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Kinetic model
Introduction and aim
Let us present our kinetic model, which is utilizing ordinary differential equations (ODE) to model the degradation pathway of elastane, which consists of a soft and a hard segment, respectively. The model can be used to find out how fast certain components can be degraded. In addition, a sensitivity analysis was made to find out which enzymes that are most affected by a change of k_m and k_cat.

Background
An ODE is a differential equation that involves derivatives with respect to only one variable per equation. ODEs can be useful for many different purposes. In biology, ODEs are a useful tool for studying dynamic systems, e.g. changes in concentrations within a cell [1].

A sensitivity analysis studies how different inputs to a mathematical model contribute to uncertainty in the output, i.e. how much is the output of a model affected by estimation of the parameters [2]. There are several methods for sensitivity analysis. The type of sensitivity analysis we have used is called “one-factor-at-a-time" (OAT) [3]. The idea is basically to analyse how the output is affected when one input variable varies while the other input variables are fixed. In our analysis it is the k_m and k_cat that is varied one by one for the hard and soft segment, respectively.

The purpose of the sensitivity analysis is to find if there are possibilities for optimization regarding the k_m or k_cat values. Let us assume that the degradation pathway would be faster if a certain enzyme would have higher k_cat value. Then it would be of interest to perform future experiments on this certain enzyme. Future experiments could be to screen for mutants or perform error-prone PCR to retrieve a mutant with higher k_cat.

Method
We created a set of equations that predict the degradation pathways for both the hard and the soft segment of elastane, see Design. The equations were derived from the Michaelis-Menten equation

(1)

where v (mol/s) is the velocity, [P] (M) is the concentration of the product, [S] (M) is the concentration of the substrate, v_max (mol/s) is the maximum rate of the system, k_m (M) is the concentration of substrate when the reaction rate is half of v_max.

For some enzymes, the k_cat was not available on BRENDA. If so, k_cat was calculated by equation 2

(2)

where k_cat (1/s) is the number of chemical conversions of substrate molecule per time unit for a given enzyme concentration and [E] (M) is the enzyme concentration [4].

Values for k_m and k_cat was retrieved from the enzyme database BRENDA [5] and is shown in table 1.

Table 1. k_m and k_cat retrieved from BRENDA. Missing values marked by “X”

Enzyme	km [mM]	kcat [1/s]
PETase	4.6	5.9
MHETase	3.9	26.8
PEG-DH	2.0	X
Oxidoreductase	0.007	X
PueA	X	X
BphA	0.018	1.1
BphB	0.0031	0.38
BphC	0.00046	115
BphD	0.0046	1300
OP4	0.041	215
Deaminase	100	0.5
H2O	Not used	10 000

Limitations
One limitation of our ODE model is that it is sensitive, meaning that a small change of an input parameter can give large change of output. This can be seen in the results section.

Some of the parameters we used were randomly set, for example the initial concentrations of the first compounds in the degradation pathways or the enzyme concentrations. In order to get the exact value for these parameters, research in the wet lab needs to be conducted first. In the future, more accurate values could be fed in the model to improve the results. Furthermore, some k_m and k_cat values could not be found on BRENDA. To try to fix this last issue we did a second parameter search on BRENDA. To query the database, we used a method developed by Ivan, a PhD student at Sysbio (you can check more of his awesome code here (https://github.com/IVANDOMENZAIN). The script gathers the EC number of the enzyme, the substrate of the enzyme according to our degradation pathway and the organism where the enzyme is expressed as input. Then, the script tries to:

• Try to match all three input arguments an entry in BRENDA.
• If no matches, let the organism vary.
• If no matches, let the substate vary.
• If no matches, let organism and substrate vary.

If there is still no match at step 4, no k_m or k_cat values was retrieved.

This parameter search was done for pH = 7.5, temperature = 298.15 and ionic strength = 0.1. k_m and k_cat values retrieved from the parameter search are shown in table 2. Even after the parameter search there are still missing values for some parameters for the enzymes of interest. This implies that these enzymes have not been studied in detail.

Table 2. k_m and k_cat retrieved from parameter search on BRENDA. Missing values marked by “X”.

Enzyme	km [mM]	kcat [1/s]
PETase	X	12.4
MHETase	X	26.8
PEG-DH	X	X
Oxidoreductase	X	X
PueA	X	3000
BphA	0.00018	4.6
BphB	X	0.9
BphC	0.00046	115
BphD	X	12

Also, the fact that some enzymes have not been studied in detail, and therefore lack many essential parameters, may suggest uncertainty in the parameters that are available. For example, if a certain parameter is found by executing an experiment only once, the outcome may be different the next time the same experiment is done.

Ordinary differential equations
A schematic overview of the degradation pathway for the hard and the soft segment is presented in figure 1. For more details, see Design.

Figure 1. The biological Degradation pathway of Elastane. This figure shows the degradation pathway and each enzyme which contribute for Elastane degradation. The left part is a pathway for the soft segment and while the right part shows degradation for the hard segment. The soft segment starts with PET or PEG and is degraded into EG, and the hard segment starts with polyurethane and is degraded into Benzoate and 2-oxopent-4-enolate. 
To simulate the degradation pathways, we created equation 3-6 for the soft segment and equation 7-14 for the hard segment. For each equation [E], which is the total enzyme concentration (mM), is set to 0.2. k_cat and k_m values for each enzyme are taken from table 1. For example k_cat(PETase) = 5.9 1/s from table 1. Initial concentrations, abbreviated c(PU), c(DAPM) etc, are taken from table 3.

Table 3. Compounds, abbreviations and initial concentrations included in equation 3-15.

