Although the models above can describe the whole process of our system in general, the experimental data still do not perform so well as our expectation because all the promoters are leaked.

According to this case, we come up with an assumption that the diffusion of the repressor results in the lower concentration around the promoter than the one in center of the gene site where the repressor protein produced. Therefore, the lower concentration of repressor causes the high probability of separation between the repressor and promoter. That explained why the promoters were leaked seriously.

Figure 3.1 | The green’s function can perfectly describe the diffusion with a delta function source which shows an instantaneous pulse in time and infinite concentration in space. If the time approaches the infinity, the concentration will be uniform distribution ultimately. However, our diffusion model with a sustained pulse can be regarded as an accumulation of a series of the green’s model.

In order to describe the change of the concentration with the distance in cell, we set up a diffusion model.

Firstly, we use Fick’s first law to define the dynamic process:

The gradient of the concentration is not always constant, so we use Fick’s second law to describe the non-equilibrium state.

Expanding to three dimension, the formula becomes:

We made following assumptions to simplify the formula:

a. The diffusivity D does not change with the position.

b. The diffusion is spherically symmetric.

Then we can get the simplified formula:

In an E.coli cell, we suppose that the repressor is produced from a point without volume, which we call it point source. And as we state in chapter III, the protein producing rate will reach a balance with the dilution and degradation rate. In addition, there is a Neumann boundary we should consider, the cytomembrane, which restricts the sphere of the diffusion.

So we can get following differential equations to describe the process

Finally, we simulated those fuctions and get the steady concentration distribution:

Figure 3.2 | The concentration distribution describes the steady relative concentration of the integrase varies with the relative position to the plasmid which we consider as the point source of repressor production in E.coli. The arrow marks out the average distance between the promoter and the point source as well as the corresponding concentration. We suppose both the average radius of the cell and the concentration of the point source as the unit 1.

The average distance can be estimated from the copy number of the plasmids. We are not able to discuss the interaction between those plasmids but the results of the diffusion model for only one point-source have already verify the conjecture and provide substantial help to our experiment.