Team:NCTU Formosa/Quorum Sensing Model


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  • Introduction
  • The Scheme of Quorum Sensing Model
  • AHL Diffusion Model
  • Simulation of Quorum Sensing Model
  • The binding reaction of AHL and LuxR
  • The binding reaction of AHL-LuxR dimer and Plux

Introduction

  Since DenTeeth is a dental bone that can switch between the sterilization and repair functions according to the concentration of bacteria. We design a quorum sensing system for it. Our E. coli with quorum sensing system can express RFP when the bacteria reach a specific concentration. While our E. coli will express GFP when the bacteria drop under this specific concentration.
  With the Quorum Sensing Model,we can know the production of LL37 under different bacterial concentrations. Since LL37 can affect the growth of E.coli and P.gingivalis. This model can also help us complete the Growth Model.

The Scheme of Quorum Sensing Model

  The Quorum Sensing system involves much interaction of compounds in and out of the cell, thus, we use the following three assumptions for our model and use differential equations to describe the rate of change of each compound. With those assumptions, we can get the relationship of red fluorescence intensity.
Three assumptions are shown below:

Assumption:

  1. The processes obey the law of mass action.
  2. Mean cell volume is a constant.
  3. Cell volume is much smaller than the total volume.


  Furthermore, we generally consider the following four factors in each element's differential equation:
1.Generation: Each intermediate product is generated from reactants. We assume that generation reactions obey the law of mass action. In our model, the symbol "C" is the reaction rate constant.
2.Degradation: We consider chemical degradation. We assume that except for the stable DNA, each chemical compound will be degraded and the degradation reactions are all first-order reactions. In our model, we use the symbol "D" as the reaction rate constant.
3.Reaction: We assume that the reactions in our model are all elementary steps. The symbol "k" represents the reaction rate constant, and the symbol "k" represents the reverse reaction rate constant.
4.Diffusion: The unique part of the quorum sensing system is the diffusion of AHL. Therefore, we cite the Fick's First Law for it. However, the algebras in Fick's First Law are in the unit of amount, which is different from our model. Thus, we restate the formulas as following:

$$Fick’s First Law:J=-D_{u}f_{1}\frac{d[S]}{dx}$$

  We assume that the cell volume is constant, thus, the cell surface area is also constant.

$$J⋅A=-D_{u}f_{1}⋅A\frac{d[S]}{dx}=-D_{u}f_{2}\frac{d[S]}{dx}$$

  Due to the cell membrane is very thin, we assume that the concentration gradient between the outer membrane and inner membrane is the same. We define Thicknessmembrane is the thickness of cell membrane.

$$\frac{dS_{in}}{dt}=-D_{u}f_{2}\frac{[s_{in}]-[S_{ex}]}{Thickness_{membrane}}=-D_{u}f_{3}([S_{in}]-[S_{ex}])$$

  In the following step, we restate the amount into concentration.

$$\frac{d[S_{in}]}{dt}=\frac{dS_{in}}{dt⋅V_{cell}}=-\frac{Duf_{3}}{V_{cell}}([S_{in}]-[S_{ex}])=-Duf_{4}([S_{in}]-[S_{ex}])$$

$$\frac{d[S_{ex}]}{dt}=\frac{-d[S_{in}]}{dt}⋅\frac{V_{cell}}{V_{out}}=+[Duf_{4}\frac{V_{cell}}{V_{out}}([S_{in}]-[S_{ex}])]⋅B_{T}$$

  We use the symbol "[Sin]" to represent the concentration inside the cell, and the symbol "[Sex]" to represent the concentration outside the cell.

AHL Diffusion Model

  As a result of the effect of toxin protein, we assume that the AHL inside the bacteria will be released after cell lysis.The following differential equations show the AHL concentration of internal and external bacteria.

$$\frac{d[AHL_{ex}]}{dt}=C_{AHL_{in}}[LuxI]-D_{AHL_{in}}[AHL_{in}]-k_{A-R}[AHL_{in}]+k'_{A-R}[A-R]-Duf_{AHL_{in}}[AHL_{in}]+Duf_{AHL_{in}}[AHL_{ex}]$$

$$\frac{d[AHL_{ex}]}{dt}=+Duf_{AHL_{in}}[AHL_{in}]-Duf_{AHL_{in}}[AHL_{ex}]⋅B_{T}⋅\frac{V_{cell}]}{V_{out}}[AHL_{in}]$$

   We simulate the AHL concentration internal and external bacteria.

 growth curve of P.gingivalis
Figure 1. The simulation of AHL concentration inside and outside the bacteria.

Simulation of Quorum Sensing Model

  We sum up the equations in quorum sensing model as following:

$$\frac{d[AHL_{in}]}{dt}=C_{AHL_{in}}[LuxI]-D_{AHL_{in}}[AHL_{in}]-k_{A-R}[AHL_{in}]+k'_{A-R}[A-R]-Duf_{AHL_{in}}[AHL_{in}]+Duf_{AHL_{in}}[AHL_{ex}]$$

$$\frac{d[AHL_{ex}]}{dt}=+Duf_{AHL_{in}}[AHL_{in}]-Duf_{AHL_{in}}[AHL_{ex}]⋅B_{T}⋅\frac{V_{cell}]}{V_{out}}[AHL_{in}]$$

$$\frac{d[mRNA_{LuxI}]}{dt}=C_{mRNA_{LuxI}}-D_{mRNA_{LuxI}}[mRNA_{LuxI}]$$

   Here,CmRNALuxI is the reaction rate constant of mRNALuxI generation.
   DmRNALuxI is the reaction rate constant of mRNALuxI degradation.

