The usual approach in synthetic biology for modeling a genetic network is to describe the interaction of inducers and repressors with promoters. This description is very detailed and very difficult to implement because of the many unknown parameters involved in the differential equations.

We tried an approach called “keep it simple”, so the goal of our model is to describe the relation between the concentration of bicarbonate and the expression of the GFP but with the possibility of make useful predictions for lab.

The model.

In the description of the project, we introduced a CO2 induced biosensor with GFP as a reporter. The biosensor is designed to sense the concentration of bicarbonate on the media. There is a complex mechanism behind for the signaling of the bicarbonate concentration to the GFP expression. Some of the parameters required for a full described model of this signaling process were not available in the literature. For these reason we decided to model only the indirect expression of GFP by the induction of bicarbonate.

Let's represent the concentrations of bicarbonate and GFP as under the time constraint .

The model is based on the next assumptions (represented en 1 and 2):
-The bicarbonate concentration is controlled in every moment, and it increments linearly.
-The bicarbonate concentration will remain between two levels, [0,K], where K is the saturation concentration.
-The GFP has a basal expression j.
-The GFP expression will remain too between the basal level, j, and the saturation J, [j,J].
Let's Suppose additionaly that with , then we can define and solve the Differential Equation:

where .

Using initial values we have

so that: .

Then we have y and x in terms of t:


The next figures represent the numerical implementation of the solution of the differential equation. The maximum level of concentration of GFP allowed is 50, and the HCO3 concentration increments linearly.
1. Simulation with initial concentration of 25 mili Mol of HCO3.

2. Simulation with initial concentration of 40 mili Mol of HCO3.

3. Simulation with initial concentration of 90 mili Mol of HCO3.

As a second approach for the simulating the model we deduced another expression of y in terms of x:

With this expression we explored the relations among GFP and HCO3 concentrations and the maximum concentration level of HCO3. This in order to explore the response of the expression of GFP at the maximum input level.

1. Simulation with maximal concentration of HCO3 25 mili Mol.

2. Simulation with maximal concentration of HCO3 40 mili Mol.

3. Simulation with maximal concentration of HCO3 90 mili Mol.

We developed a mathematical model for the expression of GFP in reponse of the stimulus of bicarbonate (HCO3). In the first set of simulations we observed that the system respond very fast to the stimulus of HCO3. This behavior was expected because it is a simple induction until saturation. This response happens even with low HCO3 concentrations, and the observation was corroborated at lab.
The second set of simulations shows how the concentration maximum level K does not affect the dynamics of the solution. And in short term, hours, the GFP expression will remain saturated.