# Modelling

**Quorum Sensing
** / PETase Transcription / Rate of PET Degradation with Biofilm / Rate of PET Degradation without Biofilm

# Modelling Towards Precise Prediction of Synthetic Biology

**1) quorum sensing time to predict when biofilm formed**

**2) the rate of PETase production**

**3) PET hydrolysis by PETase with and without biofilm.**

# Quorum Sensing

*E.coli* strain Top10, BL21 and DH5α. We modeled the growth curve of *E.coli* to determine when *E.coli* colony should be moved from to reaction flask that contains PET. We also modeled coupled ODEs to 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 our model, also confirmed through experiment, the inoculation time until *E.coli* reaches quorum sensing condition is **5-6 hours**. We also found other parameter that **affect biofilm formation significantly. Namely, specific growth rate (μ) and initial amount of bacteria that will be inoculated. This model will give** insight **to wetlab team** when constructing 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 :

Parameter | Definition | Value | Dimension | References |
---|---|---|---|---|

μDH5alpha | Specific growth rate of DH5alpha | 0.3 | h^{-1} |
This study |

X_{max}(DH5alpha) |
Maximum carrying capacity DH5alpha | 2.504 | OD_{588} |
This study |

X_{max}(BL21) |
Maximum carrying capacity BL21 | 2.645 | OD_{588} |
This study |

c_{A} |
Signaling constant | 2.5 x 10^{-3} |
OD588 | This study |

μBL21 | Specific growth rate of BL21 | 0.32 | h^{-1} |
This study |

k_{Q} |
Monod constant for AI2 production | 2.16 x 10^{-3} |
OD588 | This study |

AI2_{max} |
Maximum AI2 production | 0.088 | h^{-1} |
This study |

c_{S} |
Biofilm growth constant | 6.6x10^{-2} |
h^{-1} |
This study |

k_{B} |
Monod constant biofilm | 12.6 | h^{-1} |
This study |

B_{max BL21} |
Biofilm carrying capacity BL21 | 0.5 | OD550 | This study |

B_{max DH5alpha} |
Biofilm carrying capacity DH5alpha | 0.34 | OD550 | This study |

Next, we attempted to verify our model using data directly acquired from our Wetlab team. Our wetlab team supplied data of cells and biofilm formation growth curve. Model and data for growth curve and biofilm formation for two different strain of *E. coli* can be simultaneously seen in Fig 1 and Fig 2. As expected, μ and initial amount of bacteria has important effect to biofilm formation. *E. coli* strain BL21, which **has greater μ than DH5alpha, also has higher rate of biofilm growth than DH5alpha. This information is very precious for the wetlab team to develop the right strain to be engineered.**

# PETase Transcription

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 :_{m}= 0.011 μmin

^{-1}, α

_{c}= 0.001 μmin

^{-1}, γ

_{m}= 0.009 μ

^{-1}, γ

_{c}= 0.04 μmin

^{-1}. Using MATLAB, we obtain the graph plot as shown below:

**0.0305 mg/(liter∙h) which attained at approximately 600/60 = 10 h**. We will use this as the initial value of PETase in PET degradation.

# Rate of PET Degradation with Biofilm

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 in contact with PET.

2. Surface of PET is smooth and assume uniform at each point.

Pathway of PET degradation in engineered*E. coli*from MetaCyc is shown in Fig. 5.

*E. coli*

_{m} is the maximum adsorption of PET enzyme by unit quality PET, g; K_{a} is the adsorption dissociation constant, mL/g; E_{f} is the concentration of free PET enzyme in the solution, g/mL.

Corellation of q and q_{m},

So equation (1) can be rewritten as :

Based on assumptions that used in [TJUSLS iGEM 2016 Team, 2016], we get :

Reaction mechanisms of PET degradation are stated below [TJUSLS iGEM 2016 Team, 2016].

(E stands for PETase)

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 C, T, and S consecutively denotes the amount of PET, PET∙E, and PETE produced, E as PETase, and P is ethylene terephtalate (the product from PET degradation by PETase), each against time.

Hence, we can substitute T from equation (3) into equation (5). Thus, we have

Now, let’s analyze the parameters of the preceding equation. It’s obvious that K, k_{3}, K_{a} are constant in the system, while, in a fixed experiment, the area of the PET sheet and the concentration of the PET enzyme are unchangeable according to hypothesizes above, so the right part of the equation above is a constant, B.

Reaction is occured until value of degradation rate of PET and PETase equals to zero, so equation (9) becomes :

Insert equation (10) to equation (5), thus we have

The above differential equation is the final rate equation of the whole reaction process, and from the equation (12) we obtained that the reaction rate is constant, determined from the parameter D. And the reaction rate constants differs at varying PET concentrations and PET films.

