Team:Qdai/Model
Modeling
1.
Synthetic biology can regulate the expression levels of genes.
By predicting the expression level of a protein in advance, we can consider its function as a sensor.
In this project, we modeled the expression of the protein in response to the concentration of carbon monoxide and its ability to act as a sensor.
We decided to use modeling to predict the relationship between carbon monoxide concentration, CooA expression, and luciferase expression.
First of all, CooA reacts with CO and becomes the structure of CooA with a structural change. This is called CooAco. The relationship between carbon monoxide, CooA, and CooAco can be expressed as follows.
\[\text{CO} + CooA \overset{k_1}{\underset{k_2}{\rightleftharpoons}} CooA_{\text{CO}}\]
Based on this reaction, the rate equation for the production of CooAco is as follows.
\[\frac{[CooAco]}{dt}=k_1[CO][CooA]-k_2[CooAco] \]
CO and CooA settle into a steady state when the reaction is complete.So, we can assume that the concentrations of CooA, CO, and CooA here are in a steady state.
In other words, the concentrations of CooA, C O, and CooA here can be in the steady state.
The concentrations of CooA, CO, and CooA in the steady state are denoted as [𝐶𝑜𝑜𝐴]∗, [𝐶𝑂]∗, and [𝐶𝑜𝑜𝐴𝑐𝑜]∗, respectively.
We want to consider the concentration (expression level) of luciferase as [luciferase].
The concentration of luciferase can be expressed by the equation as follows.
\[\frac{d}{dt} \left[ luciferase \right] = \gamma \left[ CooA_{\text{CO}} \right] ^{*} - \alpha \left[ luciferase \right]\]
β can be dependent on the concentration of [CooAco]* that binds to the promoter p cooM. So, we can consider β by placing it as γ[CooAco]*.
Solving this differential equation, we obtain the following equation.
\[\left[ luciferase \right] \frac{\gamma \left[ CooA_{\text{CO}} \right] ^{*}}{\alpha} + \left( y_0 - \frac{\gamma \left[ CooA_{\text{CO}} \right] ^{*}}{\alpha} \right) e^{-\alpha t}\]
The graph showing the relationship between [CooAco]* and [luciferase] is assumed to be as follows.
The relationship between the amount of CooA in the steady state and the expression level of luciferase was predicted to be linear.
2.
We also predicted that there would be a negative impact on growth due to the constant expression of the CooA gene and the CO. However, by increasing the amount of However, by increasing the amount of CooA and luciferase, the overall amount of light will increase. We think that we can predict the right part of the balance as the balance between the intensity of light and the proliferation that is suppressed by the logistic. First, the logistic equation is given as follows
\[\frac{d}{dt} N = \left( 1 - \frac{N}{K}\right) \cdot N \cdot r \cdot f \left( \left[\text{CO}\right] \right) g\left( \left[ CooA\right] , \left[ luciferase \right] \right)\]
where each f and g is defined as the following equation.
\[f \left( X_{\text{CO}} \right) = \frac{1}{1+a_1 \cdot X_{\text{CO}}}\]
\[g \left( X_{CooA} , Y_{luciferase} \right) = \frac{1}{1+ \left( a_2 \cdot X_{CooA} + a_3 \cdot Y_{luciferase} \right)}\]
Here, "f" represents the effect of carbon monoxide. E. coli can survive under anaerobic conditions. However, it can be predicted that their ability to grow under anaerobic conditions is lower than that under aerobic conditions. "g" considers the effect of E. coli producing proteins that it does not originally produce.
For simplicity, we will represent the effect of Luciferase and the effect of CooA by a single letter, "Z".
\[g \left( X_{CooA} , Y_{luciferase} \right) = \frac{1}{1 + Z}\]
\[a_2 \cdot X_{CooA} + a_3 \cdot Y_{luciferase} = Z\]
We solve the above logistic equation.
\[N = \frac{K N_0}{N_0 + \left( K - N_0\right) exp\left( \frac{-rt}{\left( 1 + a_1 \left[ \text{CO} \right] \right) \left( 1 + Z\right)} \right)}\]
By finding Q = N x Z, we can obtain the overall amount of light emitted.
\[Q = NZ = \frac{Z K N_0}{N_0 + \left( K - N_0\right) exp\left( \frac{-rt}{\left( 1 + a_1 \left[ \text{CO} \right] \right) \left( 1 + Z\right)} \right)}\]
We took the CO, Z, and Q axes and graphed them as shown below.
Here we are considering the effects of CO and genes in terms of f and g functions, but the best may be necessary.