-----* Analysis of The Problem *-----

Firstly, we built a model in order to analyse the gene expression dynamics of our whole circuit.In our model, we simplified the actual biological process into a model that only remains the promoter, the transcription gene, mRNA and goal polymerase or molecule. In our circuit, the metal sensitive promoters we used is ars promoter with arsR repressor gene. Pollutant ions bind to the repressor protein ArsR, resulting in inability to repress the promoter and activating transcription of T7 RNA polymerase. And T7 RNA polymerase will then bind to T7 promoter and largely activate the expression of the gene of reporting molecules. In the next process, the substrate PAPG will be catalyzed by LacZ to form PAP. Also, both mRNA and polymerase have a leaky expression.

In order to describle our model, we make some assumption. Firstly, mRNA and proteins will decay following Poisson distribution. Secondly, combinations of two proteins are considered as quick reactions. Based on these, we wrote chemical kinetic equations in represent of each process.

The symbol declaration is:
X1: the pollutant ion
D1: the pollutant ion sensing promoter
X1D1: the pollutant ion- ArsR complex
X2: the mRNA of T7 RNA polymerase
X3: T7 RNA polymerase
D2: T7 promoter
X4: mRNA of the reporting molecules
X5: the reporting molecules
K1/K-1: the rate of turning of the state of pollutant ion sensing promoter
K2: the rate of formation of mRNA of T7 RNA polymerase
K3: the rate of formation of T7 RNA polymerase
K4: the degradation rate of mRNA of T7 RNA polymerase
K5/K-5: the rate of turning of the state of T7 promoter
K6: the rate of formation of mRNA of the reporting molecules
K7: the degradation rate of T7 RNA polymerase
K8: the rate of formation of reporting molecules
K9: the degradation rate of mRNA of the reporting molecules
K10: the rate of the following reaction

-----* Solutions and Implication *-----

According to the analysis, in our circuit , pollutant ions bind to the repressor protein ArsR, resulting in inability to repress the promoter and activating transcription of T7 RNA polymerase, which meets the situation that the transcription factor (either activator or repressor) binds to the site in promoter and regulates the gene expression which eventually determines the transcription rate of mRNA. Here we use Hill equation to describe the activity of the pollutant ion sensing promoter.

Here [DTot] is the total concerntration of the DNA site. Then we solve the equations.

Based on Hill Equation , we can determine the rate of transcription from a free binding site to be called as maximal transcription rate, β. The rate of mRNA production or the promoter activity is β times the probability that the binding site is free (value of β ranges from approximately 10--4 to 1 mRNA/s).

According to the analysis, we wrote down the ODEs:

Next we will do some algebra to simplify the equations and solve the differential equations. To simplify the model, we consider the rate of T7 promoter activity as certain.

We used MATLAB to solve the ODEs and drew the figures.

Figure 1. Reporting molecules-Time curve

Figure 2. Reporting molecules intensity curve

-----* Results *-----

As is shown in the figure above, we can conclude that the higher concentration of pollutant ion we set, the higher reporting molecules intensity we expect. However, under the real experimental conditions, we can see that reporting molecules intensity falls after reaching steady, in some way we consider that the death of cells will somehow affect the data.So we need to modify our model in the future work so that it can combine our experimental data.

-----* Other Models *-----

As for the analysis of LuxAB and freeze-drying, you can get the data and models in our wiki.

As for human practices,you can get more information in this wiki.

-----* References *-----

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