# Team:Paris/Modeling/FromMolReactToNLOde

From Molecular Reactions to Non-Linear ODEs

## Idea and Assumptions

We will here propose a Mathematical Modeling of the elementary molecular reactions. The idea of the Characterization Approach is that this modelization must both accounts for every small steps of the system and allow the experimental characterizations.

Therefore, the following equations do not describe properly what really happens in the cells. For exemple, we know that the transcription factor FlhD-FlhC is actually an hexamere FlhD4C2. But, as we will surely not get access to the dissociation constant of the hexamerisation, we just treat it as an abstract inducer protein "FlhDC", with an order (n) in its complexation caracterization probably around 2*4 = 8 (but perhaps completly different ! ; the estimation of the error by the Parameters Finder Programs would tell us if it stays consistent).

↓ all assumptions ↑

## Mathematical Interpretation and Simulation of the Molecular Reactions

First, we consider the complexation phenomenon. We show, under the quasi steady-state hypothesis, how it leads to Non-Linear interactions like "Hill functions".

Then, we apply these results to the Molecular Reactions of our system. That gives all the theoretical calculation of the complexations, and finally, with the previous assumptions, we get the equations of the full system of ODEs.

 Complexation Steady-State The double triangle (><) is a symbol for complexations : means n1 molecules of FlhDC complexed with a pFliA promoter ↓ Complexation Steady-State ↑ On the Example : How given amounts of FlhDC and FliA would produce FP1 ? FlhDC and FliA bind to pFliL following "Hill functions" : Equations of the Full System The double triangle (><) is a symbol for complexations : means n1 molecules of FlhDC complexed with a pFliA promoter ↓ ODEs of the Global System ↑ On the Example : How given amounts of FlhDC and FliA would produce FP1 ? FlhDC>
For complete PDF with all Reactions and Numbered References : complete list of reactions and equations