Our team used mathematical model to solve two problems in total.

A. Predicting the efficiency of integrase in vivo. It confirmed the feasibility of our circuit when we designed it.

B. Simulating the diffusion process of the repressor and trying to get a reasonable solution of the differential equations to interpret the leakage of integrases when our wet lab experiment went wrong.

All the results is calculated on MATLAB or Mathemetica.

In this part, we set up an original model of DNA recombination by integrase, the reaction function goes as follows:

Figure 1.1 | “PB” is the attP and attB sites of DNA, and “LR” is the recombination product, attL and attR sites, which are direction-changed, of the DNA. PBI is the complex of DNA and 4 integrase molecule. LRI1 and LRI2 are two kinds of complex which are conformationally distinct from PBI. The “single arrow” represents the reaction which can reach rapid equilibrium with the equilibrium constant over it. The “double arrow” represents the slow reaction with the reaction rate constant over and below it. “K(bI)” is the equilibrium constant. “k(+r), “k(-r)”, “k(+syn)”, “k(-syn)” is the reaction rate constant. The site direction change occur in the second reaction: “PBI” to “LRI(1)”.

We use “PB” and “LR” to represent the sites of DNA being recombined or not separately and “Int” to represent the integrase. The others represent the intermediates.(figure.1)

Regulating all the reaction process by appropriate mathematical functions, we figure out the efficiency of the integrase recombination in vitro, where there is no substrate production, dilution and degradation.

Then, we get the figure1.2 to interpret how the recombination rate changes over time. And as the graph shows, the transformation rate can reach almost 80 percent within 0.5 hour approximately. It confirms the possibility that our circuit based on the serine integrase possesses the capacity to make regular response to the clock signal at the hour timescale.

Figure 1.2 | Figure1.2 It separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.

Then, we improve the origin model by adding gene expressing and other biochemical reaction to predict the theoretical efficiency of the integrase in vivo.

We are not sure whether the integrase is effective as we expect in the coli, even if it prefers well in vitro, so we try to build a more integrated model. Considering that the factors in the coli are so complicated, we merely select the fairly significant factors as the component of our model, including the gene expression, the change of the promoter’s activity and the dilution and the digestion of the protein.

Finally, we get the exciting results of the model shown in the figure3. The transformation rate can reach to nearly 100 percent in 5 hours. The mathematical demonstration will explain how the factors in vivo help to shift the equilibrium forward.

Figure 2.1 | The figure separately describes how the proportion of substrate “PB” and product “LR” in total DNA varies within 10 hours in vivo. The red line represents the concentration proportion of the product, “LR”. More calculation details about this graph are attached to the back.

In this part, we use the diffusion model to solve an experimental problem.

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 explains why the promoters are leaked seriously.

In order to describe the change of the concentration with the distance in cell, we build a diffusion model. We find that the concentration of the repressor will decline rapidly due to the diffusion. However, when there is a source in the center, for example the translation of mRNA causes a steady flow of repressor produced, the concentration distribution will get constant in the end. The results are shown in figure3.2.

The average distance (relative) between the promoter and the repressor source is about 0.17 and the corresponding concentration (relative) is 0.6577 which means when the repressor gets to the promoter, its concentration decreases more than 33 percent. It gives us an reasonable explanation to the problem why all the promoters are leaked seriously. According to the enlightenment of the model, we take two kinds of measures to ease the problem.

First, we can increase the plasmid copy number so as to increase the dense of the point sources and reduce the average distance.

Second, we can use only one kind of plasmid for both repressor production and the integrase production, so the source of the repressor will get extremely close to the promoter.

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.

In general, the model not only helped us to demonstrate and analyze our project, but also gives us a vigorous tool to solve the unexpected problem. You can see more detailed mathematical demonstration at the back of each part.