From 2007.igem.org

We modeled the transcriptions factor interactions as Hill-like functions, while the protein-protein interactions are represented through a typical prey-predator model. All the hypotesis were used to get the ODE model. For the transcription factor level we have the following equations:

In the equations SA and SB indicates the inputs, i.e. if an oleate level is passed. Notice that the second assumption is used to obtain this equations form. [Pho4free]is the Pho4 concentration (the output from protein-protein interaction) that participates to transcription factor interaction. To get this value we write the equation for protein-protein interaction,:

Using the first assumption and the mass conservation law (since [Pho4]=[Pho4free]+[Pho4P] )we can obtain an expression for [Pho4free] in terms of [Pho80Pho85] and [PhO4] using steady state solution. The third equation in the ODEs system will become: