# Integrated model of PETase-MHETase cascading reaction

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To evaluate the effect of the OMPR-induced biofilm on dual enzyme degradation efficiency, we used the classic Michaelis-Menten (MM) equation to construct an enzyme kinetics model. By regarding the product concentration as the initial reaction rate, we use the Michaelis-Menten (MM) equation as the fitting model to study how the OMPR enhances the enzyme reaction. We first model the PETase and MHETase enzymes, respectively. Because in the PET plastic degradation reaction, the product of the PETase enzyme is the substrate of the MHETase enzyme, so we can get the final equation for the dual enzyme. The results showed that OMPR-induced biofilm could improve the degradation efficiency of the dual enzyme. In the overexpression system, the amount of enzyme is sufficient and the substrate is relatively low. On this premise, the following model is constructed. **

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1. Enzyme-catalyzed Model**

PETase is an enzyme or biocatalyst, which could promote the depolymerization of PET into bis(2-hydroxyethyl)-TPA (BHET), mono(2-hydroxyethyl) terephthalate (MHET) and p-benzene Dicarboxylic acid (TPA) (Figure 1). MHET is an intermediate product of PET decomposition. Then MHETase further converts MHET into monomers TPA and ethylene glycol (EG) [1]. The process of PET biodegradation by PETase and MHET biodegradation by MHETase are typical enzyme-catalyzed reactions. These reactions could be described by classic Michaelis-Menten equation.

Figure 1 PET depolymerization method [1].

Michaelis-Menten (MM) equation is a basic equation of enzyme kinetics and gives acceptable approximations of real chemical reaction processes. It is a model designed to generally explain the velocity and the gross mechanism of the reaction that is carried out by enzyme catalysts. The MM equation was first stated in 1913, where it assumes the rapid formation of a complex that is reversible in nature formed between the enzyme and its substrate. A substrate is the substance on which the catalyst acts to form a desired product. It also assumes that the concentration of the product is directly proportional to the rate of formation of the product. The MM equation reads

where the definitions of the symbols are

V0: initial reaction rate;

Vmax: the maximum reaction rate, namely reaction rate when the enzyme is
saturated with the substrate;

Km: Michaelis constant, which is only determined by the properties of the enzyme
and has nothing to do with the concentration of the enzyme. It can be used to
appraise different enzymes;

[S]: substrate concentration.

In order to use the MM equation in practice, we need to determine the constants Vmax and Km. For measured sample data of V0 and [S], we use the fit solver in Matlab to determine the parameters Vmax and Km and this will be described in detail in Section 5.

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2. Assumptions**

Before to use Michaelis-Menten equation, besides the assumptions that have been made for establishing the model and deriving the rate equation of the model [2], we made two more assumptions for our application as following:

(1) PETase and METase are relatively stable at room temperature and have a long half-life, that is, the activity of two enzymes is assumed to be almost unattenuated within 30min.

(2) PETase or MHETase and substrate are uniformly distributed in the reaction tube, therefore the enzymes may contact evenly with the substrate.

**
3. Method**

As mentioned above, PETase and MHETase have a cascading relationship during the PET polymer degradation process. MHET is not only the product of PETase degradation of PET but also the reaction substrate of MHETase. It is a little difficult to model using PET as the substrate. Thus, we used PET and MHET as the substrates to determine the enzyme activity of PETase and MHETase, respectively.

**For PETase**, we used high performance liquid chromatography (HPLC) to measure the enzyme activity. Using PET as the substrate, PETase could hydrolysis PET to BHET, MHET, and TPA. Then measure the concentration of MHET by HPLC method, we can finally get the result by series of calculations. Two experimental groups were set up in total. The first group was inoculated with E. coli expressing PETase only, and the second group with E. coli expressing both PETase and OMPR.

**For MHETase**, we also used HPLC to measure the enzyme activity. Using MHET as the substrate, MHETase could hydrolysis MHET to terephthalic acid (TPA) and ethylene glycol. Then measure the concentration of TPA by HPLC method, we can finally get the result by series of calculations. Two experimental groups were set up in total. The first group was inoculated with E. coli expressing MHETase only, and the second group with E. coli expressing both MHETase and OMPR.

