Team:NCTU Formosa/Growth model


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  • Introduction
  • E.coli Simulation
  • P.gingivalis Simulation
  • Sterilization System of LL37
  • Bacteria Growth Simulation with DenTeeth
  • LL37 tetR mRFP Production Simulation
  • BMP2 STATH GFP Production Simulation

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:

  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 can 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 [2] 0.0417 1/min
E.coliMax Maximum E.coli concentration [2] 1.5 O.D.
[E.coli] the concentration of E.coli - 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 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. 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:

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

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

Parameters Description Values Units
gP growth rate of P.gingivalis [3] 0.0025 1/min
PMax Maximum P.gingivalis concentration [3] 0.7 O.D.
[P] the concentration of P.gingivalis - O.D.
Table 2. Parameters of the growth of P.gingivalis
 growth curve of E.coli and P.gingivalis
Figure 1. The growth curve of E.coli and P.gingivalis

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
Table 3. Parameters of the sterilization system of LL37

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):

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

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

Equation 3. E.coli and P.gingivalis growth with DenTeeth
 growth curve of E.coli and P.gingivalis
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 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).

 growth curve of E.coli and P.gingivalis
Figure 3. Quorum Sensing System

  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:

  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 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:

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

 growth curve of E.coli and P.gingivalis
Figure 4. The simulation of AHL, LuxR and Plux reaction
Parameters Description Values Units
CLuxI generation rate of LuxI 0.5 O.D./min
CLuxR generation rate of LuxR 0.5 O.D./min
DLuxI degradation rate of LuxI 0.05 O.D./min
DLuxR degradation rate of LuxR 0.05 O.D./min
D(A-R)2 rate constant about AHL-LuxR complex 0.2 O.D./min
k(A-R)2 rate constant of forward reaction 0.003 O.D./min
k'(A-R)2 rate constant of reverse reaction 0.03 O.D./min
kPlux-(A-R)2 rate constant of forward reaction 0.05 O.D./min
k'Plux-(A-R)2 rate constant of reverse reaction 0.0062 O.D./min
Table 4. Parameters of Quorum Sensing System

LL37 tetR mRFP 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 analyze the concentration of these products over time. While DenTeeth will produce different proteins with different amounts of bacteria, we should match the Quorum Sensing Model.

The product prediction formula of LL37 tetR mRFP are shown below(Eq.4):

$$none $$

Equation 4. LL37, tetR and mRFP production simulation formula

And the parameters (Tab4) can be seen below:

Parameters Description Values Units
$$k_{k}$$ killing rate 0.04 1/μM·min
N LL-37 absorbed per dead cell 0.35 μM/O.D
N LL-37 absorbed per dead cell 0.35 μM/O.D
N LL-37 absorbed per dead cell 0.35 μM/O.D
N LL-37 absorbed per dead cell 0.35 μM/O.D
N LL-37 absorbed per dead cell 0.35 μM/O.D
Table 5. Parameters of LL37 tetR mRFP production simulation
 growth curve of E.coli and P.gingivalis
Figure 5. The concentration simulation of LL37, tetR and mRFP

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):

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

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

Equation 5. BMP2, STATH and GFP production simulation formula

And the parameters (Tab5) can be seen below:

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

Reference

  1. Schink, S. J., et al. (2019). "Death rate of E. coli during starvation is set by maintenance cost and biomass recycling." 9(1): 64-73. e63.
  2. 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
  3. 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
  4. 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://doi.org/10.7554/eLife.38174
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