Team:NCTU Formosa/Prediction Model
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Introduction
The Prediction Model simulates and predicts the results of DenTeeth. First, we simulate the growth curve of E.coli and P.gingivalis in dogs’ oral environments. Then, we predict the production of peptide LL37, protein BMP2 and STATH. Next, we quantify the sterilization effect of LL37 and the repair of BMP2 and STATH . In this way, we can predict the effect of DenTeeth.
E.coli Simulation
In order to complete these simulations, we first construct logical ODEs (Ordinary Differential Equations) to describe the growth curves of E.coli at 40℃ and pH value equal to 8. While this is close to the environment in dogs’ oral cavities.
Assumption:
- The nutrition of growth is sufficient to maintain a steady nutrition uptake rate.
- The cultivation environment is finite, and there is a stationary phase for the growth of E. coli.
- The bacteria mutation does not affect the growth curve.
Under these assumptions we can use the logistic function to describe the growth of bacteria(Eq.1) [1].
$$\frac{d[E.coli]}{dt}= g_{E.coli}[E.coli](1-\frac{[E.coli]}{E.coli_{Max}})$$
| Parameters | Description | Values | Units |
|---|---|---|---|
| gE.coli | growth rate of E.coli [2] | 0.0417 | min-1 |
| E.coliMax | Maximum E.coli concentration [2] | 1.5 | O.D. |
The logistic differential equation assumed the dynamic equilibrium of bacteria in the end. In order to visualize our derivation ODEs, we simulate 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 grows under the sterilization of our dental bones, we use logical ODEs again to stimulate the growth curves of P.gingivalis. [2]All the situations are the same as E.coli. Thus, the final ODE system(Eq.2) and its parameters (Tab2) of P.gingivalis can be seen below:
$$\frac{d[P]}{dt}= g_{P}[P](1-\frac{[P]}{P_{Max}})$$
| Parameters | Description | Values | Units |
|---|---|---|---|
gP
| growth rate of P.gingivalis [3]
| 0.0025
| min-1 |
|
| PMax | Maximum P.gingivalis concentration [3] | 0.7 | O.D. |
Sterilization System of LL37
To know how the bacteria in dogs’ oral cavities grow under the effect of our dental bones, we need to calculate the sterilization amount of LL37.
LL37 kill growing bacteria with a rate kk, and afterwards each dead cell quickly takes up N [LL37]. These [LL37] are 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 LL37 (1) and the time evolution of concentrations of available [LL37] (2) is 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 [4] | 0.04 | 1/μM·min |
| N | LL-37 absorbed per dead cell [4] | 0.35 | μM/O.D |
Bacteria Growth Simulation with DenTeeth
Consider the previous growth model plus the killing formula of LL37. We can write down the growth model of E.coli and P.gingivalis under the sterilization action of DenTeeth(Eq.3):
$$\frac{d[E.coli]}{dt}= g_{E.coli}(1-\frac{[E.coli]}{[E.coli_{Max}]})-k_{k}[B][LL-37]$$
$$\frac{d[P]}{dt}= g_{P}[P](1-\frac{[P]}{P_{Max}})-N⋅k_{k} [B][LL-37] $$
As we can see above, the concentration of P.gingivalis and E.coli are reduced. And finally they will achieve dynamic balance.
Quorum Sensing System
In order to make the functions of sterilization 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).[5]
Through predicting the Quorum Sensing system of DenTeeth, we can 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 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 correlation with fluorescence intensity.
Assumption:
- The processes obey the law of mass action.
- Mean cell volume is a constant.
