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.
To generate a classical Logistic growing model for our E.coli free from the effect of toxins, we have 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.
| 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 = N0, the solution to this ordinary diffrential equation is:
Model fitting
| Time(h)\Ara(%) | 0.0 | 0.1 | 0.2 | 0.6 |
|---|---|---|---|---|
| 2 | 0.11 | 0.11 | 0.11 | 0.11 |
| 3 | 0.36 | 0.48 | 0.48 | 0.50 |
| 4 | 0.55 | 0.70 | 0.70 | 0.75 |
| 5 | 0.64 | 0.74 | 0.78 | 0.81 |
| 6 | 0.72 | 0.81 | 0.82 | 0.84 |
| 7 | 0.81 | 0.87 | 0.85 | 0.87 |
| 8 | 0.92 | 0.94 | 0.91 | 0.88 |
