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).

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

Union

Deterministic Modelling: Equations and Parameters

As it was previously established, 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}

References

● 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