# Model of the APE modelisation

## What kind of Mathematical Simulation ?

We decided to use mostly Ordinary Differential Equation approach, at least for the study of the Oscillations and of the FIFO. For the Synchronisation module, we will probably use Probabilistic Differential Equations, in order to introduce the differences between the cells.

We assume in this section that the concentrations of the products of complexation reactions are at steady-states (which is reached immediatly), described as hill function (see precision on hill functions).

## Bio-Chemical General Assumptions

We know that 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 in its Hill function probably between 3 and 6 (but perhaps completly different; the estimation of the error by the 'findparam' program will tell us if we are right to do so).

For the moment, at each part of our modelisation, we reduce the expression of a gene at its transcription. The translation process is not taken into acount (see however considerations on RBS).

As we need to keep our cells in the exponential phase of growth (and since we can't use the already mentioned system of Ron Weiss, see description of the project), our system works in a chemostat. We will also be able to estimate the Cell Density, and we will have to take into acount the renewal phenomenon. Under this conditions, we assume the following constants to be true :

• Cell Division : every 35 minutes
• Dilution Rate : 0.0198 min-1
• Cell Density : XXX cell.L-1
• Average Intracellular Volume : XXX L
• Average Extracellular Volume (in the chemostat) : XXX L
• Renewal Rate : XXX L.min-1
• Dilution Rate : XXX min-1

To see more details about the modelisation and the values of the involved constants, see the bibliography and the estimation section.