# Team:NCTU Formosa/Prediction Model   dark_mode Dark Mode
• Introduction
• E. coli Simulation
• P. gingivalis Simulation
• Inhibition System of LL-37
• Bacteria Growth Simulation with DenTeeth
• Quorum Sensing System
• LL-37 tetR RFP Production Simulation
• BMP2 STATH GFP Production Simulation
• Model Validation

# Introduction

The Prediction Model simulated and predicted the results of DenTeeth. First, we simulated the growth curve of E. coli and P. gingivalis in dogs’ oral environments. Then, we predicted the production of peptide LL-37, protein BMP2 and STATH. Next, we quantified the inhibition effect of LL-37 and predict the expression of BMP2 and STATH . In this way, we could predict the effect of DenTeeth.

# E. coli Simulation

In order to complete these simulations, we first constructed logical ODEs (Ordinary Differential Equations) to describe the growth curves of E. coli at 40℃ and pH value equal to 8. While this was close to the environment in dogs’ oral cavities.

Assumption:

1. The nutrition of growth is sufficient to maintain a steady nutrition uptake rate.
2. The cultivation environment is finite, and there is a stationary phase for the growth of E. coli.
3. The bacteria mutation does not affect the growth curve.

Under these assumptions we could use the logistic function to describe the growth of bacteria(Eq.1) .

$$\frac{d[E. coli]}{dt}= g_{E. coli}[E. coli](1-\frac{[E. coli]}{E. coli_{Max}})$$

Equation 1. Final ODE system of the growth of E. coli

Parameters Description Values Units
gE. coli growth rate of E. coli  0.0417 min-1
E. coliMax Maximum E. coli concentration  1.5 O.D.
Table 1. Parameters of the growth of E. coli

The logistic differential equation assumed the dynamic equilibrium of bacteria in the end. In order to visualize our derivation ODEs, we simulated the growth curve of E. coli. at 40℃ and pH value equal to 8.

# P. gingivalis Simulation

Next, in order to know how P. gingivalis grew under the inhibition of our dental bones, we used logical ODEs again to stimulate the growth curves of P. gingivalis All the situations were the same as E. coli. Thus, the final ODE system(Eq.2) and its parameters (Tab2) of P. gingivalis can be seen below:

$$\frac{d[P]}{dt}= g_{P}[P](1-\frac{[P]}{P_{Max}})$$

Equation 2. Final ODE system of the growth of P. gingivalis

Parameters Description Values Units
gP growth rate of P. gingivalis  0.0025 min-1
PMax Maximum P. gingivalis concentration  0.7 O.D.
Table 2. Parameters of the growth of P. gingivalis Figure 1. The growth curve of E. coli and P. gingivalis

# Inhibition System of LL-37

To know how the bacteria in dogs’ oral cavities grew under the effect of our dental bones, we needed to calculate the inhibition amount of LL-37.
LL-37 killed growing bacteria with a rate kk, and afterwards each dead cell quickly took up N [LL-37]. These [LL-37] were bound to the membrane as well as to the cytoplasm of the cell and are not recycled to attack other cells. The killing formula of LL-37 (1) and the time evolution of concentrations of available [LL-37] (2) was described by the following equations:

$$(1)\frac{d[B]}{dt}= −k_{k}⋅[B][LL-37]$$

$$(2)\frac{d[LL-37]}{dt}= −N⋅k_{k}[B][LL-37]$$

And the parameters (Tab3) of this system can be seen below:

Parameters Description Values Units
kk killing rate  0.04 1/μM·min
N LL-37 absorbed per dead cell  0.35 μM/O.D
Table 3. Parameters of the Inhibition System of LL-37

# Bacteria Growth Simulation with DenTeeth

Considered the previous growth model plus the killing formula of LL-37. We could write down the growth model of E. coli and P. gingivalis under the inhibition action of DenTeeth(Eq.3):

$$\frac{d[E. coli]}{dt}= g_{E. coli}(1-\frac{[E. coli]}{[E. coli_{Max}]})-k_{k}[B][LL-37]$$

$$\frac{d[P]}{dt}= g_{P}[P](1-\frac{[P]}{P_{Max}})-N⋅k_{k} [B][LL-37]$$

Equation 3. E. coli and P. gingivalis growth with DenTeeth Figure 2. The growth curve of E. coli and P. gingivalis with DenTeeth

As we can see above, the concentration of P. gingivalis and E. coli were reduced. And finally they achieved dynamic balance.

