Modeling of the Toxin-antitoxin System
Our project aims to build up a toxin-antitoxin system of two strains of E.coli cells to realize the control on population ratio. As indicated in the former part of our modeling work, the auxotroph system is robust, but it might fail when the maximum growth rates are not suitable. What’s worse is that the maximum growth rates are difficult to change. So we seek to find another approach to tune the population ratio.

We build up a model for toxin-antitoxin system, which could be described using the ODEs below:

Explanation of the ODEs:

1) Equation (1) describes the net growing of strain 1. The first item is the growing item, which is proportional to the density of the same strain of cells, while the second item is the rate at which cells are killed by toxin. To be reasonable, the killing caused by toxin must reach saturation when the density of toxin is big enough, so we chose a hill function other than a linear function. Equation (2) describes the net growing of strain 2, because the two strains are designed symmetrically, this equation has the familiar form with Equation (1).

2) Equations (3) and (4) describe the net generating of antitoxin in each cell (Equation (3) describes strain 1, while Equation (4) describes strain 2). Each kind of antitoxin is generated constitutively by cells themselves, so the first item, say, the generating item, should be constant. The second item for each equation is the decreasing of antitoxin, it is caused by binding with the corresponding toxin.

3) Equations (5) and (6) describe the net increasing of the concentration of toxin in each cell. Assuming that the two strains are growing in a nutritionally adequate medium, but the volume of medium doesn’t change, the generating rate and export rate of toxin doesn’t change over time for each cell. Since the importation of toxin is proportional to the density of toxin in the medium, it must also be proportional to the density of cells which generate this toxin. The meaning of the second item is familiar with the second item in Equations (3) and (4).