Team:iBowu-China/Model
In our project, we hope to detect the type and density of pathogenic bacteria by detecting AHL concentration in the cell-free synthetic test paper. To achieve this goal, we designed a simple mathematical model based on the differential equations that describe the biochemical processes of the cell-free system. To optimize the model, we used GFP, which is easier to quantify, as a reporter to measure the transcriptional level of promoter lux (AHL-biosensor) with different concentrations of AHL and different copies of the template. With these data, we optimized model parameters and obtained the final model. The model can be used to predict the minimum detection concentration of AHL and optimize the DNA content. In addition, we have developed a modeling software, which can use a given system description file to calculate the dynamic behavior of the system, which will help synthetic biologists to improve their system.
In the Cell free system, the upstream gene LuxR of the biological loop is constitutively expressed, indicating that the concentration of the expression product of the LuxR gene in the system consists of simple generation and degradation. Therefore, the change rate of LuxR mRNA concentration is Eq.1:
α_m_R is the transcription rate constant of LuxR and N_R is the concentration of the LuxR gene in the system. The first term on the right side of Eq.1 is the rate of generation and the second term is the rate of degradation. To simplify the equation, we use a simple linear equation instead of the Michaelis-Menten equation to represent the rate of degradation of metabolites. β_(m_R ) is the degradation constant. After mRNA is generated, it will be translated into protein. The equation for the concentration change rate of LuxR protein is shown in Eq.2.
Like Eq.1, the first term on the right side of Eq.2 is the rate of formation and the second term is the rate of degradation. α_P_R is the translation rate constant and β_P_R is the degradation constant. The LuxR protein binds to AHL to form a complex, which will bind to the promoter of GFP gene and promotes the expression of GFP. We use Hill function to express the transcription rate of GFP.
α_leak is the transcription rate (leakage) of GFP without binding of transcription factors. α_m_B is the expression rate constant of GFP after binding of transcription factors. N_B s the concentration of GFP gene. k_B is the binding constant of transcription factor to promoter. β_(m_B ) is the degradation constant of mRNA of the GFP. A is the concentration of AHL in solution. Because a transcription factor is composed of two AHLs and two LuxR proteins, the power of Hill function is 2. Like equation 2, the change rate of GFP protein concentration can be expressed as equation 4.
Because the gene loop is a simple one-way regulatory system, the concentration of substances involved in the loop will tend to be stable without external influence (concentration does not change with time). In equilibrium, the left and right sides of equation 1-4 will be equal to 0:
Solving equation 5 ~ 8, we can get:
Equations 9-12 characterize the relationship between the concentration of LuxR's mRNA and protein, and the concentration of GFP's mRNA and protein with the concentration of LuxR's gene and the concentration of AHL in the equilibrium state of the system. We designed 8 experiments groups. Each group contained four plasmids with different concentrations. The concentration of AHL was different between the experimental groups. Then the concentration of GFP in the equilibrium state of each experiment system was measured, as shown in the table below:
| AHL(nM) | DNA(ng/uL) | FROS_1(ng/ul) | FROS_2(ng/ul) | FROS_3(ng/ul) |
|---|---|---|---|---|
| 0 | 30.3 | 311.5 | 274.3 | 260.3 |
| 0 | 20.2 | 218.3 | 208.7 | 198.7 |
| 0 | 10.1 | 190.7 | 215.5 | 196.3 |
| 0 | 5.05 | 196.7 | 187.1 | 195.1 |
| 0.1 | 30.3 | 267.1 | 245.1 | 257.9 |
| 0.1 | 20.2 | 205.9 | 200.7 | 215.9 |
| 0.1 | 10.1 | 179.5 | 185.1 | 183.5 |
| 0.1 | 5.05 | 190.7 | 167.1 | 172.7 |
| 1 | 30.3 | 523.5 | 467.1 | 484.7 |
| 1 | 20.2 | 367.5 | 354.7 | 347.5 |
| 1 | 10.1 | 202.7 | 188.7 | 221.9 |
| 1 | 5.05 | 180.7 | 170.3 | 166.7 |
| 10 | 30.3 | 4562.3 | 3941.5 | 4173.9 |
| 10 | 20.2 | 2079.1 | 2271.1 | 3023.5 |
| 10 | 10.1 | 619.5 | 762.7 | 1037.1 |
| 10 | 5.05 | 332.3 | 247.1 | 274.3 |
| 100 | 30.3 | 5970.3 | 5839.9 | 6104.3 |
| 100 | 20.2 | 3561.1 | 4017.9 | 4595.5 |
