Team:Tsinghua-A/Modeling/GILLESPIE/part3

Simulating every single trajectory

1000 genes which are in state ‘A’ are seen as a whole in the above model. To make the simulation more convincing, we then simulate the trajectory of every single molecule. After that we synthesize all trajectories to get the final result.

Result

The concentration of Cre-Loxp :

Then we use green line to denote the number of genes which are in the original state and red line to denote the number of genes which are in the state after inversion.

1 trajectory:

10 trajectories:

100 trajectories:
1000 trajectories:
From the result, we can see the process more clearly. From the figure of one trajectory, we can see the inversion is reversible so a gene may reverse a lot when Cre-Loxp exists. And the inversion stops when Cre-Loxp is degraded to zero. From all the figures we can see the randomness decreases when the number of trajectories increases. And finally 50% of the genes become the state we want.
In a Continuous markov process, when the state set is finite, there is a stationary distribution:
,and it also satisfies:(π is the distribution of every state, S is the state set, q is the transition probability )
With this theorem, we can know that after a long time, the distribution of the four states will be stable which could also be seen in the figure of 1000 trajectories.