dt=β_1 u_1 u_0- u_1 δ_1 Limited Resistance (du_2)/dt=β_2 u_2 u_0- u_2 (δ_2+γ_1' u_1+γ_2' u_2) Sensitive (du_3)/dt=β_3 u_3 u_0- u_3 ( δ_3+γ_1 u_1+ γ_2 u_2) If we set the parameters as follows β_3=3.1, β_2=3.2,? β?_1=4 γ_1=2,γ_2=0.5,? ? γ?_1'=0.1 ,γ?_2'=0.002 δ_1=δ_2= δ_3=1 and we graph this equations using a Matlab program, we get a graph which clearly shows there are stable points on each of the three axes. These stable points (u_1,0,0), (0,u_2,0) and (0,0,u_3) indicate that for the equations we have set all of the three strains may ultimately survive for infinite peiriods of time even if the initial density of the other two strains is positive. To see the difference between our model and Durret and Levin's model more clearly, we also plotted Durret and Levin's model using Matlab. Note that the parameters we have set for our equations satisfy the initail conditions of the model proposed by Durret and Levin (indicated in black font) in the sense that β_3>β_2>β_1, γ_1>γ_2 >γ_1'>γ?_2', and δ_1=δ_2= δ_3. The two new terms we added are indicated in red. From a biological perspective, this model describes the existence of two strains of bacteria that produce two different types of bacteriocin. One of these strains is not completely resistant to its own bacteriocin nor to the bacteriocin produced by its rival strain. This can be explained by thinking that the strain with this limitation in its bacteriocin resistance does not produce enough/effective resistance protein, which could be a consequence of it being a mutant of a colicin-sensitive strain. The next step is to find a set of parameters that satisfy the conditions set by the model of Durret and Levin, the above condition of allowing stable points on each of the three axes and also allow each of the three different strains of E. coli to survive in a random fashion by minimal differences on the initial concentrations of each strain. We found, by writing another program in Matlab, that the following set of parameters satisfy all of the above mentioned conditions. β_3=3, β_2=3.2,? β?_1=5.1 γ_1=2,γ_2=1,? ? γ?_1'=0.2 ,γ?_2'=0.01 δ_1=δ_2= δ_3=1 As can be seen in the graphs below, each of the strains can survive if their initial density in only tree hundredths (a.u.) greater than the other two strains' initial concentrations. The producer strain survives if u_1= 0.32, u_2= 0.29, u_3= 0.29 (a.u.).
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).