Contents

1. Overview

2. Mathematical Model

3. The Solution for Wet Lab 1

3.1. Introduction

3.2. Modeling for the 4 Different Types of Punishment in Payoff Matrix

3.3. Considering the Leakage of CmR from the Result of Wet Lab

3.4. Two Solutions for Overcoming the Leakage of CmR

3.4.1. Solution 1: Increasing the Concentration of Cm

3.4.2. Solution 2: Tagging the CmR Protein with ssrA

4. Adjusting the Suitable Cm Concentration

4.1. Introduction

4.2. Equations

4.3. When the degree of growth inhibition is different between Prisoner A and B

4.4. When the AHL concentration is different

1. Introduction

We constructed the model for replication the payoff matrix. We focused on finding a solution to the problems which happened in the wet lab experiment, and did 2 types of modeling. In one modeling, we were able to solve the problem which happened in the wet lab. In the second modeling, we could not reflect the results to our wet lab experiment, but did succeed to propose a solution to the problem.

2. Mathematical Model

We constructed the general model for the change of OD over time. The model is shown at the following equation (1).

We constructed the equation (2) ~ (5) for modeling for the 4 type of Punishment in the payoff matrix about prisoner A by developing the equation (1).

The equation (2) and (4) are the models that the term for metabolic burden is added to (1). The equation (2) and (3) are the models that the term, for the effect of the leakage of CmR is added to (1). The equation (4) and (5) are the models that the term for the effect of CmR is added to (1). If the effect of CmR is above the toxicity of Cm, and is set to 0 at these equations.

We also constructed the equation (6) ~ (9) for modeling for the 4 type of Punishment in the payoff matrix about prisoner B in the same way as prisoner A. However, the equation (7) and (9) contain not only the term of metabolic burden by production of AHL but also the term of metabolic burden by production GFP.

We also constructed the general model for production of AHL, the following equation (10) and (11).

The equation (10) is the model for production of C4HSL in prisoner A coli. The equation (11) is the model for production of 3OCHSL in prisoner B coli.

The equation (12) is the model for the leakage of CmR.

The results of replicating the payoff matrix which the ODs of 4 type of growth inhibition after 8 hours by using these equation are shown in Fig.4-1-2-1 and Fig.4-1-2-2. Fig.4-1-2-1 is the matrix of prisoner A. fig.4-1-2-2 is the matrix of prisoner B.

Fig.4-1-2-1. The payoff matrix of prisoner A

Fig.4-1-2-2. The payoff matrix of prisoner B

3. The Solution for Wet LAb 1

3.1. Introduction

From our modeling which objective was to precisely replicate the payoff matrix, we found that there was an unexpected leakage in the promoter, since the results of the modeling we had done beforehand, did not match with the results of the wet lab.

After modeling 2 different solutions to overcome the leakage, we found the most efficient solution tagging the CmR protein with ssrA, which successfully led to expected results in our wet lab and ultimately led to our precise replication of the payoff matrix.

We could not obtain positive results in our modeling by increasing the concentration of Cm, which was the other solution, so we confirmed that tagging the CmR gene with ssrA is a better solution.

We focused on the OD of the ‘middle’ and ‘none’ growth inhibition about prisoner A for modeling.

3.2. Modeling for the 4 Different Types of Punishment in Payoff Matrix

To precisely replicate the payoff matrix, we calculated the OD of the ‘middle’ and ‘none’ growth inhibition about prisoner A, but the results of the modeling did not match with the results of the wet lab.

We calculated the OD of after 480 minutes by using the following equations (13) ~ (15).

The equation (13) and (14) are the model for the ODs of each ‘middle’ and ‘none’ growth inhibition about prisoner A. The equation (15) is the model for production of CmR. We compared the result of modeling (Fig.4-1-3-1A) and wet lab (Fig.4-1-3-1B). The graph of comparision is shown at Fig.4-1-3-1.

Fig.4-1-3-1. The change of OD over time (Prisoner A)

3.3. Considering the Leakage of CmR from the Result of Wet Lab

Since the results of the modeling did not match with the wet lab, assuming that there was a leakage in the promoter of the CmR, we calculated the OD considering the leakage of the promoter of the CmR, in which the results successfully matched with the wet lab.

Given the result from the wet lab that the OD of ”none” and “low” hardly differs from the modeling, on the other hand, “middle” and “high” had huge difference, we assumed that the promoter is activated and CmR is produced even in the absence of AHL.

Therefore, we used the equations (3) and (5) instead of (13) and (14). Therefore, we revised the following equations (15) to the following (16).

We also added the equation (18), which represents the production of CmR due to the leakage of the promotor, in the absence of AHL.

We compared the result of modeling (Fig.4-1-3-2A) and wet lab(Fig.4-1-3-2B). The graph of comparision is shown at Fig.4-1-3-2. The results of the calculation were similar to the results of the wet lab.

Fig.4-1-3-2. The change of OD over time with the leakage of CmR

3.4. Two Solutions for Overcoming the Leakage of CmR

To overcome the leakage, we tried modeling two different solutions to see which one is more efficient.

3.4.1. Solution 1: Increasing the Concentration of Cm

After constructing a model in which the concentration of Cm is increased, we decided that there was no need to run any experiments in the wet lab, since the obtained results were not positive. Although up to this point, the concentration of Cm was set to 100 microg/mL, we calculated the OD, setting the concentration of Cm to 150 microg/mL.

The results are shown in Fig.4-1-3-3. We used the equations (3) and (5), (16) and (17). The OD of “high” and “middle” were decreased only for a little.

Fig.4-1-3-3. The change of OD over time with increasing the concentration of Cm

3.4.2. Solution 2: Tagging the CmR Protein with ssrA

Obtaining positive results of modeling in which we added the ssrA tag to the CmR protein, successfully led to expected results in our wet lab and ultimately led to our precise replication of the payoff matrix.

As the other solution, we tagged CmR with ssrA, which is a degradation tag.

In order to reflect the effect of the ssrA tag, we customized each of the equations (16) and (17) to the equations (10) and (12).

We compared the result of modeling and wet lab. The graph of comparision is shown at Fig.4-1-3-4.

Fig.4-1-3-4. The change of OD over time with tagging CmR with ssrA tag

From the results, tagging CmR with ssrA is a more effective solution to the leaky promoter producing CmR, compared to increasing the concentration of Cm.