Team:Jilin China/Model/feedback adjustment network

Feedback Adjustment Model

Overview

Based on the feedback adjustment principle, we designed our project. In this part of modelling, we combined the population model and the degradation model together. Phenolic pollutants are signals which could trigger the recovery of E. coli growth. As the feeding-back, the degradation rate of phenolic pollutants would increase because of the growth of E. coli. Finally the growth of our engineered bacteria would drop down as a result of phenol degradation. That is how this feedback adjustment network works.

To set up the whole feedback adjustment network, we had to work out with the population models and the degradation models. By comparing the testing results, we were able to give a general estimation of the growth rate of E. coli under/free from the effect of toxin system (see population model: free-growth model and inhibited-growth model). The general growth rate is a weighted averange of these two growth rates, since there is an anti-toxin system which would cover the inhibitation from toxin. We measured promoter activities for the antitoxin system and calculated the relationship between the weights and the promoter activities. The concentration changings of phenol pollutants would result in different promoter activities, we tested several different promoters under different phenol concentrations with luciferase expression. We selected one of the promoters in our final project for enzyme expression and antitoxin expression based on our experimental results and our modellings. The degradation kinetics would be reached from our degradation model, which would tell us the concentration changings of phenol pollutants.

Population Model Weighting

We tested with different phenol concentrations, and measured the relationship between phenol concentrations and promoter activities with luciferase, datas are listed below

Parameters are defined in the following table:

Variable Explanation
$PA$ promoter activity
$c$ concentration of phenol
Constant Explanation
$PA_{max}$ maximum of promoter activity
$c'$ concentration fixing term, in case of transcription leakage
(because of the transcription leakage, expression would happen without phenol detection, seems like there is a none zero phenol concentration)
$k$ scale factor

With model function:

$PA=PA_{max}-\dfrac{k}{c'+c}$

we fitted our data:

$PA=19.38-\dfrac{0.8723}{c+0.06177}$

as plotted below

With our experimental datas and estimations (we had to estimate because different enviroment would result in different parameters in our population models and we could not test all of those), we selected the following two model functions as our free-growth model and inhibited growth model: (parameter declarations: see population model - recovery growth)

$\begin{cases}v_f=0.4N(1.0-N)\\v_i=0.1N(0.2-N)\end{cases}$

As a weighted average of $v_f$ and $v_i$, the population growth could be wrote in the following form:

$\dfrac{\mathrm{d}N}{\mathrm{d}t}=w_iv_i+w_fv_f$

Where $w_i$ and $w_f$ would change as the promoter activity for the antitoxin system changes, and based on our experiments, we could calculate the general growth rate with:

$\dfrac{\mathrm{d}N}{\mathrm{d}t}=\dfrac{1.9cN(0.2-N)}{20c+1.2}+\dfrac{0.4(c+0.9)N(1.0-N)}{20c+1.2}\\=\dfrac{0.36N-0.36N^2+0.78cN-2.3cN^2}{20c+1.2}$
Population - Degradation Model Combination

By combining the population model and the degradation model, we were able to predict the concentration changings as well as E. coli population changings. For example, with the initial engineered bacteria population: 0.8 and the following doses:

Time 0 20 40 260 370 420 550 900
Pollutant addings 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
We were able to plot the degradation curve and the population curve:

Degradation
Population