Team:CIDEB-UANL Mexico/math overview

When Biology meets Math

In an iGEM project, mathematical models are necessary to predict the behavior of a biological machine, representing the quantitative relations between two or more variables involved in the function and expression of a gene or a set of genes in organisms like E. coli.

Our team decided to use a deterministic model to simulate and represent the function 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.

This type of mathematical model is used to include variables that considerate both, the gene expression and the physiological cycles (chemical process, transport of proteins, etc.). By using traditional differential equations, the description was able to be constructed and it also permitted the analysis of the behavior of mRNAs and protein production. The focus was to determine through a mathematical model the ideal behavior of the four modules independently.

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Deterministic Modelling: Equations and Parameters

As it was previously stated, it was necessary to work with different equations focused in the production and degradation rate of mRNAs and proteins, according to the length of the genes in each module. This was 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 post-translational variables which affect the production of the final (functional) protein.

Parameters

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

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

Bibliography/References

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

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

● 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.

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

● 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.

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

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

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