Team:CIDEB-UANL Mexico/math capture

The capture module is based in the use of NhaS gene, but as NhaS is not the only gene expressed in the circuit because we used RFP as a reporter inside it, we needed to consider that factor. The circuit is shown below:

As it is shown in the circuit, the transcription rate is affected by Uv light, so we needed to use the stablished parameters with data from NhaS and IrrE. Uv promoters, according to Shuang Li’s team (2006), have less efficiency than constitutive promoters, which is aproximately 60% of the constitutive promoters.

\begin{equation} \large \frac{d\left [ mRNA \right ]}{dt}=\alpha_{1}\cdot f_{y}-d1\left [ mRNA \right ] \end{equation} \begin{equation} \large f_{y}=0.6 \end{equation}

We assumed that “f_y”, the regulatory function that can activate or inhibit the system of x gene, was 0.6 because of the percentage given by Shuang Li’s team. Uv rays activate the promoter, but it is not as efficient as constitutive promoters in the transcription process.

We used the parameters from Singapore 2008 iGEM team for the translation and transcription rates as well as the speed at which E. coli carries out transcription and translation, assuming a transcription speed of 70nt/s and a translation speed of 40aa/s. So we used them in the equations below with the NhaS and RFP gene (1019 nt), NhaS with 69 aa and RFP with 235 aa respectively.

\begin{equation} \large \alpha_{1} =\frac{transcription speed}{gene length \cdot(nt)} \end{equation} \begin{equation} \large \alpha_{2} =\frac{transcription speed}{protein length \cdot(aa)} \end{equation} \begin{equation} \large \alpha_{1}=\frac{(70)(60)}{1019}=4.12 \end{equation} \begin{equation} \large \alpha_{2}(NhaS)=\frac{(40)(60)}{69}=34.78 \end{equation} \begin{equation} \large \alpha_{2}(RFP)=\frac{(40)(60)}{235}=10.21 \end{equation}

Then, we used the parameters for degradation rates for proteins and mRNAs from Beijing PKU 2009 iGEM team:

\begin{equation} \large D_{i}= \frac{1}{half-life} + \frac{1}{30} \cdot(min) \end{equation} \begin{equation} \large D_{p} = \frac{1}{half-life} + \frac{1}{30} \cdot(min) \end{equation}

However, since NhaS half-life has not been determined yet, we decided to search about homologous proteins with the same function as NhaS. First, we found that the half-life of membrane proteins produced in E. coli range between 2 to 20 hours (Hare, 1991), so since NhaS is a membrane protein, it should be between those values, but we decided to use 2 hours as NhaS half-life because of its size (69aa), which is relatively small. To obtain the half-life of RFP, we based the time on the GFP protein due to the similarity of their functions (fluorescence). Therefore, the half-life of RFP on which we are based is of 8 hours (Longo, 2006).

To determine the degradation rate of average mRNA, we used the information from Selinger’s team (2003). They carried out several experiments to find the average mRNA half-life in E. coli. They used mRNAs about 1100 nt concluding they have an average half-life of 5min. So with this we found that the average mRNA half-life of NhaS with RFP was 4.63 min.

\begin{equation} \large HL=\frac{1100\cdot(bp)}{5min} \end{equation}

With all these information, we were able to find the degradation rates for both, the transcription and translation of NhaS and RFP, respectively.

\begin{equation} \large d_{1}= \frac{1}{4.63} + \frac{1}{30}=0.25 \end{equation} \begin{equation} \large d_{2}(NhaS)= \frac{1}{120} + \frac{1}{30}=0.041 \end{equation} \begin{equation} \large d_{2}(RFP)= \frac{1}{480} + \frac{1}{30}=0.035 \end{equation}

For the simulation we used Simbiology, using the previous data in the equations to find the amount of proteins E. coli would produce at certain time. The simulation ended up with the next graph as a result (assuming E. coli is under UV rays which actives its promoter):

For translation, there was another factor we had to consider, the“f_post” which were the post-translational variables affecting the production of the functional protein:

\begin{equation} \large \frac{d[P]}{dt} = \alpha_{2} \cdot[mRNA] - d_{2}[P] - f_{post} \end{equation}

As NhaS’s protein needs to be expressed in the membrane of E. coli, we needed to find the average velocity at which E. coli exports its proteins. The process by which bacteria exports its proteins are divided into three phases, the “breathing” between translation and the second phase, which is the movement of a protein to the membrane and the translocation; in this phase the protein attaches to the membrane of bacteria (Peskin, 1991). Using this information, we found E. coli completes these phases in an average of 5 to 6 min, depending on the protein size (Driessen, 1990). We used 5min because of the fact that NhaS is relatively small and we assumed RFP had no posttranslational variables to consider as well as it was not the protein we wanted to measure.

With the rate of protein transport we could use Simbiology to model the functional NhaS production. The results are shown in next diagrams (assuming E. coli is under UV rays which actives its promoter):

When we compared both graphs (Graph 1 and Graph 2) we realized that not all the NhaS production was functional and that the production of NhaS was really big. We assumed this happens because NhaS is only 69aa in length. Also we concluded NhaS once they are in the membrane they are functional meaning they have the potential to capture Na⁺ ions and be able to give resistance to saline environments in E. coli. As is shown in Graph 2 the amount is of proteins produced are about 6500; we assumed those proteins are ready to bind Na⁺ ions.

Bibliography

● Arnold Driessen, W. W. (1990). Proton transfer rate-limiting for translocation of precursor proteins by the Escherichia coli translocase. Biochemistry, 2471-2475.

● Charles Peskin, S. S. (1991). What drives the translocation of proteins. Biophysics, 3770-3774.

● Diane Lonngo, J. H. (2006). Dynamics of single-cell gene expression. Molecular Systems Biology, 1-10.

● James Hare, K. T. (1991). Mechanisms of plasma membrane protein degradation: Recycling proteins are degraded more rapidly than those confined to the cell surface. PNAS, 5902-5906.

● Shuang Li, L. X. (2007). A set of UV-inducible autolytic vectors for high throughput screening. Journal of biotechnology, 647-652.

● Douglas Selinger, R. M. (2003). Global RNA Half-Life Analysis in Escherichia coli Reveals Positional Patterns of Transcript Degradation. Genome Research, 216-223.