Team:Tsinghua-A/Modeling/GILLESPIE/part2

Our model

In our model , we use Gillespie algorithm to simulate the process of inversion. Here we use Markov chain to describe the process of inversion.
We assume that the state ‘A’ , ‘B’ , ‘C’ , ‘D’ , is the original state , the state after inversion , the Intermediate state when A forms holiday junction, the Intermediate state when B forms holiday junction respectively.

(VAN DUYNE G D, 2001)[3]

(VAN DUYNE G D, 2001)[3]

The following are the reactions which are related to the process of inversion.

Then we get the following Markov chain.

Here,are the probabilities of the reaction in an Infinitesimal time. According to the property of the Markov chain , we can get:

In the model,is proportional to the concentration of Cre-Loxp andare much larger than Here we see 1000 genes which are in state ‘A’ as a whole and we get the following results.
The concentration of Cre-Loxp ：

The number of the genes which are in the original state and in the state after inversion:

From the above two figures, we can see that the rate of the inversion is proportional to the concentration of Cre-Loxp. And about 50% of the genes keep the original state and the other 50% change to the state we want. The result is exactly what we expected, which clearly demonstrates our system can truly be controlled by adding in external signal molecules (arabinose).