Compound	Abbreviation	Initial concentration
Polyethylene terephthalate	c(PET)	5
Polyethylene glycol	c(PEG)	1
Monoethylene terephthalate	c(MHET)	0
Ethylene glycol	c(EG)	0
Polyurethane	c(PU)	1
Diaminophenylmethane	c(DAPM)	0
Diphenylmethane	c(DPM)	0
(2R,3S)-3-phenylcyclohexa-3,5-diene-1,2-diol	c(Pdiol)	0
Biphenyl-2,3-diol	c(Bdiol)	0
2-hydroxy-6-oxo-phenylhexa-2,4-dienoate	c(Dienoate)	0
2-oxopent-4-enoate	c(Enoate)	0
Benzoate	c(Benzoate)	0

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

Sensitivity analysis

A sensitivity analysis was developed from the ODE model. By using a for-loop, k_m and k_cat values for each enzyme used in the ODE model were varied one by one from 0.01 to 100. A logarithmic scale (log10) was used to receive a more even distribution for the range of k_m/k_cat-values. By varying the k_m or k_cat, a molar yield was retrieved for each k-point (0.01-100). The molar yield is the output concentration (the concentration of the final product of the degradation pathway) divided by the input concentration (the concentration of the first reactant(s) of the degradation pathway). This was repeated for each enzyme for both k_m and k_cat. The molar yield for each k_m and k_cat was exported to an Excel document. The Excel document, containing molar yields for each enzyme and the range of k-values, was imported to Rstudio[6]. In Rstudio, the molar yield was plotted against the range of k-values for both k_m and k_cat for each enzyme.

Results

Ordinary differential equations

The simulations of our ODE models are shown in figure 2 and 3. Figure 2 represents the degradation pathway of the hard segment. Figure 3 represents the degradation of the soft segment. What differs the plot to the right and the plot to the right, in both figure 2 and 3, is which k_m and k_cat values that was used in the simulation. In the plots to the left k_m and k_cat values from table 1 was used. In the plots to the right, k_m and k_cat values from table 2 was used. If a parameter was not found in the parameter search, see limitations, that parameter was taken from table 1 instead.

For both the hard and the soft segment, it can be seen that the degradation pathway is faster with the parameters found in the parameter search (plot to the right in figure 2 and 3).

Figure 2. Degradation of the hard segment with parameter values from table 1 (plot to the left) respectively table 2 (plot to the right) Polyurethane (red line) is degraded, intermediates are formed and degraded. At last benzoate (cyan line) is formed and accumulated.
Figure 3. Degradation of the soft segment with parameter values from table 1 (plot to the left) respectively table 2 (plot to the right) PET (red line) is degraded to MHET (blue line), which is degraded EG. PEG (purple line) is also degraded to EG. EG does not accumulate but is diverted into the cellular metabolism.

Sensitivity analysis

In figure 4-6 the molar yield is plotted against k, which ranges from 0.001 to 100 on a logarithmic scale with base 10. The green line in each plot represents the molar yield and the dashed black line represents the k_m or k_cat value that was used in the ODE model. The dashed black line shows if the k_m or k_cat value that was used in the ODE model is the k-value that gives the highest molar yield.

Hard segment

In figure 4, the k_cat respectively k_m parameters are varied for each enzyme included in the hard segment degradation. The dashed black line is positioned at a k-value that gives the highest molar yield for every plot.

Figure 4. Sensitivity analysis for k_cat (plot to the left) respectively k_m (plot to the right) values for the hard segment The molar yield is plotted against k_cat respectively k_m, which ranges from 0.001 to 100 on a logarithmic scale with base 10, for the enzymes included in the hard segment degradation; BphA1, BphB, BphC, BphD and PueA. The dashed black line represents the k_cat respectively k_m value that was used in the ODE model.

Soft segment

In figure 5, the k_cat respectively k_m parameters are varied for each enzyme included in the soft segment degradation. For the plot to the left in figure 5, the dashed black line is positioned at a k-value that gives the highest yield for PEG-DH and Oxidoreductase and MHETase, but not for PETase. For the plot to the right in figure 5, the dashed black line is positioned at a k-value that gives the highest yield for oxidoreductase, but not for MHETase, PEG-DH and PETase. Figure 5 is a zoom in of MHETase from the plot to the right in figure 5. Figure 5 shows that the k_m value used in the ODE model does not give the highest molar yield.
Figure 5. Sensitivity analysis for k_cat (plot to the left) respectively k_m (plot to the right) values for the soft segment The molar yield is plotted against k_cat respectively k_m, which ranges from 0.001 to 100 on a logarithmic scale with base 10, for the enzymes included in the soft segment degradation; MHETase, Oxidoreductase, PEG-DH and PETase. The dashed black line represents the k_cat respectively k_m value that was used in the ODE model.
Figure 6. Zoom-in on k_m MHETase from figure 5 The molar yield is plotted against k_m, which ranges from 0.001 to 100 on a logarithmic scale with base 10, for MHETase. The dashed black line represents the k_m value that was used in the ODE model. Conclusions

From the sensitivity analysis we got several interesting results. From the plot to the left figure 5 we can see that for MHETase and PETase the dashed black line on the x-axis does not give the highest value on the y-axis. This means that the k_cat value that we used in the ODE model does not give the highest molar yield possible and can therefore be optimized. The same can be applied for MHETase, PETase and PEG-DH in the plot to the right in figure 5. Although, for PETase and PEG-DH the difference between the k_m used in the ODE model and the k_m that give the highest molar yield is quite big. For PEG-DH the highest molar yield is at k_m approximately 0.01 mM, as opposed to 2.0 mM that was used in the model. The difference is about the same for PETase. Though, for PETase there is a slight increase in molar yield just to the left of the dashed black line. Therefore, it is of interest for future experiment to perform error-prone PCR or screen for mutants to retrieve mutations of these enzymes with optimized k_m and k_cat values.

​
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