Team:uOttawa/modeling

Equations involved in modeling the biological system

The mathematical model was developed using basic rate formulas of activation and repression as shown in the system of equations above. We then expanded them to include the different parameters required for our model.

These additions are as follows

IPTG concentration to calculate the amount of Y (LacI) repressing Z (sfGFP).

ATC concentration to calculate the amount of X (rtTA) available to bind Y promoter and Z promoter.

Transfer rates of proteins between the cytosol and the nuclei to calculate transfer rates by size of each protein.

The four rate equations for the production of mRNA over time of GEV (1.1), X (rtTA) (2.1), Y (LacI) (3.1) and Z (sfGFP) (4.1) in the nucleus are show below.

The first sets of equations were used to describe the amount of mRNA transcribed. These equations also took into consideration that the mRNA would be transported out of the nuclei. In the model, we assumed that all of the mRNA molecules would be exported to the cytosol. The differential equations representing the variation in mRNA in the cytosol are represented below in equations 1.2, 2.2, 3.2 and 4.2.

Computing the variation and amounts of mRNA in the cytosol over time then allowed us to compute the variation of protein translated. The equations of translation shown below in equations 1.3, 2.3, 3.3, 4.3 represent the variation of protein in the cytosol over time. The translational rates were calculated based on the average translation rate of a ribosome (9.5 base pair per second ¹²). Here, we assume that the quantity of ribosome is not a limiting factor and that there are no translational regulations involved in the model. The degradation rates are taken from the average degradation rate of protein in yeast¹⁴.

Once the amounts of protein calculated, we assumed that they would diffuse at an average rate through the nuclear pore to then affect the transcription of the other genes in the system. The calculations leading to the transfer rates are described in the parameter section under transf. Equations 1.4, 2.4, 3.4 and 4.4 represent the variation of protein present in the nucleus over time.

To have better control on the feed-forward loop system, LacI was regulated with IPTG. Changing the amount of ATC regulated the amount of active rtTA. And the Amount of GEV available to activate production of rtTA is controlled with ß-estradiol. The following equations used to compute the amount of each compound are shown below. These numbers are imputed in the mRNA equations and vary accordingly with the amount of each transcription factors expressed in the ODE system.