Team:ASTWS-China/Model

Modeling on PETase Enhanced Degradation by OMPR

1. Enzyme-catalysed Model

1.1 Michaelis-Menten equation

PETase is an enzyme or biocatalyst, which may help degrade PET plastics to small molecules, such as bis(2-hydroxyethyl) terephthalate (BHET), mono(hydroxyethyl) terephthalate (MHET) and monomers terephthalic acid (TPA).[1] The process of PET biodegradation by PETase is a typical enzyme-catalysed reaction. This reaction could be described by classic Michaelis-Menten equation. [2]

1.2 Lineweaver-Burk equation

In order to determine Vmax and Km, we use the conversion form of Michaelis-Menten equation, Lineweaver-Burk equation, also known as the double reciprocal equation .[2]

Plot 1/V0versus 1/[S] to get a straight line. Km/Vmax is the slope of line, 1/Vmax is the intercept of the line and the y-axis, thus 1/Km is absolute value of intercept of the line and the x-axis.

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 is relatively stable at room temperature and has a long half-life, that is, the activity of PETase is assumed to be almost unattenuated within 30min.

2. PETase and substrate are uniformly distributed in the reaction tube, therefore the enzyme may contact evenly with substrate.

3. Method

We initially planned to measure the concentration of mono hydroxyethyl terephthalate (MHET) (biodegradation product of PET) by High Performance Liquid Chromatography (HPLC) to test the efficiency of our PET biodegradation system, then would use the results as data for modeling. However, the HPLC machine in our lab broke down and could not be repaired soon. We stayed up for nights and solved this problem by using method pNP-Assay and has been used by iGEM 2012 team TU Darmstadt.

pNP-assays are a common way to quantify hydrolytic activity. We would hydrolyze para-Nitrophenylbutyrate (pNPB), which has similar hydropysis properties to PET, then measure OD of its hydrolysis product p-nitrophenol, finally get the result by series of calculations.

1. 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 PETsae and OMP.
2. Both of two groups were cultured at 37℃ for 5 hours.
3. Added 7.1μl (4-nitrophenyl butyrate) into 1ml acetonitrile, mix it into 4mM mother liquor, and then diluted it to get samples to be tested with concentrations of 2/1/0.8/0.4/0.2/0.2/0.2/0.2/0.1mm. Take 10μl acetonitrile for each and add them into 96-well plate.
4. After 100μl bacterial solution was added to each well and mixed with the substrate. At 1min, 2min, 5min, 10min, 15min, and 20min, the light absorption value of 405nm was determined by a microplate reader.
5.Repeat the step 4 and recorded the light absorption value at 1min, 2min, 5min, 10min, 15min, 20min, 25min and 30min respectively.

4. Data for Modeling

The results of OD405 absorbance of pNPB hydrolyzing catalyzed by PETase and PET+OMPR are shown in Table 1. According to Lambert-Beer law, we could convert OD into concentration. Hereby, pNPB hydrolyzing products concentration are shown in Table 2. Each part date consist of two tables corresponding to two experiments respectively.

Table 1-1. OD405 absorbance of pNPB hydrolyzing catalyzed by PETase and PET+OMPR (First experiment (20mins))

Table 1-2. OD405 absorbance of pNPB hydrolyzing catalyzed by PETase and PET+OMPR (Second experiment (30mins)))

Table 2-1 pNPB hydrolyzing product concentration (First experiment (20mins))

Table 2-2 pNPB hydrolyzing product concentration (second experiment (30mins))

5. 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-particles in water, including PET microplastics as well as PETase. After the enrichment of PETase and PET near biofilm, the concentration of local PET substrate must be increased, and the combination of PETase and PET 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 PET through model calculation.

6. Test and Modeling

6.1 Acquiring initial reaction rate

Besides the data shown in Table 2, in order to use Michaelis-Menten equation to predict the concentration change of the product, one important problem is how to estimate the initial rate of the reaction. We've tried this in three ways so as to obtain better results.

