Modelling Towards Precise Prediction
Mathematical modeling acts as engineering part in Synthetic Biology to link theoretical reaction mechanisms and lab work result. Our goal in modeling is to predict system behavior and give insight to the wet lab team.
There are three main aspects of our modeling:
1) quorum sensing time to predict when biofilm formed
2) the rate of PETase production
3) PET hydrolysis by PETase with and without biofilm.
These aspects modeled, compared and fitted by the experimental data, thus giving numerical trends from the aspects before wet lab team does labwork in the lab.All models have data that needed each other. The rate of bacteria growth affects the amount of biofilm produced. According to our models, the rate of biofilm growth heavily depends on μ (specific growth rate) and the initial amount of inoculated bacteria. Bacteria produce mRNA, which influences PETase production until it reaches steady state. This steady state value of PETase production will be set as the initial amount of PETase in calculating the rate of PET degradation.
Quorum Sensing
Quorum sensing mechanism was used to form biofilm of E.coli strain Top10, BL21 and DH5α that used in labwork. We have modeled growth curve of E.coli to determine when we must move E.coli colony from inoculum flask to reaction flask that contains PET. Besides growth curve, we have modeled some coupled ODEs to model growth curve, AI-2 production that affects signaling, and biofilm formation. AI-2 production in E.coli was used as colony signal of quorum sensing until it reaches specific points and finally form biofilm that affected by its quorum sensing by AI-2 signaling. We use Hill kinetics function as our approach to model AI-2 production and biofilm formation. Based on model that we built and confirmation from wetlab team, we found inoculation time until E.coli reaches quorum sensing condition is 10 hours. Not only time that necessary for quorum sensing condition, we found from model, parameter that affect significantly to biofilm formation was specific growth rate (μ) and initial amount of bacteria that will be inoculated that insightful to wetlab team when construct their parts.
Assumption that we used in quorum sensing module is AI-2 production constant equals to AI-2 signaling constant.
Here ODEs that we used :
Growth curve :
AI-2 Production :
Biofilm Formation :
Whereas X is bacterial growth (OD), AI2 is signaling production () and B is biofilm. Parameters that we used are shown in Table 1.
Table 1. Parameters for quorum sensing modelling module
| Parameter |
Definition |
Value |
Dimension |
References |
| μ |
Specific growth rate |
0.42 |
h-1 |
This study |
| Xmax |
Maximum carrying capacity |
0.76 |
OD600 |
This study |
| cA |
Signaling constant |
2.5 x 10-3 |
h-1 |
This study |
| μ |
Specific growth rate |
0.42 |
h-1 |
This study |
| kQ |
Monod constant |
0.42 |
h-1 |
This study |
| AI2max |
Specific growth rate |
0.42 |
h-1 |
This study |
| cS |
Specific growth rate |
0.42 |
h-1 |
This study |
| kB |
Biofilm growth constant |
0.42 |
h-1 |
This study |
| Bmax |
Biofilm carrying capacity |
0.42 |
h-1 |
This study |
PETase Transcription
After we have inoculated bacteria until biofilm was formed, process that we focused is PETase production in bacterium body, or usually called transcription.
The illustration of transcription of PETase is given below.
We define M(t) and C(t) as functions versus time (in further discussions will be just written as M and C). Before going to differential equations that illustrate rate of mRNA (symbolized as M) and PETase production (symbolized by C), we made several assumptions for the model:
1. No inclusion body is produced during the transcription. Consecutively, there’s also no TetR produced during the transcription.
2. Initally, there are 0.05 μM of mRNA and zero amount of PETase.
There, the differential equations of each parameter obtained through the analysis of mass balance are :
We choose α
m = 0.011 μmin
-1, γ
m= 0.009 μ
-1, γ
C= 0.04 μmin
-1.
Each of the differential equation is solved analytically using the MATLAB. Through the graph, we can see that at the steady state, the rate of production of PETase is 1,220 mg/(liter∙h) which attained at 600/60 = 10 h. Datum of PETase production will be used as the initial value in PET degradation.
Rate of PET Degradation with Biofilm
Based on PETase production model, we use value of PETase production to be an initial value to degrade PET, based on the system we have designed.
Based on the design, assumptions that we used are :
1. Biofilm covered E. coli from the effect of nutrient solution, however, the bottom section of E. coli is contacted with PET.
2.
Enzymatic reaction of PETase is assumed to obey two mechanisms reaction, i.e. Langmuir adsorption isotherm that applied in hydrolysis reaction that using Michaelis-Menten kinetics. One of the main reason for not applying all of mechanism Michaelis Menten kinetics in PET degradation mechanism was hydrolysis involves heterogeneous reaction []. Based on Langmuir adsorption isotherm, we can derive mathematical expression that implemented to Michaelis Menten kinetics.
Langmuir adsorption isotherm equation is :
Whereas q is quality of PET enzyme adsorption by unit quality PET, g; qm is the maximum adsorption of PET enzyme by unit quality PET, g; Ka is the adsorption dissociation constant, mL/g; Ef is the concentration of free PET enzyme in the solution, g/mL.
Corellation of q and qm,
So equation 1 can be rewritten as :
Based on assumptions that used in [], we get :
K is a constant connecting the three characters and A represents the area of the PET film. Langmuir adsorption equation linked with the second step of hydrolysis reaction process, here, the equation above is the key to connect the two step, and the value of the PET•S will be used in the hydrolysis reaction.
Reaction mechanisms of PET degradation are stated below.
We can derive differential equations that we need from reaction mechanisms. Here is coupled ODEs that we used to determine rate of PETase formation and degradation of PET with biofilm forming based on assumptions that stated above.
Whereas PET defined as PET, E as PETase, and P is ethylene terephtalate, the product from PET degradation by PETase.
Rate of PET Degradation without Biofilm
Comparing to degradation rate of PET with biofilm, PETase that can break down PET must be diffused into nutrient broth so surface contacting is occured, based on our design. So our hypothesis is degradation of PET without biofilm slower than with biofilm.