# Quorum Sensing System

In order to make the functions of inhibition and repair don’t interfere with each one, DenTeeth would produce different proteins with different amounts of bacteria in dogs’ oral cavities by using the Quorum Sensing system(QS system).

Through predicting the Quorum Sensing system of DenTeeth, we could predict which function is working. Owing to the red and green fluorescence sequence in the DenTeeth, the fluorescence intensity experiment would validate the prediction.
The QS system involved much interaction of compounds in and out of the cell. Thus, we used the following three assumptions for our model and used differential equations to describe the rate of change of each compound. With those assumptions, we could get the correlation with fluorescence intensity.

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.

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

$$AHL+LuxR⇌A-R$$

$$2(A-R)⇌(A-R)_{2}$$

Furthermore, considering the change of (A-R)2 complex decided by reversible reaction and degradation. We derived and got the differential equation of AHL-LuxR dimer below:

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

Then, we write down the differential equation of Plux-(A-R)2 complex：

$$\frac{d[Plux-(A-R)_{2}]}{dt}=+k_{Plux-(A-R)_{2}}[A-R][Plux]-k'_{Plux-(A-R)_{2}}[Plux-(A-R)_{2}]$$ Figure 3. The simulation of LuxI, LuxR, AR, and AR2 reaction

And the parameters we use can seen below (Tab.4) 

Parameters Description Values Units
CLuxI generation rate of LuxI 0.5 min-1
CLuxR generation rate of LuxR 0.5 min-1
DLuxI degradation rate of LuxI 0.05 min-1
DLuxR degradation rate of LuxR 0.05 min-1
D(A-R)2 rate constant about AHL-LuxR complex 0.2 -
k(A-R)2 rate constant of forward reaction 0.003 -
k'(A-R)2 rate constant of reverse reaction 0.03 -
kPlux-(A-R)2 rate constant of forward reaction 0.05 -
k'Plux-(A-R)2 rate constant of reverse reaction 0.0062 -
Table 4. Parameters of Quorum Sensing System

# LL-37 tetR RFP Production Simulation

Because E. coli itself would also be affected by LL-37, in order to test whether this will further affect the concentration of the target product, we then used the analysis above to predict the concentration of these products over time.
The total amount of AHL was composed of the initial AHL from the quorum sensing model. The AHL-LuxR complex would activate the Plux promoter , which could lead to the production of LL-37, tetR and mRFP.
The prediction formula of LL-37 tetR RFP are shown below(Eq.4) :

$$\frac{d[mLL-37]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mLL-37}[mLL-37]$$

$$\frac{d[mtetR]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mtetR}[mtetR]$$

$$\frac{d[mRFP]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mRFP}[mRFP]$$

$$\frac{d[LL-37]}{dt}= k_{LL-37}·[mLL-37]-deg_{LL-37}[LL-37]$$

$$\frac{d[tetR]}{dt}= k_{tetR}·[tetR]-deg_{tetR}[tetR]$$

$$\frac{d[RFP]}{dt}= k_{RFP}·[RFP]-deg_{RFP}[RFP]$$

$$β=\frac{k_{a}+α[LuxR-AHL_{in}]_{2}}{k_{a}+[LuxR-AHL_{in}]_{2}}$$

Equation 4. LL-37, tetR and mRFP production simulation formula

And the parameters (Tab.5) can be seen below :

Parameters Description Values Units
KmLuxI Plasmid copy number times LuxI transcription rate 23.3230 nM*min-1
ka Dissociation rate of LuxR-AHLin2 200 nM
α Basal expression of LuxI 0.01 -
kLL-37 translation rate of mLL-37 6.52 min-1
ktetR translation rate of mtetR 0.14 min-1
kRFP translation rate of mRFP 0.54 min-1
dmLL-37 degradation rate of mLL-37 0.24 min-1
dmtetR degradation rate of mtetR 0.35 min-1
dmRFP degradation rate of mRFP 0.258 min-1
dLL-37 degradation rate of LL-37 0.011 min-1
dtetR degradation rate of tetR 0.1386 min-1
dRFP degradation rate of RFP 0.498 min-1
Table 5. Parameters of LL-37 tetR RFP production simulation Figure 4. The simulation of LL-37 and tetR

