In order to understand our system thoroughly, we constructed a model of the system that incorporates toxicity-induced cell death into consideration.
The chemical equation for this is: 
. By the law of mass action, we have 
. Solving for 
, we have 
, where 
. Applying the conservation of 
, we have the Michaelis-Menten Equation: 
, and therefore that 
This could be modeled by the promoter activity equation
.
Plugging in we have 
.
When cells die, the csoR they carry will be gone, and the promoter activity will also
decrease. Supposing that copper kills a fraction 1 - D of all cells and thus leaving D
of the
cells alive, we have 
, and so 
.
For simplicity, we can model cell death as 
.
We then use the Michaelis-Menten Equation to get 
, so
, and 
. Plugging
in we have 
In conclusion, the adjusted promoter activity is .
The following is an interactive model:
A core part of our project is based on our construction of the PIII-RBS variant collection, an effort to better our biosensor’s detection of copper.
After obtaining the experimental data for PIII-RBS variation, we tried analyzing the data in order to find the next step for our project. To accomplish this, we decided to identify the variants that more possibly can produce ideal variants. Our model successfully completed this task and showed us three variants that can be used for further mutation.
Using the 96 variants we produced from the mutation (K4494003-K4494098), we first constructed a DNA alignment (Figure 1) and deduced a minimum-evolution phylogenetic tree (Figure 2) from it. This helps us to find the genetic relations between the mutants.
In addition to the tree, we calculated the evolutionary distance of our genetic sequences compared with the original sequence (K4494196) using the Levenshtein distance, defined by the following code:
We then made a scatter plot of Blank RFU versus Foldchange, with the Levenshtein distance from the original sequence as a coloring, to find the inbuilt patterns of our results (Figure 3).
As could be seen in Figure 4, there are three peaks on our scatter plot. The leftmost one, with variant 5 (K4494007) on the peak, is evolutionally most distant from the original sequence. The middle one, with variant 1 (K4494003) on the peak, and the rightmost one, with variant 79 (K4494081) on the peak, are relatively closer to the original sequence. We conclude that there are three main families of variants that deserve to be explored more in our future project.
Therefore, through modeling, we conclude that in the future we will focus more on the evolutionary branches surrounding K4494003, K4494007, and K4494081 for further optimization.