- 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 derive and get the differential equation of AHL-LuxR dimer below:
$$\frac{d[A-R_{2}]}{dt}=-D_{(A-R)_{2}}[(A-R)_{2}]+k_{(A-R)_{2}}[A-R]^2-k'_{(A-R)_{2}}[(A-R)_{2}]-k_{Plux-(A-R)_{2}}[A-R][Plux]+k'_{Plux-(A-R)_{2}}[Plux-(A-R)_{2}]$$
Then, we write down the differential equation of Plux-(A-R)2 complex：
$$\frac{d[Plux-(A-R)_{2}]}{dt}=+k_{Plux-(A-R)_{2}}[A-R][Plux]-k'_{Plux-(A-R)_{2}}[Plux-(A-R)_{2}]$$
And the parameters we use can seen below (Tab.4) [5]：
| 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 | - |
LL37 tetR RFP Production Simulation
Because E.coli itself will also be affected by LL37, in order to test whether this will further affect the concentration of the target product, we then use the analysis above to predict the concentration of these products over time. The total amount of AHL is composed of the initial AHL from the quorum sensing model. The AHL-LuxR complex will activate the Plux promoter , which can lead to the production of LL37, tetR and mRFP. The prediction formula of LL37 tetR RFP are shown below(Eq.4) [6]:
$$\frac{d[mLL37]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mLL37}[mLL37]$$
$$\frac{d[mtetR]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mtetR}[mtetR]$$
$$\frac{d[mRFP]}{dt}= K_{mLuxI}·β·[(A-R)_{2}]-deg_{mRFP}[mRFP]$$
$$\frac{d[LL37]}{dt}= k_{LL37}·[mLL37]-deg_{LL37}[LL37]$$
$$\frac{d[tetR]}{dt}= k_{tetR}·[tetR]-deg_{tetR}[tetR]$$
$$\frac{d[RFP]}{dt}= k_{RFP}·[RFP]-deg_{RFP}[RFP]$$
$$β=\frac{k_{a}+α[LuxR-AHL_{in}]_{2}}{k_{a}+[LuxR-AHL_{in}]_{2}}$$
And the parameters (Tab.5) can be seen below [6]:
| 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 | - |
| kLL37 | translation rate of mLL37 | 6.52 | min-1 |
| ktetR | translation rate of mtetR | 0.14 | min-1 |
| kRFP | translation rate of mRFP | 0.54 | min-1 |
| dmLL37 | degradation rate of mLL37 | 0.24 | min-1 |
| dmtetR | degradation rate of mtetR | 0.35 | min-1 |
| dmRFP | degradation rate of mRFP | 0.258 | min-1 |
| dLL37 | degradation rate of LL37 | 0.011 | min-1 |
| dtetR | degradation rate of tetR | 0.1386 | min-1 |
| dRFP | degradation rate of RFP | 0.498 | min-1 |
BMP2 STATH GFP Production Simulation
When the concentration of bacteria is low, DenTeeth will start to produce BMP2, STATH and GFP. Thus, we want to predict the production of these proteins. Considering the Quorum Sensing Model, we can write down the formula(Eq.5) [7]:
$$\frac{d[BMP2]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{BMP2} ·[BMP2])$$
$$\frac{d[STATH]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{STATH} ·[STATH])$$
$$\frac{d[GFP]}{dt}= C_{ptet} ·({l_{ptet}+\frac{1-l_{ptet}}{1+(\frac{[tet]}{k_{tet}})^{n_{tet}} } })-(d_{GFP} ·[GFP])$$
And the parameters (Tab5) can be seen below [7]:
| 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 |
In order to observe the switching between sterilization and restoration of our engineered bacteria, we add RFP after the sterilization sequence and GFP after the restoration sequence. Next, we simulate the relative fluorescence intensity of RFP and GFP to know the actual operation of our engineering E.coli. The result is shown in the figure below. (Fig.7):
Reference
- https://2020.igem.org/Team:NCTU_Formosa/Model
- Kim C, Wilkins K, Bowers M, Wynn C and Ndegwa E., et al. (2018). "Influence of Ph and Temperature on Growth Characteristics of Leading Foodborne Pathogens in a Laboratory Medium and Select Food Beverages.” Austin Food Sci. 2018; 3(1): 1031
- Kriebel K, Biedermann A, Kreikemeyer B, Lang H , et al(2013). “Anaerobic Co-Culture of Mesenchymal Stem Cells and Anaerobic Pathogens - A New In Vitro Model System.” PLoS ONE 8(11): e78226. doi:10.1371/journal.pone.0078226
- Snoussi, M., Talledo, J. P., Del Rosario, N. A., Mohammadi, S., Ha, B. Y., Košmrlj, A., & Taheri-Araghi, S., et al (2018). Heterogeneous absorption of antimicrobial peptide LL37 in Escherichia coli cells enhances population survivability. eLife, 7, e38174.
- https://2019.igem.org/Team:NCTU_Formosa/QS_Model
- https://2019.igem.org/Team:HZAU-China/Model
- https://2013.igem.org/Team:TU-Delft/Timer_Plus_Sumo
