# Diffusion model

## The model about the diffusion of the repressor

The models described on the ‘model-intagrase model’ page simulate the entire recombination procedure in our in vivo system. These models laid the solid theoretic basis for the feasibility of our project. However, our wet lab results are far less satisfying than the expected data from the prediction of these models. The serious leakage of the promoters is the main cause here.

In order to get a better understanding of the problem, we came up with a new model that focuses especially on the concentration of the repressor for the promoter at different site in the cell. The model wants to prove 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 is produced. Therefore, the lower concentration of repressor causes the high probability of separation between the repressor and promoterwhich then explains the extremely high leakage of the promoter.

**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 from the original transcription and translation site 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 assume 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 assume both the average radius of the cell and the concentration of the point source as the unit 1.

The average distance mentioned above can be estimated from the copy number of the plasmids.Though further discussions about the the interactions between those plamids can not be made yet, our hypothesis about the diffusion of the repressor causes the leakage has already been proved by this model. Futher improvements to eliminate the impact caused by diffusion may help to solve our leakage problem. .