Modeling

1. Overview

In terms of design principle, the modeling greatly contributes to obtain the improved part required for the project. We constructed the model for replication the payoff matrix, and applied the model to solve two problems found in wet lab experiments. The first application is model-based improvement of a previously existing part. Our project requires the improvement of the previously existing part for Chloramphenicol resistance (CmR) coding sequence to decrease unintended antibiotic effect by leaky expression, because we had to diminish such effect for implementation of a payoff matrix in Prisoners’ Dilemma. The second application is confirmation of tunability of our payoff matrix.

2. Mathematical Model

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

The parameters used in the equations are shown at Table.1

Table.1 The parameters in the equations

We then set specific parameters to draw the equations (2) ~ (5) for modeling for the 4 type of Punishment in the payoff matrix about prisoner A by developing the 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 CmR is excepted from (1). If the effect of CmR is above the toxicity of Cm, is set to 0 at the equation (4) and (5).

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. Note that, 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.

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

3. Modeling selected a solution for leaky expression problem

3.1. Introduction

Although our initial modeling successfully replicate the payoff matrix, Our wet lab found an unexpected leakage in the promoter. In other words, the results of the modeling we had done beforehand did not match with the results of the wet lab. We adjusted the model to include leaky expression of CmR protein.

We then planned two modifications of the circuits, each of which is evaluated by modeling, to circumvent the effect of leaky expressions. The first modification is addition of ssrA degradation tag to the C-terminal of the chloramphenicol resistant protein (CmR). The other is increase of Cm concentration. We here 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

o 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 wet lab results showing leaked expression of CmR.

We calculated the OD of after 480 minutes by using the following equations (3), (5) and (10). 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.

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 (12) and (13). Therefore, we revised the following equations (10) to the following (14).

We also added the equation (15), 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

In contrast to the above models with 100 microg/mL Cm, 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 (12) ~ (15). Unfortunately, the OD of “high” and “middle” were decreased only for a little. We thus decided that there was no need to run any experiments in the wet lab, since the obtained results were not positive.

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

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 (14) and (15) to the equations (16) and (17).

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. In terms of design principle, the modeling thus greatly contributes to obtain an improved part required for the project.

4. Adjusting the suitable Cm concentration

4.1. Introduction

Although we planned to replicate a symmetrical payoff matrix, one replicated in our wet lab experiment was asymmetrical, in which the growth inhibition differs between Prisoner A and B. The reason why the growth inhibition differed between Prisoner A and Prisoner B can be two types of difference. One is that the degree of the metabolic burden is different between producing rhlI and lasI, and the other is that the concentration of AHL given to the E. coli is different.

For future wet experiments, we need to confirm, by modeling, whether we can tune the asymmetry of not. In the following section, we here show that the tunableness of our pay off matrix implemented by E. coli. Even though large unintended difference in growth occurs between Prisoner A and B, suitable concentrations of Cm in each prisoner can minimize such difference in growth.

4.2. Equations

We used the equations the following (18) ~ (28).

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

We found the optimal Cm concentration for when the degree of metabolic burden is different between Prisoner A and B.

We obtained the optimal concentration of Cm to make the payoff matrix symmetrical when the concentration of C4HSL and 3OC12HSL is the same, but the metabolic burden of producing proteins is different.

We set the concentration of C4HSL and 3OC12HSL to 500 nM.

The OD of the 4 types of growth inhibition for Prisoner A and B is shown in Fig.4-1-4-1 Here, the maximum difference in growth is when the growth inhibition is high, and it is over 30 %. E shows the absolute value of the difference in growth between the ODs of A and B, per growth inhibition.

Fig.4-1-4-1．The ODs of the 4 types of growth inhibition at asymmetrical matrix

Next, we fixed the concentration of Cm for Prisoner A, and by changing the concentration of Cm for Prisoner B, processed the difference in growth of all the 4 types of growth inhibition to around 10%. The results are shown in Fig.4-1-4-2.

Fig.4-1-4-2．The ODs of the 4 types of growth inhibition at symmetrical matrix

The difference in growth of “middle” increased, but only increased to around 10 %. Instead, when the Cm concentration of Prisoner B is 69 µg/mL, the difference in growth of “high” was reduced to less than 1/3 compared to before adjusting the Cm concentration, and was below 10 %. From this result, we can conclude that we have made the difference in growth small for all the 4 types of growth inhibition between Prisoner A and B.

4.4. When the AHL concentration is different

Next we obtained the optimal concentration of Cm to make the payoff matrix symmetrical when the metabolic burden of producing proteins is the same, but the concentration of C4HSL and 3OC12HSL is different.

Fig.4-1-4-3. The ODs of the 4 types of growth inhibition at asymmetrical matrix

Compared to Fig.4-1-4-1., although the difference in growth is smaller, the difference of “low” and “none” is above 10 %. Therefore, we fixed the concentration of Cm of Prisoner A, and by changing the concentration of Cm for Prisoner B, adjusted the concentration of Cm so that the difference in growth of all the 4 types of growth inhibition was processed to be within 10%. The results are shown in Fig.4-1-4-4.

Fig.4-1-4-4. The ODs of the 4 types of growth inhibition at symmetrical matrix

When the concentration of Cm of Prisoner B is 72, the difference in growths of all 4 types of growth inhibition was within 10%.

From the results of 4.3 and 4.4, in order to make the payoff matrix symmetrical, adjusting the concentration of Cm is an effective solution.

Team:Tokyo Tech/Project/Modeling