Team:CIDEB-UANL Mexico/math overview

When Biology meets Math

For iGEM projects, the use of mathematical models is performed to describe the behavior of biological systems, representing the quantitative relations between two or more variables involved in the function of a gene or a set of genes in organisms as E. coli.

Our team decided to use a deterministic model to simulate and represent the functioning of the four modules assuming that the variables (mRNA and protein concentrations) adopt a continuous behavior and obey kinetic rules that can be represented with constant values (Loera, 2012).

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

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.

This type of mathematical model is used to involve variables that abstract both the gene expression and physiological cycles (chemical process, transport of proteins, etc.). By the use of traditional differential equations we were able to construct the description and analyze the behavior of mRNAs and proteins production. We were focused in determine through a mathematical model of the four modules (one per each): Capture, Aroma, Resistance and Union.

Deterministic Modelling: Equations and Parameters

As it was established previously, is necessary work with different equations focusing in the production rate and degradation rate according to the length of genes in each module of the project. This is performed in order to obtain both the concentration rates of mRNA and protein based on system (gene) length and protein length respectively.

Equations

● mRNA

Generally, to describe the amount of mRNA produced over t time, the equation implemented is shown below:

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

Where: “α₁” means the transcription rate of a given gene; “f_y” represents a regulatory function (if there is) that can activate or inhibit the system; and “d₁ [mRNA]” the degradation rate of the mRNA produced.

● Protein

The same happens with the protein production, but differs in the formula because it also involves a post-translational variable:

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

Where: “α₂ [mRNA]” means the translation rate of a protein based on the amount of mRNA available; “d₂”[P] represents the degradation rate of that protein; and “f_post" the posttranslational variables which affect the production of the final (functional) protein.

Parameters

To determine both the transcription (1) and translation (2) rates, we used the parameters from Singapore 2008 iGEM team, as well as the degradation rates from Beijing PKU 2009. We used the parameters for the speeds at which E. coli carries out transcription and translation from Singapore 2008 iGEM, assuming a transcription speed of (70nt/s) and a translation speed of (40aa/s). We multiply the speeds by 60 because we use minutes as units in the simulations of the modules. We assumed that E. coli division time was 30min based on PKU Beijing 2009 iGEM team.

\begin{equation} \large \alpha_{1} = \frac{transcription speed}{gene length \cdot(nt)} \large \alpha_{2} = \frac{translation speed}{protein length \cdot(aa)} \end{equation}
\begin{equation} \large \alpha_{1} = \frac{70 \frac {nt}{s} \cdot(60)}{gene length \cdot(nt)} \large \alpha_{2} = \frac{40 \frac {aa}{s} \cdot(60)}{protein length \cdot(aa)} \end{equation}
\begin{equation} \large d_{1} = \frac{1}{half-life} + \frac{1}{division time} \cdot(min) \end{equation} \begin{equation} \large d_{2} = \frac{1}{half-life} + \frac{1}{division time} \cdot(min) \end{equation} \begin{equation} \large d_{1} = \frac{1}{half-life} + \frac{1}{30} \cdot(min) \end{equation} \begin{equation} \large d_{2} = \frac{1}{half-life} + \frac{1}{30} \cdot(min) \end{equation}

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.

● iGEM CIDEB UANL. (2012). Modelling: Equations. Retrieved from: https://2013hs.igem.org/Team:CIDEB-UANL_Mexico/Math-Equations

● 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.

● NTU Singapore. (2008). Modelling: Parameters Retrieved from: https://2008.igem.org/Team:NTU-Singapore/Modelling/Parameter

● PKU Beijing. (2009).Modelling: Parameters. Retrieved from: https://2009.igem.org/Team:PKU_Beijing/Modeling/Parameters

● Vorackova Irena, S. S. (2011). Purification of proteins containing zinc finger domains using Immobilized Metal Ion Affinity Chromatography. Protein Expression and Purification, 88-95