The fur-lac-cecropin AD system

In order to measure the ability of engineered bacteria to kill iron bacteria, we modeled the production of the cecropin AD and the sterilizing ability of the cecropin. We want to model the fur-box fell the concentrate of Fe2+ to drive the LacI to reverse the signal to produce the cecropin AD which play a bactericidal effect. So our model used in the project includes four parts: the iron sensor, the inverter system, the sterilizing system and the chelator system.

The first part focuses on the sensor system to find the Fe2+ needed by our system. And the strength of the best promoter with the fur-box of the three kinds of fur-box was determined. Second, the inverter system our team implemented in the bacteria focused on the cecropin AD produced with time. The third part modeled the sterilization rate of the cecropin AD. The last part integrate those previous parts leading to the cecropin production and sterilization. The model show the time it takes for our bacteria system to work.

1.Sensor model

1.1 Introduction

We first modeled the sensing system using ODEs with the help of experimental results to determine one of our parameters ki1.We had three kinds of fur-box designs. We model our three kinds of fur-box (shown in the figure 2) to find the optimal fur-box and the strength of the promoter with the best kind of fur-box. We finally corrected our model through the experiments. We make this framework like figure 1.

1.2 Methods and materials:

1.2.1 The dynamic simulation of sense iron to FBS:

(1) the iron-FUR complex formation:

$$2\cdot FUR+2\cdot Fe\leftrightarrow Fe_{2}FUR_{2}$$

We think this equation to:

$$FUR+Fe\leftrightarrow FeFUR:K_{FeFUR}$$

We just want to use differential equations more easily. And we can easily divide our [FeFur] by two to get the real complex concentration.

(2) We can easily make the formation (v) and the dissociation (v') speeds:

$$V=K_{FeFUR}\cdot \left[ FUR\right] \cdot \left[ Fe\right]$$

$$V'=d_{ff}\cdot \left[ FeFUR\right] $$

• K_FeFUR : Formation constant of FeFur complex (m^-1∙s^-1)

• d_ff : FeFUR degradation rate (min^-1)

We model the iron input in the bacteria using a linear function of the external iron concentration Ferext with the factor p which is the cell-wall permeability for iron.

$$\dfrac {d\left[ Fe\right] }{dt}=p\cdot Ferext-K_{FeFUR}\cdot \left[ FUR\right] \cdot \left[ Fe\right] +d_{ff}\cdot \left[ FeFUR\right]$$

$$\dfrac {d\left[ FUR\right] }{dt}=FurO-K_{FeFUR}\cdot \left[ FUR\right] \cdot \left[ Fe\right] +d_{ff}\cdot \left[ FeFUR\right]$$

• p : Permeability of cell wall (min^-1)

• d_ff : FeFUR degradation rate (min^-1)

We track the free Fe-FUR complex but not those attached to a Fur Binding Sites in our model.

$$\dfrac {d\left[ FeFUR\right] }{dt}=K_{FeFUR}\cdot \left[ FUR\right] \cdot \left[ Fe\right] -d_{ff}\cdot \left[ FeFUR\right] -\dfrac {1}{N_{A}V}.\dfrac {dFBS}{dt}$$

• NA : Avogadro’s constant (mol^-1)

• V : Volume of a bacterium (m³)

• FBS : the number of inhibited Fur Binding Sites

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

$$\dfrac {dFBS}{d\left[ FeFUR\right] }=\dfrac {K_{i}1}{K_{f}}\cdot FBS\left( \left[ FeFUR\right] \right) \cdot \left( 1-\dfrac {FBS\left( \left[ FeFUR\right] \right) }{N_{pla1}}\right) $$

• K_f : fixation rate of FeFUR (min^-1)

• K_i1 : constant repesents the inhibition power (min^-1)

• N_pla1 : pET28-a plasmid number (nb/cell)

K_i1 is a non-dimensional parameter which repesents the inhibition power, and K_f is the fixation rate of the Fe-FUR on the FBS. N_pla1 is the number of pasmids containing the sensor system.

1.3 Result

We want to know the fittest ki1 for the model to sense the iron and the concentrate of iron. We make three kinds of fur-box for our sensor system. We want to know which is our best choice. Our experiment result show in the figure 3.

Then we try to change the value of K_i1 to model different strength of promoter with fur-box in our experiment which show in the figure 4. We want our system to make sense in the high level of Fe²⁺, so we choose the fur-2 system. And we finally set the KI1:6.3*10^-5.