Team:NCTU Formosa/Growth Model
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Overview
IPTG Inhibition
Toxin Inhibition
Chemical Effects
Computational Growth Model
Introduction
The purpose of building a computational growth model is to simulate and analyze the experiment result. We use Matlab to simulate the genetic regulation and fit the experiment data into our model. After modeling, we can calculate mutation frequency of any chemical compound.
Overview
To calculate the exact mutation caused by testing objects, we need to isolate its cause to bacteria growth from other factors including IPTG inhibition, natural mutation, and chemical toxicity. First of all, while using IPTG to induce toxin gene the addition of IPTG may inhibit the growth of E. coli. Hence, we isolate the toxin gene to deal with this problem. Second, we calculate the effect of the toxin gene to get the natural mutation frequency of the bacteria. Third, we also take the inhibition effect of the chemical into consideration as a constant. In the end, we combine the parameters we have got to get the mutation parameter caused by chemicals.
The following flow chart shows the steps of building our computational growth model.
Figure 1: The steps of building computational growth model
IPTG Inhibition
Isopropyl β-D-1-thiogalactopyranoside (IPTG) is a molecular biology reagent, which is closely similar to lactose structurally. However, bacteria cannot consume it metabolically, and it also inhibits the growth of the bacteria. In order to check if the dose of IPTG we used in our experiment will inhibit the growth curve and how it affects the growth curve, we compare the growth curve ofE. coli with and without IPTG as following:
$$Statistical\: Significance:\: p-value=0.504679$$
Figure 2: Comparison of IPTG toxicity
According to the definition of statistical significance, once the p-value is lower than 0.05, the two data have “statistical significance,” which indicates that the two data are not from the same experimental sample. As we can see above, the statistical significance p-value is above 0.05, which means the two data are not from the different experiment samples, so we can prove that the bacterial growth curve will not be influenced by IPTG inhibition statistically.
Toxin Protein Inhibition
Bacterial Growth Parameters with Different Inserted Gene
Bacteria with different inserted genes may have different growth rates due to the reason that the bacteria have to distribute the energy to the genes to generate proteins instead of doing mitosis. We observe the bacterial growth curve with different toxin inserted as follows:
Figure 3: Comparison of different inserted gene
Table 1: The p-value of the pSB1A3 growth curve to each toxin gene
| Toxin | p-value |
|---|---|
| ccdB | 0.15528744 |
| MazF | 0.6107677 |
| yafQ | 0.05239678 |
| ydfD | 0.06656952 |
| ChpBK | 0.87345055 |
The p-values of pSB1A3 with each toxin gene are all higher than 0.05, as shown in the table. In other words, statistically, the bacterial growth data cannot be regarded as a different experiment sample. Thus, we can take the difference of the growth curve as experiment error and use the general parameters to model the growth curve.
Growth Curve with Toxin Protein Induced
In this model, we simulate the effect of toxin protein and analyze the natural mutation rate, or we called the background mutation rate.
Before building our mutation model, it is essential to define what exactly mutation is in our model. We assume that mutation happens randomly in the DNA in E. coli. Once the toxin gene cannot generate toxin protein functionally, we define it as "mutation happens."
Figure 4: An overview of IPTG-LacI mechanism
We constructed a general ODE model of the gene circuit and growth model.
$$\frac{d[toxin]}{dt}=C_{toxin}-D_{toxin}\cdot [toxin]$$
$$\frac{dB_{N}}{dt}=[ g\cdot B_{N}(1-\frac{B_{T}}{B_{Max}})](1-M_{1})-T_{toxin}\cdot B_{N}\cdot [toxin]$$
$$\frac{dB_{M}}{dt}=[ g\cdot B_{N} (1-\frac{B_{T}}{B_{Max}})] (M_{1})+[g\cdot B_{M}(1-\frac{B_{T}}{B_{Max}})]$$
Figure 5: The demonstration of mutation rate “M.”
We assume that non-mutated bacteria(BN) have a natural mutation rate(M1) transforming into mutated bacteria(BM), and the possibility of mutated bacteria transforming into non-mutated bacteria is too low so that we don’t count it in.