# BMP2 STATH GFP Production Simulation

When the concentration of bacteria was low, DenTeeth would start to produce BMP2, STATH and GFP. Thus, we wanted to predict the production of these proteins. Considering the Quorum Sensing Model, we could write down the formula(Eq.5) :

$$\frac{d[BMP2]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{BMP2} ·[BMP2])$$

$$\frac{d[STATH]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{STATH} ·[STATH])$$

$$\frac{d[GFP]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{GFP} ·[GFP])$$

Equation 5. BMP2, STATH and GFP production simulation formula

And the parameters (Tab.5) can be seen below :

Parameters Description Values Units
Ctet maximum transcription rate of ptet 2.79 min-1
Iptet leakage factor of ptet 0.002 -
ktet dissociation constant of ptet 6 -
ntet hills coefficient 3 -
dBMP2 degradation rate of BMP2 0.05 min-1
dSTATH degradation rate of STATH 0.0000248 min-1
dGFP degradation rate of GFP 0.347 min-1
Table 6. Parameters of BMP2, STATH and GFP production simulation Figure 5. The simulation of BMP2 and STATH

In order to observe the switching between inhibition and restoration of DenTeeth, we added RFP after the inhibition sequence and GFP after the restoration sequence. Next, we simulated the relative fluorescence intensity of RFP and GFP to know the actual operation of DenTeeth. The result is shown in the figure below. (Fig.6): Figure 6. The relative intensity of GFP and RFP

# Model Validation

In order to ensure that our model’s predictions match the real situation, we did the experiment to verify the model. We used DenTeeth as the experimental group and E. coli without engineered as the control group. Then, incubated them at 37 degrees Celsius and measured the O.D. value, GFP and RFP expression an hour at a time in the time of 24 hours.
After the experiment, we found that it was necessary to consider the dead E. coli because it influenced the O.D. value. The following picture(Fig.7) is the adjusted growth curve of E. coli. Figure 7. The fitting result of the growth of E. coli

As you can see, the red line is the prediction growth curve of dead E. coli [deadE. coli(prediction)]. The blue line is the E. coli which is still alive [liveE. coli(prediction)]. And the black line is all the E. coli include living and dead, which is the prediction O.D. value [totalE. coli(predict)].

Next, we considered the expression of RFP. We subtracted the control group data from the experimental group and verified with the model. We found that the environment of the Erlenmeyer Flask was different from the paper. The degradation of RFP was lower than expected. Thus, we lowered the degradation rate and verified it with the experimental results again. The following picture(Fig.8) is the result. Figure 8. The fitting result of the intensity of RFP

Then, we fit the data of GFP. We also subtracted the control group data from the experimental group and verified with the model. We found that the expression of GFP exceeded expectations. So, we raised the translation rate of GFP and lowered the degradation rate. Although we did the same experiment for 24 hours, since the GFP expression had exceeded the detection range of the machine, the measured values were maintained at the maximum. Therefore, we only took the fata for the first 10 hours. The result is shown below. (Fig.9) Figure 9. The fitting result of the intensity of GFP

After the validation, we compared the expression of GFP and RFP. As you can see in the picture (Fig.10), because E. coli concentration was low at the beginning of the experiment, DenTeeth expressed GFP first. As E. coli continued to grow over time, it started to inhibit the expression of RFP. Then, the concentration of E. coli decreased due to the inhibition. DenTeeth turned to express GFP and started the restoration function. According to this experiment, we confirmed that the Quorum Sensing System of DenTeeth worked successfully. Figure 10. The fitting result of the intensity of GFP and RFP

After finishing the whole validation, we predicted the LL-37, BMP2, and STATH expression again (Fig.11~Fig.12). Figure 11. The prediction of LL-37 Figure 12. The prediction of BMP2 and STATH

And we also predicted the new growth curve of E. coli and P. gingivalis with DenTeeth. As you can see in the picture (Fig.13), compared to the growth of P. gingivalis without DenTeeth (Fig.1), the final O.D. value of P. gingivalis decreased from 0.7 to 0.08, which showed that our DenTeeth could effectively kill 88% of the pathogenic bacteria in dogs' oral cavities. Figure 13. The prediction of the growth of bacteria
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DenTeeth can kill up to 88% of the pathogenic bacteria in dogs' oral cavities.

# Reference

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