Team:Jilin China/Model/population model
Since environmental factors (such as nutrients, temperature, etc.) and the toxin\anti-toxin system would both affect the growth of E. coli. Our first aim was to estimate how toxin system decreases the growing speed of the engineered bacteria, for that, firstly we set up a population model free from toxins (free-growth model), which was supposed to help cancel the environmental factors. Then, we developed another model (inhibited-growth model) to fit the growth curve of E. coli in the same environment but affected by the CbtA protein (in another word, the promoter for the toxin gene were activated). Comparing the two models, we could finally know how the toxin system itself (or, we can say promoter activity for the toxin gene) affected the growth of E. coli – which could also help us to select the appropriate strength of promoters (we hoped to choose a promoter which would lead to a suitable level for CbtA expression – not too little to hold back the growth of E. coli, nor too much to kill all the engineered bacteria). For the recovering process (the anti-toxin system), since the anti-toxin protein (YeeU) will neutralize the effect of CbtA, the growth of the engineered bacteria would return to normal (free from toxin) in ‘some level’ – this should be described as a weighted average of the two models – the free-growth model and the inhibited-growth model, and the certain weight show its ‘level’. However, during our experiments, we found that the inducer for the anti-toxin gene (IPTG) could also hold back the growth of E. coli, so we set up another model (IPTG-inhibited-growth model) to cancel the effect of IPTG and the IPTG induced recovering process was described as a weighted average of the IPTG-inhibited-growth model and the inhibited-growth model, and the weights are related with the promoter activity for the anti-toxin gene. Noticing that the inducer for the anti-toxin protein used in our final project was DmpR-phenol complex but not IPTG, so the growth model of our fully-functional engineered bacteria was still the average of the free-growth model and the inhibited-growth model.
Assumptions
To generate a classical Logistic growing model for our E. coli free from the effect of toxins, we had to involve the following assumptions:
- Environment capacity for our engineered bacteria is a constant through out the whole experiment.
- If the plasmid transformed into the bacteria doesn't transcribe, the bacteria could be seemed the same as the bacteria without plasmid.
Mathematical Modelling
| Variable | Explanation |
|---|---|
| $N$ | current population of E. coli, counted in OD600 |
| $t$ | time |
| Constant | Explanation |
| $N_0$ | initial population of E. coli, counted in OD600 |
| $N_{ec}$ | environment capacity for E. coli, fixed under specific environment |
| $k$ | the speed coefficient, fixed under specific environment |
The bigger the population is, the faster the population will grow. The more the capacity is saturated, the slower the growth will be. These simple regulations can be described with the following Logistic equation:
with initial point: $t=0$, $N=N_0$, the solution to this ordinary diffrential equation is:
Model fitting
We tested four different arabinose concentrations for the model fitting, experimental datas are listed below 
With the non-linear least squares fitting method, the fitting results are listed below
The fitting results are plotted below
Assumptions
Based on the free-growth model, we set up this model with the following assumptions:
- The inhibition from the toxin protein can only affect the growth of E. coli in two aspects:
1.slow down the growth speed, 2.kill E. coli - With a fixed toxin gene promoter activity, the slowing-down effect can be described as " v'=p1v "
(while v is the original growth speed and p1 represents the degree of the slowing down effect). - With a fixed toxin gene promoter activity, the killing effect can be described as " v''=v'-p2N "
(while N is the current population and p2 represents the dying speed of E. coli)
Mathematical Modelling
| Variable | Explanation |
|---|---|
| $N$ | current population of E. coli, counted in OD600 |
| $t$ | time |
| Constant | Explanation |
| $N_0$ | initial population of E. coli, counted in OD600 |
| $k$ | the speed coefficient if not affected by CbtA |
| $N_{ec}$ | enviroment capacity for E. coli if not affected by CbtA |
| $p_1$ | growth speed decreasing coefficient, represents how much the growth is slowed down |
| $p_2$ | suiside speed coefficient, represents how fast the CbtA protein kill E. coli |
The following equation was generated based on the former free-growth model and our assumptions:
with initial point: $t=0$, $N=N_0$, the solution to this ordinary diffrential equation is:
Let $k'=kp_1-p_2$ and $N_{ec}'=\frac{kp_1-p_2}{kp_1}N_{ec}$, we could change this function into the following form:
Model fitting
We tested four different arabinose concentrations for the model fitting, experimental datas are listed below
With the non-linear least squares fitting method, the fitting results are listed below
The fitting results are plotted below
Purpose
Since we used IPTG as the trigger for the expression of the antitoxin protein, we had to cancel the side effect of IPTG -- we found out that we could not ingore it based on our experiment. IPTG showed a strong inhibition on the growth of the engineered bacteria. This model was developed to cancel the effect of IPTG, and was aimed to tell us how T-A system itself control the growth of E. coli.
