Team:CIDEB-UANL Mexico/math union

The union module is based in the use of a fusion protein composed by L2 and AIDA. L2+AIDA is a protein which is not affected by external factors during its transcription as well as its translation, so we needed to use the established parameters but with the data we obtained from it:

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

We used the parameters for translation and transcription rate from Singapore 2008 iGEM team as well as the speeds at which E. coli carry 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 L2+AIDA gene length (2620nt) and protein length (856aa) 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)}{2620} = 1.6 \end{equation}
\begin{equation} \large \alpha_{2} = \frac{(40)(60)}{856} = 2.8 \end{equation}

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

\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}

As the protein was the fusion of two we need to search for each half-life. The half-life of membrane proteins range between 2 to 20 hours in E. coli (Hare, 1991), and as AIDA-I is a membrane protein its half-life must be between that range since it is not determined the specific half-life of AIDA. To find the half-life of L2 we assumed it was 7.8 hours (Bergant, 2010). Bergant’s team made test with a homologous protein but found in the minor capsid of the Human Papillomavirus (HPV). Although the function of the L2 strand in HPV is viral, and in E. coli is ribosomal, both share similar structures and sequences. Once we have decided to use the half-life from the homologous L2 we determined to use it as the half-life for the fusion protein because it was between the range of AIDA-I, and also because it was the lower half-life assuming as E. coli start the L2 degradation, it would degrade the whole protein.

For determining the degradation rate of average mRNA we used the information from Selinger’s team (2003). They carried several experiments for finding average mRNA half-life in E. coli. They used mRNAs about 1100nt concluding they have an average half-life of 5min. So with this we found the average mRNA half-life of L2+AIDA was 11.9min.

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

With all these information we could find the degradation rates for both transcription and translation of L2+AIDA

\begin{equation} \large d_{1} = \frac{1}{11.9} + \frac{1}{30} = 0.11 \end{equation}
\begin{equation} \large d_{2} = \frac{1}{468} + \frac{1}{30} = 0.035 \end{equation}

For the simulation we used Simbiology using the previous data in the equations for finding the amount of proteins E. coli would produce at certain time. The results obtained are the following:

But for translation there was another factor we had to consider, the “f_post” which were the posttranslational 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 the fusion 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 movement of a protein to the membrane and 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 determined to use 5.5min because L2+AIDA are not too small or too big in length (2620nt).

According to Ikeda y Kuroda (2011) L2 carries out an unfolding process to become functional. We found that the 50S ribosomal proteins L2, L3, L14, L23, L24, and L32, as well as the 30S ribosomal proteins S12 and S18 were native premolten globules in their free forms but adopted rigid well-folded conformations during the formation of a functional ribosome. They exhibit some amount of ordered secondary structure; the unfolding of a protein molecule results in an essential increase in its hydrodynamic volume. For instance, there is a well-documented 15–20% increase in the hydrodynamic radius of globular proteins upon their transformation into the molten globule state (Unversky, 2002). Also, we used the equation from Unversky to find the L2 unfolding rate in E. coli which is shown below:

\begin{equation} \large [H]boundary = \frac{[R]+1.51}{2.785} \end{equation}

This equation gives the estimation of the "boundary" mean hydrophobicity value, “[H]boundary”, below which a polypeptide chain with a given net charge “[R]” will most probably be unfolded. Thus, sequences of natively unfolded proteins may be characterized by a low sequence complexity and/or high net charge coupled with low mean hydrophobicity (the values are specified for globular proteins). According to Ikeda and Kuroda (2011) the net charge “[R]” of L2 is 10.9, so we substituted it in the equation below:

\begin{equation} \large [H]boundary = \frac{10.9 + 1.51}{2.785} = 4.45 \end{equation}

With the unfolding value and the rate of membrane transport in E. coli we could use it in Simbiology for modelling the functional L2+AIDA production. The results obtained were the following:

Comparing both graphs (Graph 1 and Graph 2) we realize that although E. coli needs to transport the L2+AIDA proteins to its membrane, the rate at which E. coli does it is slower than the production of the fusion protein, but one thing we noticed and was great is that according to Simbiology, almost all the proteins once they are inserted in the membrane unfold correctly leaving less than 25 nonfunctional proteins.

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.

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

● 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.

● Martina Bergant, N. M. (2010). Modification of Human Papillomavirus Minor Capsid Protein L2 by Sumoylation. Journal of Virology, 11585-11589.

● Takeshi Ikeda, A. K. (2011). Why does the silica-binding protein "Si-tag" bind strongly to silica surfaces? Implications of conformational adaptation of the intrinsically disordered polypeptide to soli surfaces. Colloids and Surfaces, 359-363.

● Uversky, V. (2002). Natively unfolded proteins: A point where biology waits for physics. Protein Science, 739-756.

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