

1. Motivation 
In our story, Bank has to change its state from collection state to distribution state, and vice versa depending on the concentration of C4HSL. 
We analyzed the parameter range which can realize the mechanism. 



2. Analysis 
2.1. Problem definition 
The main components of Bank circuit are shown below. 

Fig. 4321. C4HSL dependent switch 

The equations for the components are described in Fig. 4322. RhlR is ignored in the equations because it is constantly expressed in the cell. Also, AiiA and RhlI are ignored because it is not related to the following analysis. 


Fig. 4322. equations for C4HSL dependent switch 

What we want to achieve is 
1. In the high concentration of C4HSL, Bank circuit is stably in collection state. TetR expresses highly and LacI doesn’t express much. 
2. In the low concentration of C4HSL, Bank circuit is stably in the distribution state. LacI expresses highly and TetR doesn’t express much. 

2.2 Nullclines and stable state 
Since our aim is to estimate stable state, nullclines for each equation play crucial role. Equations for nullclines of each protein are described below. 


Fig. 4323. nullcline equations for LacI and TetR 

In terms of these equations, our aim is translated as follows. 
When the value of [C4] is high, these nullclines have to have one intersection which has high value on [TetR] and low value on [LacI]. 


Fig. 4324. Two nullclines intersect on the point which is on high TetR concentration and low LacI concentration. Green line shows the equation (1) and blue line shows the equation (2). 

To meet this requirement, we have to make the maximum expression rate of Prhl/lac should be high enough to make the nullclines have only one intersection. 
When the value of [C4] is low, these nullclines have to have one intersection which has low value on [TetR] and high value on [LacI] 


Fig. 4325. Two nullclines intersect on the point which is on low TetR concentration and high LacI concentration. Green line shows the equation (1) and blue line shows the equation (2). 

This condition can be satisfied with any parameter values because the C4HSL concentration can be 0. This make the intersection of the two nullclines to have one intersection with any parameter values. 
With the optimized parameter values, the system was confirmed to work as we expected. 


Fig. 4326. LacI concentration depending on C4HSL 

When the C4HSL concentration gradually decreases, LacI will be expressed following the green line as shown in Fig. 4326. On the other hand, when the C4HSL concentration increases, LacI will be expressed following the blue line as shown in Fig. 4326. 