Assumptions
We made the following assumptions in this model:
- The inhibition from IPTG can only affect the growth of E. coli in two aspects:
1.slow down the growth speed, 2.kill E. coli - With a fixed concentration of IPTG, the slowing-down effect can be described as " v'=p1v "
(while v is the original growth speed and p1 represents the degree of the slowing down effect). - With a fixed concentration of IPTG, the killing effect can be described as " v''=v'-p2N "
(while N is the current population and p2 represents the dying speed of E. coli). - IPTG would affect the growth of our engineered bacteria with and without T-A system, the inhibition would be the same if IPTG didn't induce the expression of antitoxin.
Mathematical Modelling
| Variable | Explanation |
|---|---|
| $N$ | current population of E. coli, counted in OD600 |
| $t$ | time |
| Constant | Explanation |
| $N_0$ | initial population of E. coli, counted in OD600 |
| $k$ | the speed coefficient if not affected by IPTG |
| $N_{ec}$ | enviroment capacity for E. coli if not affected by IPTG |
| $p_1$ | growth speed decreasing coefficient, represents how much the growth is slowed down |
| $p_2$ | suiside speed coefficient, represents the dying speed of E. coli |
This model is quite similar with the inhibited-growth model under the effect of the toxin protein. Let $k'=kp_1-p_2$ and $N_{ec}'=\frac{kp_1-p_2}{kp_1}N_{ec}$, again, with the initial point: $t=0$, $N=N_0$, we could get following equations:
Model fitting
We tested different IPTG concentrations, experimental datas are listed below
With the non-linear least squares fitting method, the fitting results are listed below
The fitting results are plotted below
Assumptions
This model is a weighted average of the free growth-model and the inhibited-growth model, which would give us a description of how our engineered bacteria recover its growth (when the expression of antitoxin being actived). We set up this model with the following assumptions:
- Not all the bacteria could recover from the affect of toxin, the recovery could only happen when the expression of antitoxin reach a specific level.
- CbeA is a intracellular protein which would not have any effect on other cells -- the two models (free-growth and inhibited-growth) are independent of each other
In this way, we could generate this model from the free-growth model and the inihibited-growth model.
Mathematical Modelling
| Variable | Explanation |
|---|---|
| $t$ | time |
| $v$ | the growth speed in the recovery model (free from IPTG) |
| $v_f$ | the growth speed in the free-growth model |
| $v_i$ | the growth speed in the inhibited-growth model |
| Constant | Explanation |
| $w_f$ | the weight of the free-growth model in this recovery model |
| $w_i$ | the weight of the inhibited-growth model in this recovery model |
We used IPTG as the inducer to trigger the expression of antitoxin in our testing bacteria (which was used to test the T-A system, but not the degradation of phenol). To cancel the side effect of IPTG, we had to set up the IPTG-inhibited-growth model, and with the following steps, we were able to cancel the effect of IPTG (theoretically):
- Fit experimental datas with the equation: $N=\dfrac{N_{ec}'e^{k't}\frac{N_0}{N_{ec}'-N_0}}{1+e^{k't}\frac{N_0}{N_{ec}'-N_0}}$.(Variable definitions: see the models above)
- Calculate the function $v'$ with fitting results: {$N_{ec}'$, $k'$, $N_0$}, $v'=k'N\dfrac{N_{ec}'-N}{N_{ec}'}$.
- Compare the IPTG-inhibited-growth model with the free growth model, calculate $p_1$ and $p_2$ (defined in the IPTG-inhibited-growth model).
- Fix $v'$ to $v$ with $p_1$ and $p_2$: $v=\dfrac{v'+p_2N}{p_1}$, this $v$ could be regarded as the recovery-growth speed free from the affact of IPTG.
We were then able to calculate the weights with the following equation set:
Model fitting
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