**
4. Preliminaries**

According to the enzymatic reaction model, under the condition of low substrate concentration, the rate of enzymatic reaction is positively correlated with substrate concentration. This is because when the substrate concentration is very low, there are redundant enzymes that do not bind to the substrate. As the substrate concentration increases, the concentration of the intermediate complex increases continuously. When the substrate concentration is high, the enzymes in the solution are all bound to the substrate to form intermediates, although increasing the substrate concentration will not produce more intermediates [2]. Since PET is practically insoluble in water [5], the concentration of PET substrate in water must be extremely low. Therefore, trying to increase the substrate concentration in the reaction system will definitely help to improve the reaction rate and thus the degradation efficiency.

In our project, we choose the OMPR to regulate biofilm production that would allow Curli monomers to be exported and form curli fibers and biofilm. Biofilm can effectively capture micro-plastics in water. That is, the concentration of local PET substrate must be increased, and at the same time, overexpressed PETase and MHETase will be around the biofilm. Thus, the combination of PETase and PET, MHETase and MHET will increase the probability of forming intermediate complexes, so as to improve the reaction efficiency. This result has been confirmed by our experiments, and we hope to understand the improvement effect of biofilm on the degradation efficiency of dual enzyme (PETase and MHETase) through model calculation.

**
5. Test and Modeling**

We used the Michaelis-Menten equation (1) for modeling. In particular, we will use the command fit in Matlab of version 2017b as follows:

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myfittype = fittype('Vmax*S/(Km+S)','dependent',{'V'},'independent',{'S'},
'coefficients',{'Vmax','Km'});
[MMeq,G]=fit(S,V,myfittype);
**

where S and V are row vectors which collect the sample data of the experiment. The return symbol ‘MMeq’ is a function representing the MM equation with found Vmax and Km, that is

**
MMeq(S)= VmaxS/(Km+S).
**

How to get the value of Vmax and Km from such a Matlab function MMeq(S) ? The following is a simple idea to fixed Vmax and Km by using the value of MMeq(S). By noticing

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MMeq(1)= Vmax/(Km+1), MMeq(2)= 2Vmax/(Km+2),
**

we let a= MMeq(1) and b= MMeq(2) (known quantities) and then we have

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Vmax/(Km+1)=a, (2)
2Vmax/(Km+2)=b. (3)
**

From this we have (by (3) ÷ (2))

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2(Km+1)/(Km+2)) =b/a,
**

which gives

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Km=(2b-2a)/(2a-b).
**

For Vmax, from (2) we get

**
Vmax=a(Km+1) = ab/(2a-b).
**

**
5.1 Modeling for PETase and PETase+OMPR
**

The data used for numerical experiments is listed in Table 1, where the product concentration is measured after 30 minutes. In our numerical experiments, we regard this value as the initial reaction rate V0.

Based on the data in Table 1, the fitting results for the MM equation (1) are shown in Figure 2 and Figure 3 (the parameters Vmax and Km listed in the title of each figure and are summarized in Table 2).

Figure 2. Measured data and the fitted data for the case of PETase in Table 1.

Figure 3. Measured data and the fitted data for the case of PETase+OMPR in Table 1.

From the results in Table 2, we see that the Michaelis constant Km for PETase+OMPR is 3.2985, smaller than that of PETase only (4.0179). As we all know, Km is a measure of the affinity of a substrate to an enzyme. The smaller the Km value, the greater the affinity between the substrate and the enzyme. Therefore, the results indicated that the OMPR induced biofilms could improve the affinity of PET and PETase which meets our expectations.

**
5.2 Modeling for MHETase and MHETase+OMPR
**

The data used for numerical experiments is listed in Table 3, where the product concentration is measured after 1 minute. In our numerical experiments, we regard this value as the initial reaction rate V0.

