The original model of DNA recombination by integrase
At the beginning of our dry lab work, we use a simple model to describe the main process of the integrase recombination. (figure1.1)
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. 
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, 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)
As the graph shows, the transformation rate can reach almost 80 percent in general.
The improved model with gene expression
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 (except for recombinases) shall decrease as a result of both degradation and dilution.
The maximum promoter activity can be estimated by the following equation
For the serine integrase, we assume the decrease in concentration is solely caused by dilution. And the dilution rate can be estimated as follows.
Last but not least,with the cell division, the new DNA produced without integrase combined should also be taken into consideration. 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.