However, in real process, the reaction rate will be decreased as the PET decreased during the reaction. So, the damping factor It included in the equation to contemplate the effect of substrate reduction.

To simplify the equations, define the constant K_{0} below

Thus, we have

and the differential equations become

Using Maple18, we make a plot graph of ethylene terephtalate production to observe the reaction rate. We have the parameters value of the equation as follows:

Parameter | Definition | Units | Values | References |
---|---|---|---|---|

K_{0} |
A constant of the equation, K_{0} = Kk_{5}k_{3}/k_{4} |
mg/(mL.h.mm^{2}) |
1.43 | TJUSLS (2016) |

A | The area of the PET film | mm^{2} |
28.27 | TJUSLS (2016) |

K_{a} |
Adsorption constant of the PET enzyme | mL/mg | 7.89 x 10^{-2} |
TJUSLS (2016) |

I | The damping factor for the adsorption process of all reaction products | mg/(mL.h) | 2.15 x 10^{-4} |
TJUSLS (2016) |

E_{0} |
The initial PET enzyme concentration | mg/mL | 0.0305 | This study |

From the parameter values provided, a graph of ethylene terephtalate production has been plotted in Maple18 for the initial amount of ethylene terephtalate being zero as shown below:

Notice that the ethylene terephtalate production stops after approximately **18-20 hours** before starting to decrease. This means we can collect data for **PET degradation after 18-20 hours.** Data that we have can be expanded to design PET degradation bioreactor that we have design here

# 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. Molecular weight of PETase is 30,247 g/mol, that relatively larger than oxygen (16 g/mol) or albumin (5,200 g/mol). Larger molecular weight makes value of diffusivity coefficient smaller. After diffusion, enzyme must create contact to PET surface so PET degradation will occur. Modeling of PETase diffusion and *E. coli* motility should modeled as stochastic model like Brownian motion, and we lack of data that we need.
Biofilm that we used as media of *E. coli* to attach at PET surface based on our design should be evaluated as channel to PETase can flow because molecular weight relatively large. Large molecular weight also makes diffusion of PETase will occur in slow rate. So, there is possibility that biofilm can slower PET degradation.
But, constraint that we have explained above enable us to make some hypothesis. Our hypothesis are :

**1. PET degradation without biofilm is slower than PET degradation with biofilm, with assumption no biofilm at bottom section of surface that contact with PETase.**

**2. PET degradation without biofilm is faster than PET degradation with biofilm, because biofilm cover all contact surface of PET, and PETase can't use it as channel to degrade PET**

Results from our wetlab team has

**proven our hypothesis (2) is true**and this explain how

**mathematical model can be used as tool to assist wetlab team make decision and predict final result**of the experiment.

# Conclusions

To sum up, we made several conclusions regarding the modeling:

- We choose to use BL21 strain of Eschericia coli as the host.
- The maximum theoretical steady rate of PETase production is 0.0305 mg/(liter.h). The steady state attained in 10 h.
- PETase degradation occurs in 18-20 h.

# References

Klipp, Edda, Wolfram Liebermeister, Christoph Wierling, Axel Kowald,Hans Lehrach, and Ralf Herwig.
(2009): *Systems Biology*. Weinheim: WILEY-VCH Verlag GmbH & Co. KGaA.

MetaCyc Reaction: 3.1.1.101. Retrieved November 01, 2017, from https://biocyc.org/META/NEW-IMAGE?type=REACTION&object=RXN-17825

Rachmananda, Faisal (2015): *Models of PET Degradation and Conversion by E-Coli Bacteria*,
Bachelor’s Program Final Project, Institut Teknologi Bandung.

Shuler, Michael L., Fikret Kargi (2002): *Bioprocess Engineering Basic Concepts*. 2nd ed. New Jersey:
Prentice Hall PTR.

Shen, Y., Zhao, J., De La Fuente-núñez, C., Wang, Z., Hancock, R. E., Roberts, C. R., ... & Wang, Q. (2016). Experimental and
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Silmi, Melia (2015): *Models of LC-Cutinase Enzyme Regulation with Feedback System in PET
Biodegradation Process*, Bachelor’s Program Final Project, Institut Teknologi Bandung.

Talib, T. (2016): *Modelling Biodegradation of PET Involving The Growth of Factor E-Coli Bacteria
Measure*, Master’s Program Thesis, Institut Teknologi Bandung.

TJUSLS iGEM 2016 team. Retrieved November 01, 2017, from https://2016.igem.org/Team:TJUSLS_China/Modeling