Revision as of 14:12, 17 October 2018

Iron Model

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

3.Goals

Our goal of this model is to create a generic quorum sensing model so that:

• We can determine the effect of afeR promoter and predict the production of DspB and enterobactin.

• We can predict hao long our engineered bacteria would take to remove the biofilm and rust.

4.Materials and Methods

4.1 HSL Transfer

HSL is produced by iron bacterias and realeased into the water environment. So the first step of our sensing is HSL transfering into our engineered E.coli from the water. And a passive transusion model is used for this process that the transfer rate of HSL can be described as this:

• K_HSL,W-C : transfer coefficient through the membrane (s−1)

• We can predict hao long our engineered bacteria would take to remove the biofilm and rust.

4.2 AfeR-HSL Complexation

AfeR is produced by engineered E.coli and functions in cell and its concentration is obtained approximating the number of protein per cell, using the E.coli concentration (cell/L) and the Avogadro number.

The AfeR-HSL complexation is simply formed that way:

Assuming kinetics of AfeR-HSL complexation complexation is fast compared to the rest of the system, we assumed that the free and complexed forms are at equilibrum.

• K _{eq, AfeR-HSL} : equilibrum constant of the AfeR-HSL complexation (mol/L)

4.3 DspB Production

The production of the DspB from the DspB gene includes transcription and translation after activation. In addition, we should also consider its transport and degradation.

4.3.1 DspB Gene Activation

This process is modeled using a Michaelian formalism depending on its activator (AfeR-HSL complexation) concentration. The promoter strength is also taken into account.

• DspB _DNA,0/cell : total number of DspB DNA per cell

• DspB _DNA/cell : number of activated DspB DNA per cell

• K _{a, AfeR-HSL} : activation constant of the AfeR-HSL complexation (mol/L)

• k _{p, afeR} : afeR promoter influence

4.3.2 DspB Transcription

The DspB transcription depends on the transcription rate of the strain and the length of the DspB gene. The Avogadro number is used to express the transcription velocity in molar concentration in one cell per time unit.

• k_transcript : E.coli transcription rate (nucleotides/s)

• RNA polymerase/gene: number of RNA polymerase per gene

• DNA length (DspB): number of nucleotides on the DspB gene

• V _intracell: volume of a bacterial cell (L)

For the convenience of mathematical operation, we merged the ktranscript、RNA polymerase/gene and "V" intracell to a constant.

4.3.3 DspB Translation

The DspB translation depends on the translation rate of the strain, the mRNA length and the quantity of mRNA. The translation velocity is expressed in molar concentration in one cell per time unit.

• k_translation : E.coli translation rate (nucleotides/s)

• Ribosomes/RNA: number of ribosomes per mRNA

• RNA length (DspB): number of nucleotides on the DspB mRNA

• [DspB mRNA] : DspB mRNA concentration in one E.coli cell

For the convenience of mathematical operation, we merge the ktranslation and Ribosomes/RNA and to a constant.

4.3.4 Degradation

Some of the DspB protein and mRNA are degraded. A degradation constant is used to model the degradation velocity.

• K_deg,DspB: DspB degradation constant (s−1)

• K_{deg,DspB mRNA}: DspB mRNA degradation constant (s−1)

4.3.5 DspB Transfer

DspB protein needs to be transferred to the water environment to function. This process is taken into account through a passive transusion model.

• K_DspB,C-W : transfer coefficient through the membrane (s−1)

4.4 Biofilm Removel

The biofilm is removed by the DspB and the process is modeled assuming a Michaelis-Menten kinetics.

• kcat,DspB : catalytic constant of the DspB enzyme (s −1)

• "KM,D" : Michaelis constant of the DspB enzyme (mol/L)

4.5 EntE Production

We treat enterobactin enzymes gene cluster as a whole gene (EntE gene). The production of the enterobactin enzymes from the EntE gene includes transcription and translation after activation. In addition, we should also consider its degradation. Because the enterobactin enzymes function in the cell, we don't need to consider its transport to the water environment.

4.5.1 EntE Gene Activation

This process is modeled using a Michaelian formalism depending on its activator (AfeR-HSL complexation) concentration. The promoter strength is also taken into account.

• EntE DNA,0/cell : total number of EntE DNA per cell

• EntE DNA/cell : number of activated EntE DNA per cell

• K a, AfeR-HSL : activation constant of the AfeR-HSL complexation (mol/L)

• k p, afeR : afeR promoter influence

4.5.2 EntE Transcription

The EntE transcription depends on the transcription rate of the strain and the length of the EntE gene. The Avogadro number is used to express the transcription velocity in molar concentration in one cell per time unit.

• "EntE DNA,/cell" : number of EntE gene per cell

• ktranscript : E.coli transcription rate (nucleotides/s)