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A Kinetic Model in MATLAB

The creation of our electrochemical sensor presented us with an interesting question: How fast will it be? Instead of trying many timecourses in the lab we turned to modelling to give us an answer. We have created a kinetic model for one of our systems, the uidA system, to give us an estimate of the time needed to observe a result.

How did we build it?

Using the SimBiology toolbox for MATLAB we created a framework of the electrochemical reporter system. We started at the genetic level, modelling transcription, translation, and the enzyme catalysis. We also included the transport of the substrate into the cell and the diffusion of the product out of the cell. The degradation reactions for the RNA transcript and the enzyme were also included. The reaction scheme is shown below in figure 1.

Figure 1: Diagrammatic representation of the reactions involved in converting the PNPG substrate into the electrochemical analyte PNP. This was created in the SimBiology toolbox for MATLAB

After creating the basic framework for the model we needed to create mathematical equations for each reaction with appropriate rate constants. These equations and the corresponding values are shown below in tables 1 and 2.

Table 1: Mathematical representation of electrochemical reporter gene expression and the subsequent production of the electrochemical analyte. Constants are described in Table 2.
Table 2: Values assigned to kinetic parameters described in Table 1.

What did it show?

Before running the model we needed to decide what an appropriate endpoint would be. As we were trying to determine how quickly we could get a result we chose 2µM as our endpoint. This was chosen because initial electrochemical testing of PNP showed strong responses at concentrations as low as this. With our endpoint chosen the model was run and the data is shown below in figure 2.

Figure 2: PNP produced by a cell expressing uidA according to our mathematical model. The 2µM endpoint is reached in just over 4 minutes.

Our model showed that under strong expression we would be able to see a response in under 4 minutes. This was subsequently tested in the lab to determine the accuracy of our model. The data from that test is shown below.


Future plans

There are ways that this model could be improved for future uses. One main avenue for further exploration is that of modelling the remaining two electrochemical systems (lacZ and bglX). The system modeled here was chosen because of the availability of data for the compound, both electrochemically and enzymatically. A second avenue for improvement would be to model varying promoter strengths to see how dramatically the timecourse would change over varying concentrations of an inducer. This could be expanded upon to predict appropriate promoters for a final system.

This model has shown us a timecourse and successfully informed the wetlab experiments. The iterative process of modelling and experimenting will give future versions of this model more data to use and increase its accuracy.

References for now...

Vogel U, Jensen KF. The RNA chain elongation rate in Escherichia coli depends on the growth rate. J Bacteriol. 1994 May;176(10):2807-13.
Young R, Bremer H. Polypeptide-chain-elongation rate in Escherichia coli B/r as a function of growth rate. Biochem J. 1976 Nov 15;160(2):185-94.
Lim HN, Lee Y, Hussein R. Fundamental relationship between operon organization and gene expression. Proc Natl Acad Sci U S A. 2011 Jun 28;108(26):10626-31.
2009 iGEM Calgary,
Phillips R, Kondev J, Theriot J. Physical Biology of the Cell. 1st ed. Garland Science. 2008.
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Xiong AS, Peng RH, Cheng ZM, Li Y, Liu JG, Zhuang J, Gao F, Xu F, Qiao YS, Zhang Z, Chen JM, Yao QH. Concurrent mutations in six amino acids in beta-glucuronidase improve its thermostability. Protein Eng Des Sel. 2007 Jul;20(7):319-25. Epub 2007 Jun 8.
Wright JK, Overath P. Purification of the lactose:H+ carrier of Escherichia coli and characterization of galactoside binding and transport. Eur J Biochem.1984 Feb 1;138(3):497-508.