Team:CIDEB-UANL Mexico/math aroma

The aroma module is based in the production of SAM/salicylic acid methyltransferase (BSMT1opt) in order to generate methyl salicylate, leaving wintergreen odor. Since this module is not affected by external factors during its transcription, the established formula of mRNA was used.
\begin{equation} \large \frac{d\left [ mRNA \right ]}{dt}=\alpha_{1}-d_{1}\left [ mRNA \right ] \end{equation}

In the case of translation, a RNA thermometer affects the production of the protein BSMT1opt. When the temperature reaches 32°C, the RBS allows the translation of BSMT1 gene, but below 32° C does not. \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}; RBS=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 rates from Singapore 2008 iGEM team were used, as well as the transcription and traslation speeds carried out by E. coli assuming a transcription speed of 70nt/s and a translation speed of 40aa/s. They were used in the equations below with the Wintergreen gene length (1198nt) and protein length (358aa) respectively:

\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}

After, it was needed to use the parameters for degradation rates of proteins and mRNAs obtained from Beijing PKU 2009 iGEM team:

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

Since the half-life of BSMT1 has not been determined, a research was made and according to Zubieta (2003), the average half-life for salicylic acid methyltransferases are about 40min.

Later, was used the information from Selinger’s team (2003) to determine the mRNA desgradation. They performed several experiments for finding the average mRNA half-life in E. coli . They used mRNAs of about 1100nt concluding they have an average half-life of 5min. Using the previous relation was found mRNA's half-life from BSMT1opt which was about 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.

“K_m” and “K_cat” were established values for SAM (Zublileta, 2003)of 23 and 0.092, respectively. Based on their results, the oprimal induced concentration of salicylic acid is 2mM, but in the performed expreimentation, the predicted value (at which E. coli could survive) was considered as 10mM. 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 ( 10mM \right )\left [ P \right ]}{23 + 10mM} \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 ( 10mM \right )=0.92 \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/References

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

● SELINGER, Douglas 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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