# Team:Evry/Modelmeta1

### From 2013.igem.org

# Sensor Model

## Introduction

In order to determine our **Iron Coli**'s enterobactin production rate, we first have to know how much time our bacteria will take to sense the ambient iron. So, this first part of the *Enterobactin production model* focuses on the synthetic sensing system our team implemented in the bacteria.

## Observations

Naturally, when there is a large amount of iron in the environment, 2 iron ions complex with 2 FUR proteins and bind to the FUR Binding Site (FBS). This interaction regulate negativelly the genes downstream.

As shown on the Figure 1, our sensing system relies on a FBS. In order to easily model the sensing delay and analyse its results, the first construction is composed of a GFP placed right after the FBS.

## Goals

Our goal in this part of the model is to create a generic FBS-related sensing model so that:

- We can determine the iron-sensing delay of our bacteria
- The model can be reused by other projects that include a FBS sensor

## Materials and methods

**From Iron to FBS:**

The iron-FUR complex is simply formed that way:

We reduced this equation to:

Which is not annoying, since we just have to divide our [FeFur] by two to get the real complex concentration.

We can easily write down both the formation (v) and the dissociation (v') speeds:

We chose to model the iron input in the bacteria using a linear function of the external iron concentration *Ferext*, the factor *p* being the cell-wall permeability for iron.

The FUR on the other hand is produced by the bacteria. Its evolution can also be considered as linear, using a mean production rate of *Fur0*.

In this model, we only track the free Fe-FUR complex and not those attached to a FUR Binding Site. *FBS* is the number of inhibited Fur Binding Sites.

**GFP expression:**

To simulate the inhibition phenomenon, we chose to use our Logistic function under its differential form. Since it is the Fe-FUR that represses it, the LacI can be expressed as a logistic fuction of the Fe-FUR:

K_{i1} is a non-dimensional parameter that repesents the inhibition power, and K_{f} is the fixation rate of the Fe-FUR on the FBS. Finally, N_{pla1} is the number of pasmids containing the GFP.

**GFP Production:**

The *[mRNA]* and *[GFP]* equations are alike. The prodction rates are K_{r} for the mRNA and K_{p} for the GFP. Since *FBS* represents the number of inhibited Fur Binding Sites, we have to substract it from N_{pla1}.

Both variables also have a negative degadation term:

## Tuning the model

The only parameter we couldn't find in the litterature is K_{i1}. Nevertheless, thanks to our construction and the experimental results we obtained with a plate reader, we can tune this parameter to obtain a **realistic sensing model**!

The Figure 2 is a set of experimental results showing the sensing efficiency for different iron concentrations. We can notice a substantial behaviour change between 10^{-4}M and 10^{-5}M of iron : the activity of our bacteria leaps at 10^{-5}M.

Therefore, we ran simulations with different K_{i1} values to try to determine the most fitting K_{i1}.

The Figure 3 shows that our model can lead to 3 different leaps, depending on the K_{i1} value. The leap we are interested in is the 10^{-4}M 10^{-5}M leap.

We thus zoomed in, and performed the simulations with a more adapted range of K_{i1}.

The Figure 4 shows a more precise graph. It is with those simulation values that we calculated the most realistic K_{i1}, by maximizing the leap it creates. We also figured that the best fitting case requires the GFP reaching a plateau right after 10^{-5}M of iron.

Finally, we set **K _{i1}** at

**6,3.10**.

^{-5}## Results

The following results were computed using those parameters:

Name | Value | Unit | Description |
---|---|---|---|

p | 0.1 | min^-1 | Permeability of cell wall |

K_{FeFUR} |
0.01 | M^{-1}.s_{-1} |
Formation constant of FeFur complex |

D_{ff} |
0.001 | min^{-1} |
FeFUR degradation rate |

K_{p} |
0.5 | min^{-1} |
translation rate |

N_{A} |
6.02*10^{20} |
mol^{-1} |
Avogadro's constant |

V | 6.5*10^{-16} |
m^{3} |
Volume of a bacterium |

D_{mRNA} |
0.001 | min^{-1} |
mRNA degradation rate |

K_{f} |
10^{-4} |
min^{-1} |
fixation rate of FeFUR |

Fur_{0} |
0.01 | mM.min^{-1} |
Fur Production rate |

K_{i1} |
6,3.10^{-5} |
- | Sensor efficiency |

As shown in the Figure 5, there is a significant change in GFP production after 1000 seconds between the curve at 10^{-4}M of iron and the one at 10^{-5}M. The bacteria can thus sense low concentrations of iron.

## Conclusion

The first construction of our team is fully modeled : we made a **generic iron sensing model** (with the FBS system).

Thanks to the **experimental results** we obtained with a plate reader, we were able to **tune an unknown parameter** to obtain a more realistic model.

This sensing model allowed us to continue the Enterobactin production model, the next step being the Inverter model.

## Models and scripts

This model was made using the Python language. You can download the python script here.