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 from 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 10 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 :

Whereas X is bacterial growth (OD600), AI2 is signaling production and B is biofilm (OD600). 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 the biofilm is formed, we can focus on the mRNA transcription from DNA and mRNA translation to PETase process that will lead to PETase production. Transcription process of PETase is illustrated in Fig. 2:

Fig. 2. Reaction pathway of mRNA translation into PETase

We will construct a model based on the final project written by Silmi (2014) and thesis by Talib (2016). 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 :

A thorough analysis provide us with the critical point of the production, which is reached when

We choose α_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:

Fig. 3. PETase production from mRNA transcription

Each of the differential equation is solved analytically using MATLAB and then plotted in graph as follows. Through the graph, for M(0) = 0.05 and C(0) = 0, we obtain at the steady state, the rate of production of PETase is 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 PETase production model, we use value of PETase production as the initial value of PETase during PET degradation, 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 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. 4.

Fig. 4. Degradation pathway of PET in engineered E. coli

Our modelling does not look all reaction in pathway, but only PET degradation into ethylene terephtalate and 4-[(2-hydroxyethoxy)carbonyl]benzoate.

Enzymatic reaction of PETase is thermodynamically favored (δG = -37.87 kcal/mol), so PET degradation will occur spontaneously without any trigger substance. PETase reaction 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 the involvement of heterogeneous reaction during hydrolysis [TJUSLS iGEM 2016 Team, 2016]. Based on Langmuir adsorption isotherm, we can derive mathematical expression that implemented to Michaelis Menten kinetics. Langmuir adsorption isotherm equation is :

... (1)

Whereas q is quality of PET enzyme adsorption by unit quality PET, g; q_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 :

... (2)

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

... (3)

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 [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.

... (4)

... (5)

... (6)

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

... (7)

Now, let’s analyze the parameters of the preceding equation. It’s obvious that K, k₃, 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.

... (8)

... (9)

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

... (10)

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

... (11)

... (12)

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 K₀ 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:

Table 2. Parameters for rate of PET degradation with biofilm modelling module

Parameter	Definition	Units	Values	References
K0	A constant of the equation, K0 = Kk5k3/k4	mg/(mL.h.mm2)	1.43	TJUSLS (2016)
A	The area of the PET film	mm2	28.27	TJUSLS (2016)
Ka	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)
E0	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:

Fig. 5. Ethylene terephtalate production versus time

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.

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 slower than PET degradation with biofilm, with assumption no biofilm at bottom section of surface that contact with PETase.

2. PET degradation without biofilm 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.

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 theoretical investigation of multispecies oral biofilm resistance to chlorhexidine treatment. Scientific reports, 6, 27537.
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