From 2007.igem.org

1.toggle model

First,the ordinary differential equations(ODEs) of the toggle switch were derived as

These equations were normalized as follows:

In the steady state,time derivatives are zero:

As a result,the nullclines of this system were derived as

Therefore,the phase plane of this system can be plotted as Fig●

About parameters,we use three sets of parameters.
　　1)the maximum expression rate of repressor A and repressor B is balanced,and hill coefficient of both A and B is three.
　　2)the maximum expression rate of repressor A and repressor B is equal,and hill coefficient of A is one.
　　3)the maximum expression rate of repressor A and repressor B is not balanced,and hill coefficient of both A and B is three.

we correlate phaseplane analysis and simulation results.
First,we simulate about the phaseplane of two stable equilibrium points(the upper left figure) and use three kinds of initial values.
1. (Ra:low , Rb:high) 2. (Ra:high , Rb:low) 3. (Ra:middle , Rb:middle)

安定点B付近から始めるとB状態で安定し，安定点A付近から始めるとA状態で安定しているのが分かる． 不安定点付近から始めるとどちらかで安定化する．

Next, 次に，安定点が一つしかない場合のシミュレーション結果は下のようになる．

安定点が一つしかない場合は，安定点B付近から始めてもA状態で安定化してしまうのが分かる．
As a result,taking two stable status need the phaseplane of two stable equilibrium points and we have to set proper parameters.