P.gingivalis Simulation

  Next, in order to know how P.gingivalis grew under the sterilization of our dental bones, we used logical ODEs again to stimulate the growth curves of P.gingivalis. _[2]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:

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

Sterilization System of LL37

  To know how the bacteria in dogs’ oral cavities grow under the effect of our dental bones, we needed to calculate the sterilization amount of LL37.
  LL37 killed growing bacteria with a rate k_k, and afterwards each dead cell quickly took up N [LL37]. These [LL37] 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 LL37 (1) and the time evolution of concentrations of available [LL37] (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:

Bacteria Growth Simulation with DenTeeth

  Considered the previous growth model plus the killing formula of LL37. We could 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 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 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：

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

LL37 tetR RFP Production Simulation

  Because E. coli itself would also be affected by LL37, 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 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]:

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) _[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 (Tab.5) can be seen below _[7]:

Model Validation

  In order to ensure that our model’s predictions match the real situation, we used experimental data to fitting the model. 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.