Laccases and peroxidases are known to have a ping pong bi-bi mechanism.Typically, free oxygen and hydrogen peroxide (H2O2) oxidize the enzymes, modifying it to be receptive to a second substrate such as guaicol, ABTS, syringaldazine, and many more.

Schematic diagram of laccase catalytic activity (Kumar and Chandra, 2020)

Example of Peroxidase Activity: Guaiacol Oxidation (Radulescu, et al., 2019)

This is a typical ping pong bi-bi equation, where substrate A represents either free oxygens or H2O2.

However, to simplify the model, it assumes that [A] << [B]. Meaning, it assumes a “substrate depletion” condition, where substrate A acts as a catalyst and the reaction rate will rely on the concentration of substrate B. Thus, it simplifies the typical ping pong bi-bi equation to a pseudo-first order reaction (eq. below).

To account for the multi-substrates (PE and PFAs) that are catalyzed by one enzyme, this model followed the suggestion reflected in Pocklington and Jeffrey‘s (1969) paper. They suggested for if either of the substrate doesn’t form any ternary complex with the enzyme or modifies the substrate and no additional reaction happens between the two substrates, then they can be modelled as independent reactions. However, each substrate must be considered as a competitive inhibitor of each other.

However, for MnP and LiP , there is limited literature on PFAs degradation pathway. For the MnP and LiP models, PFA degradation was not considered; however its inhibitory effect on PE was still considered. More details on the ODEs are found in this PDF .

1. The chosen ODE solver during simulation is ode15s.
  a. Initially, ode45(Dormand-Prince) was chosen as an ODE solver; however, when running the parametric scan for
    LiP, the resulting graph wasn’t as expected. Thus, to make it uniform ode15s was used.

2. Due to time constraint of the wet lab experiments and limited literature review on the kinetics (esp on Kcat) of the enzymes when interacting with PE and PFAs, different substrates were chosen as chemical analogs.

  a. Due to the polymeric nature of PE, the chemical analog will be lignin. However, it is also difficult to
   characterize the kinetics when lignin as substrate. The closest derived kinetic values will be from lignin
   monomers, which are veratryl alcohol and vanillic acid.

   i. Additionally, the enzymatic systems of fungi can only accommodate compounds with 10-50 carbons in
    their active site (Vassilev, 2022). Thus, for this modeling, dodecane will be used.

  b. For PFAs, although ABTS has a ring structure, ABTS will be chosen due to the similarity of elements
   found in both compounds. Note, fluorine is the most electronegative atom, and this can totally change the
   kinetic values of PFAs.

3. Due to the limited kinetic studies, the enzymes were selected from microorganisms from the same order Polyporales.
Note: We tried to keep it in the same genus or even family as we know that the enzyme may diff from one species to another (Elisashvili & Kachlishvili, 2009).

  a. The derived values were mainly from the BRENDA system. Doubled-checked in literature as well for
   further confirmation

  b. Trametes Hirsuta, Bjerkandera adusta, and Phanerochaete chrysosporium for laccase, LiP, and MnP,
   respectively.

4. For the laccase-mediated reaction, it assumes that half of the enzyme will catalyze PE and the other as PFOA.

5. The culture media of the enzymes is set to 100mL.

First, simulation was performed on three models, one for each enzyme. The following tables and figures show the initial conditions, chosen kinetic values, and the results.

Table 1. Initial Conditions of the Species in these 3 Models

Table 2. Chosen Kinetic Values

Figure 1. Laccase-mediated Polyethylene-Perfluoroalkoxy Alkanes Degradation

Figure 2. Manganese Peroxidase-mediated Polyethylene-Perfluoroalkoxy Alkanes Degradation

Figure 3. Lignin Peroxidase-mediated Polyethylene-Perfluoroalkoxy Alkanes Degradation

In the above figures, 10uM (0.6g) of laccase can degrade PE and PFa in 62 and 42 hours, respectively. For 10uM (0.4g) LiP and MnP, the model only considered the degradation of PE, which had 146.5 hours and 27 hours, respectively. In these three models, sensitivity analysis showed that the degradation of polyethylene to propionic acid is sensitive to the molarity of the enzymes present, with laccase to be the most sensitive and MnP to be the least sensitive. To confirm this, parametric scan was performed on the models with values from 10uM to 20uM. Below are the results.

Figure 4 and 5. Sensitivity Analysis and Parameteric Scan Results of the Laccase Model

Figure 5 and 6. Sensitivity Analysis and Parameteric Scan Results of the MnP Model

Figure 6 and 7. Sensitivity Analysis and Parameteric Scan Results of the LiP Model

If there was wet lab data, parameter estimation would have been also performed to fit the experimental data to the model. Nonetheless, the above results do show that it is possible that a multi-enzymatic treatment may result in faster degradation if the ratio of the enzymes are optimized. Note that multi-enzymatic treatment can be in terms of 2 or 3 enzymes. For the multi-enzymatic model, we modelled the presence of three enzymes in the media as this was the only available degradation pathway in literature (Santacruz-Juárez, et al., 2021). Although the parameter estimation isn't performed, the information from the previous three models are still useful for gauging a rough estimation of the initial conditions of the enzymes for this multi-enzymatic model. The molar concentration could be used also by the wet lab team as a baseline to conduct their experiments. The following graphs show that for initial conditions of 15uM MnP, 40μM laccase, and 100μM LiP, the degradation of PE to propoinic acid is around 22 hours while PFOA degradtion is around 10 hours.

Figure 8. Multi-Enzyme-mediated Polyethylene-Perfluoroalkoxy Alkanes Degradation

For the parametric scan, the runs are as follows:
1. 20μM of laccase; 7.5 μM of MnP; 50 μM of LiP
3. 70μM of laccase; 18.75μM of MnP; 125μM of LiP
5. 120μM of laccase; 30μM of MnP; 200μM of LiP

Figure 9. Parameteric Scan Results of the Multi-Enzymatic Model

A. Modelling

    1. Inclusion of Fluid Dynamics: Since the final scenario where the modelling should be used is for the chemical reactor, the effect of fluids (hydrodynamics) on the enzymatic activity must be included.

    2. Parameter Estimation and Model Validation: It is obvious that the chosen substrates for the kinetic values of this model have differences in the chemical properties with PE and PFOA. Thus, experimental results is needed to fit and validate this model.

B. Bioreactor

    1. Immbolization of Enzymes (Polyalginate beads): Experiments must be done to characterize and evaluate strategies to create the beads and the amount of enzymes able to interact with the substrate.

    2. Membrane: Membrane must be selected to filter the desired compounds.

    3. Prototype: Prototype and testing must be done to improve the proposed design.