Team:NCTU Formosa/QS Model

Measuring OD value consistently to get the bacterial growth curve requires the high sensitivity of the machine. However, we want to simplify the cost and size of our device to improve the convenience. Therefore, we design a quorum sensing system for it. Our E. coli with quorum sensing system can express abundant RFP when the bacteria reach a specific concentration. This way, once our device gets the time of red fluorescence, we can get the mutagenicity of the analyte.

Quorum Sensing Mechanism

As we can see from the mechanism figure below, our quorum sensing system lets our bacteria express abundant RFP when it reaches a specific concentration. Want to know the detail? Click the link below.

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Future Prospect

Figure 1: The mechanism of quorum sensing system

The Scheme of QS Model

Figure 2: The scheme of QS model

The QS 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:

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:

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.

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 reaction. In our model, we use the symbol "D" as the reaction rate constant.

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.

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:

$$Fick's\, First\, Law: J = -Duf_{1}\frac{d[S]}{dx}$$

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

$$J\cdot A = -Duf_{1}\cdot A\frac{d[S]}{dx}=-Duf_{2}\frac{d[S]}{dx} $$

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 Thickness_membrane is the thickness of cell membrane.

$$\frac{dS_{in}}{dt}= -Duf_{2}\frac{d[S]}{dx}=-Duf_{2}\frac{[S_{in}]-[S_{ex}]}{Thickness_{membrane}}=-Duf_{3}([S_{in}]-[S_{ex}])$$

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

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

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

We use the symbol “[S_in]” to represent the concentration inside the cell, and the symbol “[S_ex]” 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.

Figure 3: The demonstration of AHL releasing after cell lysis.

The following differential equations show the AHL concentration internal and external bacteria.

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

$$\frac{d[AHL_{ex}]}{dt} = +[{Duf_{AHL_{in}}}[AHL_{in}]-{Duf_{AHL_{in}}}[AHL_{ex}]\cdot B_{T}\cdot \frac{Vcell}{Vout}+T_{toxin}\cdot [toxin]\cdot B_{N}\cdot [AHL_{in}]\cdot \frac{Vcell}{Vout}$$

We simulate the AHL concentration internal and external bacteria.

Figure 4: 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:

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

$$\frac{d[AHL_{ex}]}{dt} = +[{Duf_{AHL_{in}}}[AHL_{in}]-{Duf_{AHL_{in}}}[AHL_{ex}]\cdot B_{T}\cdot \frac{Vcell}{Vout}+T_{toxin}\cdot [toxin]\cdot B_{N}\cdot [AHL_{in}]\cdot \frac{Vcell}{Vout}$$

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

Here, C_{mRNA_LuxI} is the reaction rate constant of mRNA_LuxI generation.

D_{mRNA_LuxI} is the reaction rate constant of mRNA_LuxI degradation.

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

In the formula, C_LuxI is the reaction rate constant of LuxI generation .

D_LuxI is the reaction rate constant of LuxI degradation .

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

C_{mRNA_LuxR} is the reaction rate constant of mRNA_LuxR generation.

D_{mRNA_LuxR} is the reaction rate constant of mRNA_LuxR degradation.

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

C_LuxR is the reaction rate constant of LuxR generation.

D_LuxR is the reaction rate constant of LuxR degradation.

The following figure shows the simulation of LuxI and LuxR generation.

Figure 5: 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:

$$AHL+LuxR\rightleftharpoons A-R$$

where k_A-R is the rate constant of forward reaction while k’_A-R is the rate constant of reverse reaction.

$$2 (A-R)\rightleftharpoons(A-R)_{2}$$

where k_(A-R)₂ is the rate constant of forward reaction while k’_(A-R)₂ is the rate constant of reverse reaction.

$$A-R \rightarrow \phi$$

D_A-R is the rate constant about AHL-LuxR complex

We derive and get the differential equation of AHL-LuxR complex below:

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

The binding reaction of AHL-LuxR dimer and Plux

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

$$2 (A-R)\rightleftharpoons(A-R)_{2}$$

where k_(A-R)₂ is the rate constant of forward reaction while k’_(A-R)₂ is the rate constant of reverse reaction.

$$(A-R)_{2}+Plux\rightleftharpoons Plux-(A-R)_{2}$$

where k_{Plux-(A-R)₂}is the rate constant of forward reaction while k’_{Plux-(A-R)₂} is the rate constant of reverse reaction.

$${(A-R)}_2\rightarrow \phi$$

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-(AHL-LuxR)2 complex is decided by a reversible reaction.

$$(A-R)_{2}+Plux\rightleftharpoons Plux-(A-R)_{2}$$

where k_{Plux-(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:

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

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

Figure 6: The simulation of AHL, LuxR and Plux reaction

$$\frac{d[mRNA_{RFP}]}{dt} = C_{mRNA_{RFP}}[Plux-(A-R)_{2}]-D_{mRNA_{RFP}}[mRNA_{RFP}]$$

Here, C_{mRNA_RFP} is the reaction rate constant of mRNA_RFP generation.

D_{mRNA_RFP} is the reaction rate constant of mRNA_RFP degradation.

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

Here, C_RFP is the reaction rate constant of RFP generation.D_RFP is the reaction rate constant of RFP degradation.

Finally, we get the intensity equals to the product of [RFP] and B_T .

$$Intensity = [RFP] \cdot B_{T}$$

We combine our quorum sensing model and computational growth model to simulate the red fluorescence intensity with different mutation rate M₂.

Figure 7: The simulation of different mutation rate to RFP intensity

As we can observe above, the higher the mutation rate M₂, the earlier the red fluorescence light up.

Table 1: The following table shows the parameters we used in the simulation.

g	0.07146
B_Max	320
IPTG	2.5E-7
T_IPTG	2.110648
C_t	0.000108
D_t	0.004806
T_t	7.2
T_chem	0.0
M ₁	3.0E-8
C_{mRNA_LuxI}	0.1
D_{mRNA_LuxI}	0.03
C_LuxI	0.5
D_LuxI	0.05
C_{mRNA_LuxR}	0.3
D_{mRNA_LuxR}	0.03
C_LuxR	0.5
D_LuxR	0.05
C_{AHL_in}	0.005
D_LuxR	0.05
C_{AHL_in}	0.65
D_{AHL_in}	0.05
Duf_{AHL_in}	0.005
V_cell / V_out	0.00001
k_A-R	0.005
k’_A-R	0.05
D_A-R	0.2
K_(A-R)₂	0.003
k’_(A-R)₂	0.03
D_(A-R)₂	0.02
K_{Plux_(A-R)2}	0.05
k’_{Plux_(A-R)2}	0.0062
C_mRNARFP	0.03
D_mRNARFP	0.8
C_RFP	0.03
D_RFP	2.40228

Data Fitting

Figure 8: The RFP intensity with respect to time

As we can observe from the chart, the experiment data of RFP is corresponded to our simulation, and it proves our concept is right.

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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Mutagenicity Prediction