| 100 | 10.1 | 862.7 | 691.1 | 1151.1 |
| 100 | 5.05 | 549.9 | 581.9 | 391.5 |
| 500 | 30.3 | 8887.5 | 10196.7 | 6416.7 |
| 500 | 20.2 | 5480.3 | 6117.5 | 4792.7 |
| 500 | 10.1 | 848.7 | 1189.9 | 813.5 |
| 500 | 5.05 | 547.1 | 463.9 | 447.5 |
| 1000 | 30.3 | 8141.9 | 7137.9 | 6092.7 |
| 1000 | 20.2 | 7403.1 | 5264.7 | 4765.9 |
| 1000 | 10.1 | 1095.5 | 753.9 | 1280.3 |
| 1000 | 5.05 | 537.9 | 519.9 | 501.5 |
| 1500 | 30.3 | 6789.5 | 7589.9 | 6439.9 |
| 1500 | 20.2 | 5504.7 | 3591.1 | 4427.1 |
| 1500 | 10.1 | 1283.5 | 1046.3 | 1187.9 |
| 1500 | 5.05 | 643.5 | 451.1 | 476.3 |
(Table. 1. The concentration of GFP protein in equilibrium period under different concentration of AHL and DNA.)
The fluorescence intensity of GFP was measured directly in the experiment. The relationship between fluorescence intensity and concentration was as follows:
(Figure. 1. The relationship between GFP concentration and fluorescence brightness.)
Because LuxR and GFP genes are located in the same plasmid in the experiment, the concentration of LuxR and GFP in the system is the same. In the experiment, some other plasmids were containedinto the plasmid, so a variable was needed to express the proportion of the target plasmid in the equation.
γ is the content (percentage) of the target plasmid in the added DNA. N is the added DNA concentration. The least square method is used to fit the above equation with Table.1 (See simple_model_optimize.py for the code). The parameters of the equation are as follows:
The experimental data and the fitting curve are as follows:
(Figure.2. The curved surface of the concentration of GFP during equilibrium with DNA concentration and AHL concentration was fitted by least square method. The black point was the experimental data point and the blue grid line was the fitting curve.)
From the fitting curve, it can be seen that the equilibrium concentration of GFP increases with the increase of DNA concentration when the concentration of AHL is constant. When the concentration of DNA is constant, the equilibrium concentration of GFP changes obviously with the concentration of AHL when the concentration of AHL is low, but when the concentration of AHL is high, the equilibrium concentration of GFP hardly increases with the increase of the concentration of AHL(see Fig.3.). The code can be seen in equilibrium.py.
Taking the parameters fitted above into the ordinary differential equations (1-4), we can draw the concentration curve of each substance (see Cell_free.py code) under the given concentration of AHL and DNA, as shown in Fig.3(Suppose AHL was added to the system from the beginning).
(Figure.3. The concentration curves of four substances (LuxR, GFP, and LuxR) were plotted using the fitted parameters. The amount of AHL and DNA in the curve was 1000 nM and 20.2 ng/ul respectively.)
It can be seen that the concentration of LuxR products increases rapidly and can reach a steady state within 5 hours. As the downstream gene of the loop, GFP has a certain time delay, and the concentration of GFP increases slowly, and finally tends to be stable. In the real situation, Cell-free test strips are generally produced first, and there is also a case where LuxR is preferentially expressed before joining AHL. Therefore, we simulated by adding AHL in a delayed manner. The following figure shows the case where AHL is added to the reaction up to 5 hours.
(Figure.4. he concentration curve of each metabolite of AHL was added to the reaction at 5 hours.)
It can be seen that the concentration change of LuxR is similar to that of Fig.3 before adding AHL, but the concentration of GFP is always at a low level. After adding AHL, the concentration of GFP increased rapidly and finally stabilized.
An important role in building a model is to explore how the minimum amount of AHL can be detected with minimal raw material consumption. The raw material here refers to DNA (the active ingredient is the target plasmid). Given a detectable threshold for GFP, we can very easily calculate the lowest detectable AHL concentration for each DNA concentration. Fig.5 shows a plot of the minimum AHL concentration at each concentration of DNA when the detectable GFP concentration is varied from 1000 to 3000. equilibrium_mini.py).
(Figure.5. The relationship between the lowest detectable AHL concentration and the corresponding raw material DNA concentration for a given GFP concentration.)