1) Average rate calculated from product concentration at a certain point as V0

Relative to the entire degradation process, the first 30 minutes can be regarded as the initial stage of degradation. If we assume that the degradation rate is constant over the first 30 minutes, V0 can be calculated in a simple and efficient way: the initial rate is calculated directly from the final concentration of the product divided by time. That is, take the product concentration of 20mins (for experiment 1 data) and 30mins (for experiment 2 data) and divide by 20mins and 30mins respectively, to get the initial rate. In addition, the initial rate calculated at the above two points has been averaged as well to reduce as much data uncertainty as possible. In order to meet the requirements of lineweave-Burk equation modeling, the reciprocal of [S] and V0 in each group of data are taken respectively. The data obtained by this method are shown in Table3 to Table 5.

Table 3 Data for Lineweaver-Burk equation, V0 from 20mins point average reaction rate of “20 mins Experiment group”

Table 4 Data for Lineweaver-Burk equation, V0 from 30mins point average reaction rate of “30 mins Experiment group”

Table 5 Data for Lineweaver-Burk equation, V0 from an average of Table 3 and 4

2) Linear slope fitting to calculate the average value as V0

The method of calculating the average reaction rate from the product concentration at 30 minutes or 20 minutes is simple, but its accuracy is doubtful. Then we tried the second method: plot the product concentration and time of each sample, and then perform linear fitting on the data to obtain the average reaction rate (slope of the line) as V0.The following fig.1 and 2 show our calculations for samples with substrate concentrations of 0.1mM and 4mM in the 30-minute experimental group. Compared with the methods that we mentioned above, the average rate values obtained by this method are all small. See Table 6 to Table 8 for the complete 20min and 30min groups of data used for lineweaver-Burk equation modeling and the average of the two groups of data.

Fig. 1 Average reaction rate from linear fitting of substrate concentration 0.1mM samples

Fig.2 Average reaction rate from linear fitting of substrate concentration 4mM samples

Table 6 Data for Lineweaver-Burk equation, V0 from linear fitting average reaction rate of “20mins experiment group”

Table 7 Data for Lineweaver-Burk equation, V0 from linear fitting average reaction rate of “30mins experiment group”

Table 8 Data for Lineweaver-Burk equation, V0 from an average of Table 6 and 7

3) Fitting V0 by piecewise approximation method

For the initial rate V0 data, the team still has another opinion: the initial rate should be instantaneous rate, rather than the average rate of the stage. Based on this consideration, it is assumed that the reaction rate of the product has some continuity on the time axis, and the average rate of the phase at the far point in time gradually approaches the rate at the initial time point. We tested the data of PET only sample in the 30mins experimental group and the results were shown in Fig.3

Fig.3 V0 approached by piecewise approximation (PET only sample)

Unfortunately, due to insufficient data acquisition frequency, it is difficult to evaluate the accuracy of the approximation results when the approximation method is used to fit V0, thus affecting the accuracy of the linear regression equation carried out with this data. We hope that if there is an opportunity to conduct the experiment again in the future, increase the frequency of data acquisition, and try again to fit V0 with the approximation method.

6.2 Model solution

By plotting 1/[S] and V0 scatter plots of PETase and PETase+OMPR data, it can be intuitively found that the data has a linear trend, so the dependent variable may be set y =1/V0 and independent variable x=1/([S]), the linear regression equation equivalent to Lineweaver-Burk equation is

y=ax+b

a=Km/Vmax is the slope of line, b=1/Vmax is the intercept of the line and the Y-axis. After linear fitting by Excel, the parameter a and b can be obtained. And the model parameter Vmax and Km can be determined. Goodness of Fit R2 measures the fitness of the regression equation as a whole, and represents the population relation between dependent variables and all independent variables. R2 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.

The plots of PETase and PETase+OMPR data from Table 3 – 8 are shown in Fig.4 – 9. Model parameters Vmax and Km as well as Goodness of Fit R2 for are shown in Table 9.