$$\frac{d[LuxI]}{dt}=C_{LuxI}[mRNA_{LuxI}]-D_{LuxI}[LuxI]$$

   In the formula, CLuxI is the reaction rate constant of LuxI generation.
   DLuxI is the reaction rate constant of LuxI degradation.

$$\frac{d[mRNA_{LuxR}]}{dt}=C_{mRNA_{LuxR}}-D_{mRNA_{LuxR}}[mRNA_{LuxR}]$$

   CmRNALuxR is the reaction rate constant of mRNALuxR generation.
   DmRNALuxR is the reaction rate constant of mRNALuxR degradation.

$$\frac{d[LuxR]}{dt}=C_{LuxR}[mRNA_{LuxR}]-D_{LuxR}[LuxR]$$

   CLuxR is the reaction rate constant of LuxR generation.
   DLuxR is the reaction rate constant of LuxR degradation.
   The following figure shows the simulation of LuxR and LuxR generation.

 growth curve of P.gingivalis
Figure 2. The simulation of LuxI and LuxR Generation

The binding reaction of AHL and LuxR

  The change rate of AHL-LuxR complex is decided by the following two reversible reactions and degradation reaction:

$$AHL+LuxR⇌A-R$$

  Where kA-R is the rate constant of forward reaction while k'A-R is the rate constant of reverse reaction.

$$2(A-R)⇌(A-R)_{2}$$

  Where k(A-R)2 is the rate constant of forward reaction while k'(A-R)2 is the rate constant of reverse reaction.

$$A-R→ ϕ$$

  DA-R is the rate constant about AHL-LuxR complex.
  We derive and get the differential equation of AHL-LuxR complex below:

$$\frac{d[A-R]}{dt}=-D_{A-R}[A-R]+k_{A-R}[AHL_{in}][LuxR]-k'_{A-R}[A-R]-2⋅k_{(A-R)_{2}}[A-R]^2+2⋅k'_{(A-R)_{2}}[(A-R)_{2}]$$

The binding reaction of AHL-LuxR dimer and Plux

  The change of (A-R)2 complex is decided by two reversible reaction and degradation.

$$2(A-R)⇌(A-R)_{2}$$

  Where k(A-R)_{2} is the rate constant of forward reaction while k'_{(A-R)_{2}} is the rate constant of reverse reaction.

$$(A-R)_{2}+Plux⇌Plux-(A-R)_{2}$$

  Where kPlux-(A-R)2 is the rate constant of forward reaction while k'Plux-(A-R)2 is the rate constant of reverse reaction.

$$(A-R)_{2}→ϕ$$

  D(A-R)2 is the rate constant about AHL-LuxR complex.
  We derive and get the differential equation of AHL-LuxR dimer below:

$$\frac{d[A-R_{2}]}{dt}=-D_{(A-R)_{2}}[(A-R)_{2}]+k_{(A-R)_{2}}[A-R]^2-k'_{(A-R)_{2}}[(A-R)_{2}]-k_{Plux-(A-R)_{2}}[A-R][Plux]+k'_{Plux-(A-R)_{2}}[Plux-(A-R)_{2}]$$

  Plux-(A-R)2 complex is decided by a reversible reaction.

$$(A-R)_{2}+Plux⇌Plux-(A-R)_{2}$$

  Where kPlux-(A-R)2 is the rate constant of forward reaction while k'Plux-(A-R)2 is the rate constant of reverse reaction. We derive and get the differential equation of Plux-(A-R)2 complex below:

$$\frac{d[Plux-(A-R)_{2}]}{dt}=+k_{Plux-(A-R)_{2}}[A-R][Plux]-k'_{Plux-(A-R)_{2}}[Plux-(A-R)_{2}]$$

  The following figure shows the simulation of the binding reaction of AHL, LuxR and Plux.

 growth curve of P.gingivalis
Figure 3. The simulation of AHL, LuxR and Plux reaction

  The following table(Tab.1) shows the parameters we used in the simulation.

Parameters Values Units
CmRNALuxI 0.1 -
DmRNALuxI 0.03 -
CLuxI 0.5 -
DLuxI 0.05 -
CmRNALuxR 0.3 -
DmRNALuxR 0.03 -
CLuxR 0.5 -
DLuxR 0.05 -
CAHLin 0.005 -
DLuxR 0.05 -
CAHLin 0.65 -
DAHLin 0.05 -
DufAHLin 0.005 -
Vcell/Vout 0.00001 -
kA-R 0.005 -
k'A-R 0.05 -
DA-R 0.2 -
K(A-R)2 0.003 -
k'(A-R)2 0.03 -
D(A-R)2 0.02 -
KPlux_(A-R)2 0.05 -
k'Plux_(A-R)2 0.0062 -
Table 1. Parameters we used in the simulation.

Reference

  1. Boada, Y., et al. (2017). "Engineered Control of Genetic Variability Reveals Interplay among Quorum Sensing, Feedback Regulation, and Biochemical Noise." ACS Synth Biol 6(10): 1903-1912.
  2. Diggle, S. P., et al. (2007). "Quorum sensing." Curr Biol 17(21): R907-910.
  3. Hartmann, A. and A. Schikora (2012). "Quorum sensing of bacteria and trans-kingdom interactions of N-acyl homoserine lactones with eukaryotes." J Chem Ecol 38(6): 704-713.
  4. Hong, S. H., et al. (2012). "Synthetic quorum-sensing circuit to control consortial biofilm formation and dispersal in a microfluidic device." Nat Commun 3: 613.
  5. Wagner, V. E., et al. (2003). "Microarray analysis of Pseudomonas aeruginosa quorum-sensing regulons: effects of growth phase and environment." J Bacteriol 185(7): 2080-2095.

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