Model

Overview

Our circuit was composed of three sub-circuits, each functioning to produce AHL, Lux R, and luciferase.

We plotted 8 different graphs:

1. Time vs. AHL production graph: We aimed to determine whether AHL or Lux R is the limiting factor, so we plotted the concentration of AHL for time.
2. Time vs. Lux R production graph: We aimed to determine whether AHL or Lux R is the limiting factor, so we plotted the concentration of Lux R for time.
3. Concentration of fusaric acid vs. AHL: We plotted the amount of AHL that will be produced depending on the concentration of fusaric acid.
4. The concentration of fusaric acid vs. the amount of complex: We plotted the concentration of the complex that will be formed depending on the fusaric acid concentration, as its concentration will depend on the limiting substance’s concentration (AHL concentration).
5. Concentration of fusaric acid vs. rate of luciferase production: We plotted the rate of luciferase production depending on the fusaric acid concentration, which showed a similar shape as the 4th graph, as the rate of luciferase production will depend on the concentration of the complex formed.
6. 3D graph: We plotted the 3d graph, with its x-axis showing time, its y-axis the concentration of fusaric acid, and its z-axis showing the amount of luciferase produced. This allowed us to view how the time and concentration of luciferase affect the amount of luciferase produced in one view.
7. The concentration of fusaric acid vs. the amount of luciferase: We compared the 5th graph to the graph of the total amount of luciferase produced after 300 minutes depending on the concentration of fusaric acid.
8. Each concentration of fusaric acid vs. the amount of luciferase produced: Finally, we plotted the graph of the total amount of luciferase produced after 300 minutes again, but this time the graph shows 6 lines for 6 different concentrations of fusaric acid, and we compared how those different concentrations impact the amount of luciferase produced after 300 minutes.

Objective

Since we will detect the light generated by the luciferase present in E. coli, our objective is to figure out:

1. Whether AHL or Lux R is the limiting factor
2. The amount of limiting factor (from step 1) produced depending on the concentration of fusaric acid
3. Since the amount of complex produced will depend on the amount of limiting factor, find the amount of complex depending on the concentration of fusaric acid
4. The rate of luciferase production depending on the concentration of fusaric acid
5. Amount of luciferase depending on the concentration of fusaric acid (100-300 min)
6. A 3D graph that allows us to see how time and concentration of fusaric acid affect the amount of luciferase produced

Pre-Processing Data

Since we didn’t have real-world experimental values, we researched for each factor which will be used to determine the graphs that we plotted.

Parameters

• Vmax: 100μM/h (Maximum rate of luciferase production)
• Km: 50μM (Michaelis constant)
• K rod: 0.2h^-1 (Rate of production)
• K degr: 0.1h^-1 (Degradation rate)
• t: 300 minutes (Total time for production - may change from 0 to 300 min for graphs that have changing time as x-axis) The reason we chose this time is to find out if our protein will denature
• [Fusarium] : (Concentration of fusaric acid - ranging from 100 ~ 600μM)
• a LuxR: 150μM/h (Rate constant for transcription/translation of Lux R in μM/h)
• B LuxR: 0.05h^-1 (Degradation rate constant for Lux R in h^-1)

OD Modeling

The relationship between the amount of luciferase generated depending on the amount of Lux R produced and variation can be represented by various mathematical models. We used the Michaelis-Menten Kinetics model since it is commonly used to describe enzyme kinetics and can be applied to the production of luciferase, which depends on the concentration of Lux R and AHL.

