Team:CIDEB-UANL Mexico/math capture

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

As it is shown in the circuit, the transcription rate is affected by Uv light. Then, was needed the use of established parameters with data from NhaS and RFP. 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}

It was assumed that “f_y”, is the regulatory function that can activate or inhibit the system of a given gene, it was used 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.

The parameters from Singapore 2008 team were used 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 (1019nt), NhaS with 69aa and RFP with 235aa respectively.

\begin{equation} \large \alpha_{1} =\frac{transcription speed}{gene length (nt)} \end{equation}
\begin{equation} \large \alpha_{2} =\frac{trasnlation speed}{protein length (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, the parameters for degradation rates for proteins and mRNAs from PKU Beijing 2009 team were used :

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

However, since NhaS half-life has not been determined yet, it was decided to search for homologous proteins with the same function as NhaS. We found out that the half-life of membrane proteins produced in E. coli range between 2 to 20 hours (Hare, 1991); since NhaS is a membrane protein, it's half life should be between those values. It was decided to use 2 hours as NhaS half-life because of its relatively small size (69aa) . To obtain the half-life of RFP, the team basethe 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 average mRNA degradation rate, information from Selinger’s team (2003) was used. The team carried out several experiments to find out the average mRNA half-life in E. coli. They used mRNAs about 1100nt in lenght, and they concluded that it has an average half-life of 5min. Based on this information, it was concluded that the mRNA average half-life of NhaS with RFP was 4.63min.

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

With all these information, the team was 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 Simbiology® was used , using the previous data in the equations to calculate the amount of proteins E. coli would produce at certain time. The next graph shows the results of the simulation (assuming E. coli is under UV rays which actives its promoter):

For translation, there was another factor taken in consideration, 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}

Since NhaS protein needs to be expressed in E. coli's membrane, the average velocity at which E. coli exports its proteins needed to be calculated. The process by which bacteria exports its proteins is divided into three phases, the “breathing” between translation, the second phase, which is the movement of a protein to the membrane and finally, the translocation in which the protein attaches to the membrane of bacteria (Peskin, 1991). Using this information, we found E. coli completes these three phases at an average of 5 to 6 min, depending on the protein size (Driessen, 1990). 5min were plugged into the data, because NhaS protein is relatively small and it was assumed that RFP had no post-translational 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 both graphs were compared (Graph 1 and Graph 2) the data shown that NhaS production was functional and that the production of NhaS was really big. It was assumed that this happenned because NhaS is only (69aa) in length. Also it was concluded that NhaS is functional when it is located in the membrane, (by "functional" meaning that it has 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/References

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

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

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

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

● HARE, James 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.

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

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