At the beginning of our dry lab work, we use a simple model to describe the main process of the integrase recombination. (figure1.1)

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)”.

Previous researches have shown that the combination and dissociation process between DNA and the integrase can be very fast. Therefore, we assume that rapid equilibrium can be reached in the first and forth step in our model. Both of the reaction share the same equilibrium constant, KbI. [1]

Thus, we are able to describe the whole process above with four equations.

The original model composed of all the functions above can perfectly predict the recombination reaction in vitro[1], where the concentration of integrase is considered to be constant, because the concentration of integrase required for efficient recombination is much higher than the one of the DNA substrate. The simulation results are shown here. (figure1.2)

Figure 1.2 | 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.

As the graph shows, the transformation rate can reach almost 80 percent in general.

Despite the great accuracy showed by the original model under in vitro conditions, we still want to figure out how the recombinases might work in in vivo experiments with gene expressing at the same time, for an obvious reason that we will finally carry out our circuit inside a living changing E. coli.

Considering the fact that the combination between the inducer and promoter are much faster than the production of integrase, we we assume that the activation of the promoter happens instantaneously..

Furthermore, the concentration of the protein shall decrease as a result of degradation and dilution.

The maximum promoter activity can be estimated by the following equation

For the serine integrase, we suppose only the dilution exists. And the dilution rate can be estimated as follows.

Last but not least, with the cell division, the new DNA is produced without integrase combination. So the origin model can be adapted to a new form as follows.

As the graph shows(figure2.1), the transformation rate can reach almost 100 percent in general. The critical difference between the vivo and vitro environment is the replication of DNA which decrease the concentration of LRI₁ and LRI₂ and increase the one of LR relatively and partially. Such process is a special kind of “transformation” which can skip the slow equilibrium process., ”syn”, to achieve a higher transformation rate in a short time.

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.