We sum up the differential equation of BN and BM to get the total bacterial population (BT).
$$\frac{dB_{T}}{dt}= g\cdot B_{T} (1-\frac{B_{T}}{B_{Max}})-T_{toxin}\cdot B_{N}\cdot [toxin]$$
Prediction
We substitute the parameters in the following table into the ODEs to predict the experiment result as following. As we can see below, the prediction curve is correlated to our experiment result, which proves both of our model and experiment are successful.
Figure 6: Prediction of toxin induction.
Table 2: The parameters we used in the prediction.
| Parameter | Value | Unit |
|---|---|---|
| g | 0.008 | min-1 |
| BMax | 3 | O.D. |
| Ctoxin | 0.01491 | nM1．min-1 |
| Dtoxin | 0.02 | min-1 |
| Ttoxin | 0.00028569 | nM-1．min-1 |
| M1 | 3．10-8 | relative unit |
Chemical Effects
As a result of the effect of the chemicals on our growth curve, not only by altering a mutation rate but also by killing some bacteria in the process, we firstly compare the growth curve of the bacteria with and without chemical to get the parameter of chemical inhibition (Tchem). Then, we fit the growth curve with a mutation rate (M2), which caused by chemicals.
We assume that the chemical inhibition effect is constant, and we fit our experiment data with the formula below to get the toxicity parameter of the chemical Tchem.
$$\frac{dB}{dt}=g\cdot B(1-\frac{B}{B_{Max}})-T_{chem}\cdot B$$
We take the mutation rate of chemicals (M2) into consideration, and restate the ODEs below.
$$\frac{dB_{N}}{dt}=[ g\cdot B_{N}(1-\frac{B_{T}}{B_{Max}}) -T_{chem}\cdot B_{N}](1-M_{1}-M_{2})-T_{toxin}\cdot B_{N}\cdot [toxin]$$
$$\frac{dB_{M}}{dt}=[g\cdot B_{N}(1-\frac{B_{T}}{B_{Max}})-T_{chem}\cdot B_{N} ] (M_{1}+M_{2})+[g\cdot B_{M}(1-\frac{B_{T}}{B_{Max}})-T_{chem}\cdot B_{M}]$$
We sum up the differential equation of BN and BM to get the total bacterial population (BT).
$$\frac{dB_{T}}{dt}=[g\cdot B_{T}(1-\frac{B_{T}}{B_{Max}})-T_{chem}\cdot B_{T} ] -T_{toxin}\cdot B_{N}\cdot [toxin]$$
Simulation
We simulate our model with different estimated mutation rate M2 as following:
Figure 7: The simulation of bacterial growth curve under M2 from 0.01 to 0.07
The deeper color means it has the higher mutation rate M2
As we can see above, the higher the mutation rate M2, the higher the growth curve. Due to the increasing of mutation rate M2, the mutated E. coli will grow faster, which influences the whole bacteria O.D. value significantly.
Proof of Concept
Figure 8, 9, and 10: Growth curve of different concentration of EtBr with ydfD gene induced
After we have build up our model and get the experiment data, we do the analysis of those data in order to prove our concept. The figures above show the growth curve under different concentration of EtBr with ydfD gene induced. The curve firstly drops due to the expression of ydfD protein as we expected, and the curve rises up because of the mutation caused by EtBr. With the higher concentration of EtBr, the curve after the local minium has the higher slope, which reflects on the changeing of mutation rate M2.
Validation of Our Simulation
Figure 11: The validation of our simulation and experiment result
Figure 11 shows the comparison of the experimental data curve and our model simulation. Because the expression time of toxin protein and the initial O.D. value were all different from each experiment, the time and the value of the local maximum of each curve would not be identical. On top of that, as we can observe from the simulation result, mutation rate M2 influences the slope after the local minimum significantly. Therefore, we compared the slope after the local minimum in the curve in experimental data and the simulation results were highly correlated, which proved that our model could analyze the mutation rate, and our experiment was successful as expected.
Reference
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4. Lee, H., et al. (2012). "Rate and molecular spectrum of spontaneous mutations in the bacterium Escherichia coli as determined by whole-genome sequencing." Proc Natl Acad Sci U S A 109(41): E2774-2783.
5. McCardell, R. D., et al. (2017).
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9. Williams, A. B. and P. L. Foster (2012). "Stress-Induced Mutagenesis." EcoSal Plus 5(1).
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