Graphs

Time vs. AHL Production Graph

Since AHL is one of the factors that will affect the amount of complex produced, which will then affect the amount of luciferase produced, we aimed to determine whether AHL or Lux R is the limiting factor, so we plotted the concentration of AHL for time. The equation is shown as follows:

Where the variables are:

• AHL(t): Concentration of AHL produced at time t (in μM)
• Vmax: 100μM/h (Maximum rate of luciferase production)
• Km: 50μM (Michaelis constant)
• [Fusarium]: 200μM (Kept constant for this graph)

Time vs. Lux R Production Graph

Since Lux R is one of the factors that will affect the amount of complex produced, which will then affect the amount of luciferase produced, we aimed to determine whether AHL or Lux R is the limiting factor, so we plotted the concentration of Lux R for time. The equation is shown as follows:

Where the variables are:

• [Lux R](t) : Concentration of Lux R produced at time t (in μM)
• a LuxR : 150μM/h (Rate constant for transcription/translation of Lux R in μM/h)
• B LuxR : 0.05h^-1 (Degradation rate constant for Lux R in h^-1)
• t : Time in hours

Concentration of Fusaric Acid vs. AHL Graph

By comparing time vs. AHL production graph and time vs. Lux R production graph, we found out that AHL is the limiting factor in producing luciferase, which allowed us to plot the third graph. The equation is shown as follows:

Where the variables are:

• [AHL] : Concentration of AHL produced (in μM)
• [Fusarium] : Concentration of fusaric acid, ranging from 0 - 600μM
• Vmax : 100μM/h (Maximum rate of luciferase production, representing the maximum concentration of AHL that can be produced when fusaric acid is abundant)
• Km : 50μM (Michaelis constant, which is the concentration of fusaric acid at which the production rate is half of Vmax. It indicates how effectively the system responds to changes in fusaric acid concentration)

Concentration of Fusaric Acid vs. Amount of Complex Graph

Since AHL is the limiting factor in producing luciferase, the amount of complex produced by AHL and Lux R bonding 1:1 will entirely depend on the AHL, allowing us to plot the graph of the amount of complex produced depending on the concentration of fusaric acid. The equation is shown as follows:

Where the variables are:

• [C](Fusarium): Concentration of the complex formed (in μM)
• [AHL](Fusarium): Concentration of AHL produced based on fusaric acid concentration
• [Lux R]: Concentration of Lux R available for binding

Concentration of Fusaric Acid vs. Rate of Luciferase Production

Since the amount of luciferase produced will be directly proportional to the amount of complex produced, its rate will be basically the same as the amount of complex produced. This allows us to plot the graph of the rate of luciferase produced depending on the concentration of fusaric acid. The equation is shown as follows:

Where the variables are:

• v: Rate of luciferase production
• [Fusarium]: Concentration of fusaric acid ranging from 0 to 600 μM

3D Graph: Time, Concentration of Fusaric Acid, Amount of Luciferase Produced

Since have figured out the relationship between concentration of fusaric acid and the rate of luciferase production, we can figure out the amount of luciferase produced depending on the time as well, which is just multiplying the rate of luciferase production in specific concentration of fusaric acid to the specific time. This allows us to plot a 3D graph, with the x-axis being time, the y-axis being concentration of fusaric acid, and the z-axis as amount of luciferase produced.

Concentration of Fusaric Acid vs. Amount of Luciferase Graph

The amount of luciferase produced will vary depending on the concentration of fusaric acid, which we figured out with concentration of fusaric acid vs. the rate of luciferase production graph, and we have plotted the 3D graph of its relations. The below graph shows the amount of luciferase produced at time = 300 min, which will be seen in the 3D graph from the right-hand side.

Time vs. Amount of Luciferase Produced

The amount of luciferase produced will vary depending on the concentration of fusaric acid, which we figured out with concentration of fusaric acid vs. the rate of luciferase production graph, and we have plotted the 3D graph of its relations. The below graph shows the amount of luciferase produced depending on the time. One important thing about this graph is that there are 6 different lines, each representing the concentration of fusaric acid, ranging from 100 to 600 μM with an interval of 100 μM. This will be seen in the 3D graph from the left-hand side.

Conclusion

We were able to figure out that between AHL and Lux R, AHL is the limiting factor, which tells us that the amount of complex formed will depend on the AHL. From our research, we know that the complex formed will be directly proportional to the amount of luciferase produced, thus we were able to figure out the rate of luciferase production depending on the concentration of fusaric acid. Finally, we plotted the 3D graph, which shows how both factors of time and concentration of fusaric acid affect the amount of luciferase produced.