Now, by using the values for the substrate concentration [S] and the initial reaction rate V0, i.e., the first and last column in Table 3, we show in Figure 4 and Figure 5 the fitting results for the case of MHETase and MHETase+OMPR respectively, where the parameters Vmax and Km in the MM equation (1) are shown in the title of each figure. Moreover, for each figure we also show the R2-value of the fitting, namely the value of ‘goodness’, i.e., the index of fitting quality. Such a R2-value is equal to the ratio of the sum of squares returned to the total sum of squares, which is the percentage of the variance of the dependent variable explained by the regression equation. If R2 is close to 1, it indicates that the fitting effect is very good.

Figure 4. Measured data and the fitted data for the case of MHETase in Table 3.

Figure 5. Measured data and the fitted data for the case of MHETase+OMPR in Table 3

According to our fitting results (from Figure 4 and Figure 5), the crucial parameters Vmax and Km for the MM equation are listed in Table 4.

From the results in Table 4, we see that the Michaelis constant Km for MHETase+OMPR is 1.4289, smaller than that of MHETase only (2.3249). Therefore, the results indicated that the enhanced biofilms could also improve the affinity of MHET and MHETase. That is, OMPR-induced biofilm could improve the enzyme activity of MHETase.

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5.3 Modeling for PETase–MHETase cascading reaction (with or without OMPR)
**

To evaluate the effect of the OMPR-induced biofilm on dual enzyme degradation efficiency, we combined the MM equation of the two-stage enzymatic reaction constructed before. The process is as followed:

For the two-stage enzymatic reaction, we have

and the product of the first stage is the substrate of the second stage, i.e., S2=V0,1. This implies the following MM equation for the whole enzymatic reaction process from S1 to V0,2:

Let

Then, the mathematic model for the two-stage enzymatic reaction is

Using the data in Table 2 and Table 4, we can calculate the parameters and , as shown in Table 5. Then, we finally conduct the Michaelis-Menten equation of our dual enzymatic reaction. The equations are shown as below:

For PETase-MHETase:

For PETase-MHETase enhanced by OMPR::

where the definitions of the symbols are

V0,2: initial reaction rate;

[S1]: PET concentration.

Compared the final Km of PETase–MHETase cascading reaction with or without OMPR, we found that the addition of OMPR could help to lower the Km of the dual enzymes. That is, OMPR-induced biofilm could improve the degradation efficiency of the dual enzymes.

**
Conclusion**

In the subsequent experiments (see Proof of Concept), we used HPLC to test the degradation effect of our system using PET power. The result is listed in Table 6. Comparatively, although the calculated value is not exactly the same as the experimental value, but basically in line with the tested reaction rate. In other words, the MM equation predicts the measured data very well. This deviation may be related to the following reasons. First, our reaction system is not a uniform concentration system. PET power cannot be completely dissolved in water. Second, as the reaction progresses, the enzyme activity will gradually decrease. Third, we use Spy system to couple the two enzymes (PETase-MHETase). This coupling effect is not as perfect as theoretically.

**
Reference**

[1] Furukawa, M., Kawakami, N., Tomizawa, A. et al. Efficient Degradation of Poly(ethylene terephthalate) with Thermobifida fusca Cutinase Exhibiting Improved Catalytic Activity Generated using Mutagenesis and Additive-based Approaches. Sci Rep 9, 16038 (2019). https://doi.org/10.1038/s41598-019-52379-z

[2] Rongwu Yang et al. Principals of Biochemistry (Third Edition), Chapter 9 Enzyme kinetics, Section 2 Michaelis-Menten kinetics, P161-165, Higher Education Press, Beijing，2018

[3] Quinn, D.M., Shirai, K., Jackson, R.L., and Harmony, J.K., (1982) Biochemistry 21, 6872-6879

[4] Shirai, K. and Jackson, R. L. (1982) Journal of Biological Chemistry 257, 1253-1258

[5] Record of Polyethylenterephthalat in the GESTIS Substance Database of the Institute for Occupational Safety and Health, accessed on Oct. 26 2020.

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