Team:CIDEB-UANL Mexico/math aroma

The aroma module is based in the production of SAM/salicylic acid methyltransferase (BSMT1) in order to generate methyl salicylate, leaving a physical evidence of wintergreen odor. Since this module is not affected by external factors during its transcription, the established formula of mRNA was used with their parameters, but with the data we obtained from BSMT1.

\begin{equation} \large \frac{d\left [ mRNA \right ]}{dt}=\alpha_{1}-d_{1}\left [ mRNA \right ] \end{equation}

In the case of translation, a ribo-switch affects the production of the protein BSMT1. When the temperature reaches 32°C, the ribo-switch allows the translation of BSMT1 gene, but below 32° C, the translation is not allowed.

\begin{equation} \large RBS\left\{\begin{matrix} T<32=0 & \\ T\geq 32=1& \end{matrix}\right. \end{equation}
\begin{equation} \large \frac{d[P]}{dt} = \alpha_{2} \cdot[mRNA] - d_{2}[P] - f_{post}\left\{\begin{matrix} T<32 \left\{\begin{matrix} \alpha_{2}=\frac{2400\cdot 0}{358}; R=0 & \\ No protein degradation& \end{matrix}\right. & \\ T\geq 32 \left\{\begin{matrix} \alpha_{2}\frac{2400}{358}=6.7& \\ d_{1}\frac{1}{40min}+ \frac{1}{30min}=0.058 & \end{matrix}\right.& \end{matrix}\right. \end{equation}

The parameters for translation and transcription rate from Singapore 2008 iGEM team were used, as well as the speeds at which E. coli carry out both, the transcription and the translation, assuming a transcription speed of 70nt/s and a translation speed of 40aa/s. These data were used in the equations below with the Wintergreen gene length being 1198nt and the protein length being 358aa:

\begin{equation} \large \alpha_{1} = \frac{transcription speed}{gene length (nt)} \end{equation}
\begin{equation} \large \alpha_{2} = \frac{translation speed}{protein length (aa)} \end{equation}
\begin{equation} \large \alpha_{1} = \frac{(70)(60)}{1198} = 3.5 \end{equation}
\begin{equation} \large \alpha_{2} = \frac{(40)(60)}{358} = 6.7 \end{equation}

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

\begin{equation} \large d_{1} = \frac{1}{half-life} + \frac{1}{30} min \end{equation}
\begin{equation} \large d_{2} = \frac{1}{half-life} + \frac{1}{30} min \end{equation}

Since the half-life of BSMT1 has not been determined by MIT Team 2006, an investigation was made and according to Zubieta (2003), the average half-life for salicylic acid methyltransferases are about 40min.

To determine the degradation rate of the average mRNA, the information from Selinger’s team (2003) was used. Several experiments to find the average mRNA half-life in E. coli were carried out. They used mRNAs of about 1100nt length, leading to the conclusion that they have an average half-life of 5min.With this information, it could be determined that the average mRNA half-life of BSMT1 was 5.44min.:

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

With all these information, the degradation rates for both, the transcription and the translation of BSMT1, could be calculated:

\begin{equation} \large d_{1} = \frac{1}{5.44} + \frac{1}{30} = 0.21 \end{equation}
\begin{equation} \large d_{2} = \frac{1}{40} + \frac{1}{30} = 0.058 \end{equation}

Simbiology was used for the simulation. The previous data were used in the equations to find the amount of proteins E. coli would produce at certain times. The results are shown in the next graph:

For the translation, another factor was needed to be taken into 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}

During the research of information, it was found out that BSMT1 is a special type of enzyme called Michaelis-Menten enzyme. As BSMT1 will perform an enzymatic reaction, it was needed to know at which rate it will be carried on producing methyl salicylate (Zubieta 2003). For that reason, the post-translational function considers the rate of methyl salicylate production as the variable that directly affects the production of the final protein. The formula used was also called Michaelis-Menten Equation:

\begin{equation} \large f_{post}=\frac{k_{cat}\left [ S \right ]\left [ E \right ]}{K_{m} + \left [ S \right ]} \end{equation}

Where: “[S]” means the substrate concentration; “[E]”, the enzyme concentration (obtained by the rest of the translational formula); “K_cat”; is the turnover number; and “K_m”, the substrate concentration needed to achieve a half-maximum enzyme velocity.

Zubieta et al. (2003) established “K_m” and “K_cat” values for SAM (BSMT1) of 23 and 0.092, respectively. All this values are used when a salicylic acid concentration of 2mM is induced. As “[E]” stands for enzyme concentration, the protein produced (by the rates of translation and degradation of the protein) will introduce this value in the equation. By substituting these values in the Michaelis-Menten Equation, it will change as follows:

\begin{equation} \large f_{post}=\frac{k_{cat}\left [ S \right ]\left [ E \right ]}{K_{m} + \left [ S \right ]}=\frac{0.092\left ( 2mM \right )\left [ P \right ]}{23 + 2mM} \end{equation}

Also, this formula is used in order to get the maximum rate of methyl salicylate production. This value is given by the product of “K_cat” times the substrate concentration “[S]”.

\begin{equation} \large V_{max}=K_{cat}\cdot \left [ S \right ]=0.092\left ( 2mM \right )=0.184 \end{equation}

The enzymatic rate were used in Simbiology to model the functional BSMT1 production. The results are shown in the next graphs:

With the analysis from both graphs (Graph 1 and Graph 2) , it was established that the enzymatic reaction was too slow compared to the BSMT1 production, leaving almost ¼ nonfunctional of the total amount of proteins produced. It was assumed that the functional BSMT1 is the final product, which releases the Wintergreen odor and it would be relatively concentrated with the amount of proteins produced.

Bibliography

● David Dibden, J. G. (2005). In vivo cycling of the Escherichia coli transcription factor FNR between active and inactive states. Microbiology, 4063-4070.

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

● Zubieta, C. (2003). Structural Basis for Substrate Recognition in the Salicylic Acid Carboxyl Methyltransferase Family. Plant Cell, 1704-1716.

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