# Team:Paris/Modeling/hill approach

### From 2008.igem.org

## Contents |

# Model of the *APE modelisation*

## What kind of Mathematical Simulation ?

One of the strength of the synthetic biology is that precise knowledge and caracterisation of certains interactions allow very good predictions and simulations. So, our second model intends to get the best precision in the modelisation, consistent with the simpliest (but still logical) hypothesis possible.

To determine interactions like MichaĆ«lis-Menten's or Hill's, we start from the basical chemicals equations and try to caracterise their consequences on the behaviour of the system with few parameters. For instance, each *complexation reactions* will be caracterised at their steady-state, for all sets of initial concentrations (see complexations).

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.

## 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* FlhD_{4}C_{2}. 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 caracterisation* probably around 2*4 = 8 (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).
Besause of that, our caracterisations doesn't allow us to know the *real concentrations* of the proteins we produce, but only their "real times effects" on the promoters they influence.

Every complexations we deal with are the supposed to reach *immediatly* their **steady-states**. So, the only phenomenon we are observing along time is the protein production ; that's why this model can't really simulate the possible delay a complexation could introduce, and which could be important for the stiffness of our oscillations...

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 : cell.L
^{-1}

→ Average Intracellular Volume : L

→ Average Extracellular Volume (in the chemostat) : L

- Renewal Rate : L.min
^{-1}

(→ Dilution Rate : min^{-1})

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

## Incrementally detailed Parts of our Project