Fig.4 Lineweaver-Burk equation linear fitting for V0 from 20mins point average reaction rate data of “20 mins Experiment group” (Table 3)

Fig.5 Lineweaver-Burk equation linear fitting for V0 from 30mins point average reaction rate data of “30 mins Experiment group” (Table 4)

Fig.6 Lineweaver-Burk equation linear fitting for V0 from average of 20mins and 30mins group (Table 5)

Fig.7 Lineweaver-Burk equation linear fitting for V0 from linear fitting average reaction rate of “20mins experiment group” (Table 6)

Fig.8 Lineweaver-Burk equation linear fitting for V0 from linear fitting average reaction rate of “30mins experiment group” (Table 7)

Fig.9 Lineweaver-Burk equation linear fitting for V0 from an average of linear fitting average reaction rate of “20mins and 30mins experiment group” (Table 8)

Table 9 Results of model solution

6.3 Model accuracy and confidence interval

After the fitting parameters of the linear regression model, an important and interesting question is: for calculating the parameters of the equation, the calculation is the essence of point estimation of unknown parameters (calculated from one or more groups of experimental data), how to evaluate the approximate degree between point estimate and the true parameters of unknown approximation as well as the error range? This is actually an interval estimation problem in mathematical statistics. An interval of the parameter is constructed, an interval centered on the point estimate (known as the confidence interval), which contains the parameter with a certain probability (known as the confidence level, which is assumed to be 95% in this work). In other words, for the probability of a and b falling within the range of 0.9173±1, 0.2798 ± 2 was 95%.The details are shown in the figure 10.

Fig. 10 Example of confidence interval

7. Conclusion and Future Improvements

7.1 Biofilms improve the biodegradation efficiency of PET

From the results in Table 9, we may conduct the several Michaelis-Menten equation depending on different V0. The two group data which are averaged from two experiments have a R2 > 0.99, so are both significant models. For the V0 from average value of 30mins and 20mins points product concentration, the equations are shown as below:

For PETase only:

For PETase enhanced by OMPR:

By comparing the above two equations, we can find that when the substrate concentration is lower than 0.135mol/L (since PET is practically insoluble in water, the concentration of PET substrate in water must be lower than this range), OMPR improves the degradation effect of PET, and can accelerate the reaction speed to about 48.4% at most. This predicted enhancing rate are varied between different Km and Vmax which are calculated based on different V0. The highest enhancing rate is predicted by the model that V0 calculated from product concentration at 30 mins point; and the value is 62.2%.

In the subsequent experiments (see Proof of Concept), we used HPLC to test the degradation effect of our system, and found that in the presence of OMPR, the degradation efficiency increased about 66%, which exceeded the prediction of our model. This deviation may be related to the data we collected for modeling. For example, in the experiment, the substrate concentration of the sample we configured was close to the geometric series, and the proportion of the low concentration point was too large. After taking the reciprocal, the points were concentrated at the lower left of the coordinates. This result increased the fitting error and the calculation of Km and Vmax was inaccurate. In future experiments, we can match the substrate concentration to the concentration difference of 1/[S] instead of the concentration range of [S], so as to average the point distance, and then use the least square linear regression analysis. Of course the V0 calculation method plays an important role as well. V0 based 30 mins product concentration gave the most closer prediction. It may because that longer reaction time reflects the process of the reaction better. After all, the difference of PET degradation process and para-Nitrophenylbutyrate (pNPB) hydrolyzation process may be another reason as well.

7.2 Model improvement : Improved Michaelis-Menten equation

In our project, we introduced biofilm OmpR234 to the system which should improve the efficiency of classic enzyme-catalysed reaction. Considering this effect, we modified the classic equation and obtain the improved Michaelis-Menten equation as below, which introduced 3 variables ϒ1,ϒ2 and Bf.

V0, [S], Vmax and Km are the same as in classic equation.

Bf: State variable, which may take 0 or 1. When Bf=0, there is no ompR in the system; and when Bf=1, the ompR exists in the system.

ϒ1: The increase value of the reaction rate after OMP enhancement who equals to VmaxOMP － Vmax , and VmaxOMP represents maximum reaction rate when ompR exists.

ϒ2: The increase value of the reaction rate when the reaction rates achieved 1/2 of the maximum rate after OMP enhancement who equals to KmOMP - Km , and KmOMP represents improved Michaelis constant when ompR exists.

To solve the parameters of improved Michaelis-Menten equation, assuming: 1) the common parameter values of double factor and classic single factor are identical; 2) a group of new data can be obtained by experiments; then the solution can be obtained by using the nonlinear fitting method, where Bf, [S] and V0 are known variables, and Vmax, Km, and are parameters to be solved.

References:

[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.

Appendix

The program code of Matlab7.0