Team:USTC-China/modeling

Toggle Switch without designed genetic route

The chemical reactions are:

According to the Law of mass action, we can easily write down the equation approximations to reduce these equations.

First of all, we do quasi-steady state approximation for dimerization reactions

where [cIT] and [croT] are the total concentration of cI and cro.

Besides, we decide to take the degradation into account in the very end step. So, considering the conservation of binding site

Where R is the total binding site number, we get

The above equations give us the expression of [cro2 �OR3 ];[cI2 �OR1 ];[cI2 �OR1 �cI2 �OR2 ]. The �nal step is to write down the evolution equations for [cI] and [cro].

where

Toggle Switch with designed genetic route

There is a di�erence if we add our designed genetic route:

where �mcro refers to �mcro.

Three additional equation should be taken into account

The above gives us the expression for [img], which is a function of [img]. It is then straightforward to derive [mcro] from the following

Once obtained [mcro], substitute it into the following to get the �nal equation

Result

We found that for the parameters in the value table, there is no real solution for both the two part in the following equation The reason of this problem probably stems from the[img] arameters|�1 ;�2 di�ers too much wich �3 ;�4 ;�5 , thus it is impossible for the equation to have real solution. We lot the stream diagram for [cIT] and [croT]:

After adding our designed route, the stream flow diagram is

We can see that after adding our designed part, the system goes through double steady state to single steady state.