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Determination of key-thermodynamic parameters using a kinetic modeling approach to describe the post-consumer poly(ethylene terephthalate) hydrolysis catalyzed by cutinase from Humicola insolens

The search for a straightforward technology for post-consumer poly(ethylene terephthalate) (PC-PET) degradation is essential to develop a circular economy. In this context, PET hydrolases such as cutinases can be used as bioplatforms for this purpose. Humicola insolens cutinase (HiC) is a promising biocatalyst for PC-PET hydrolysis. Therefore, this work evaluated a kinetic model, and it was observed that the HiC seems not to be inhibited by any of the main PET hydrolysis products such as terephthalic acid (TPA), mono-(2-hydroxyethyl) terephthalate (MHET), and bis-(2-hydroxyethyl) terephthalate (BHET). The excellent fitting of the experimental data to a kinetic model based on enzyme-limiting conditions validates its employment for describing the enzymatic PC-PET hydrolysis using two-particle size ranges (0.075-0.250, and 0.250-0.600 mm) and temperatures (40, 50, 55, 60, 70, and 80 ºC). The Arrhenius law provided a reliable parameter (activation energy of 98.9 ± 2.6 kJ mol −1 ) for enzymatic hydrolysis, which compares well with reported values for chemical PET hydrolysis. The thermodynamic parameters of PC-PET hydrolysis corresponded to activation enthalpy of 96.1 ± 3.6 kJ mol -1 and activation entropy of 10.8 ± 9.8 J mol -1 K -1 . Thus, the observed rate enhancement with temperature was attributed to the enthalpic contribution, and this understanding is helpful to the comprehension of enzymatic behavior on hydrolysis reaction.

PET enzymatic hydrolysis is described as a heterogeneous reaction system in which the biocatalyst in an aqueous environment catalyzes with the insoluble and macroscopic polyester, containing highly ordered (crystalline) and less ordered (amorphous) regions [24], being the last ones more accessible. Therefore, PET erosion probably occurs similarly to cellulose degradation [24]. This phenomenon consists of chain disruption and loosening, making the individual substrate molecule more accessible and available for interactions with degrading enzymes [25]. Moreover, enzymatic hydrolysis of PET occurs with the generation of new free hydroxyl and carboxyl groups at the polymer surface through the cleavage of ester backbones, which results in increased PET hydrophilicity [26]. The resulting low molecular weight products of the enzymatic hydrolysis comprise ethylene glycol (EG), and terephthalic acid (TPA) that are the PET synthesis monomers, and beyond these, mono (2-hydroxyethyl) terephthalate (MHET), and bis(2-hydroxyethyl) terephthalate (BHET) that are the main intermediate reaction products.
The depolymerization of water-insoluble substrates using enzymes, such as cellulases, chitinases, xylanases, and depolymerases, occurs via two-steps: adsorption and hydrolysis. In the first moment occurs the adsorption of the enzyme on the hydrophobic surface of the substrate. The second step is the hydrolysis of the polymer chain at the active site of the enzyme [27]. Therefore, the hydrophobicity of the enzyme surface influences the interaction with hydrophobic polyesters [28]. Thus, enzymatic hydrolysis is a route for recycling PET waste that depends strongly on the substrate surface availability [29,30].
The mobility of PET chains is also a relevant factor for the biodegradability of polyesters [31]. The chain mobility of the polymer is determined by the crystallinity of the polymer, and it increases at temperatures close to its glass transition temperature (Tg) [26,32,33]. The increase of the reaction temperature of PET hydrolysis increases polymer chain flexibility, which favors the diffusion of water molecules between the polymer chains through the weakening of the hydrogen bonds, as well as randomizing and increasing the flexibility of the PET chain, thus, enhancing the enzyme accessibility to the ester bonds [26].
In light of the challenges mentioned above to describe PC-PET enzymatic hydrolysis, this study initially evaluated the stirring speed effect and the availability of the substrate concentration on the reaction rate. Thus, a kinetic analysis was performed through a model based on enzyme-limiting conditions to determine key-kinetic parameters, which are relevant for developing an industrial process of PET depolymerization. The product inhibition was also evaluated. Although some studies have suggested a mathematical approach to describe PET hydrolysis, to the best of our knowledge, none of them determined the activation parameters for this reaction. Therefore, the investigation of Humicola insolens cutinase (HiC) ability to undergo efficient catalysis is reported here over the temperature range of 40 to 80 o C. The aim of this study is not only to determine the kinetics of the enzymatic reaction but also to provide critical parameters that must be considered in the design of a scalable process.
The post-consumer PET (PC-PET) was originated in an industrial mechanical recycling plant and was composed of different (i.e., different thicknesses, colors, etc.) packages. The PC-PET presented high crystallinity (41.1%) and high molar mass (43,379 g mol -1 ), as described by Castro and collaborators [18]. The material was received as flakes, grinded in a knife mill, and sieved into two fractions with different granulometric size ranges: F1: 0.075-0.250 mm, and F2: 0.250-0.600 mm.

Quantification of the enzymatic hydrolysis products
The quantification of TPA, MHET, and BHET concentrations was carried out in a Waters High-Performance Liquid Chromatograph equipped with a binary pump (1525 model), and a UV/Visible detector (2489 model) at 254 nm wavelength. The injection volume employed was 20 μL, and the column, an Eclipse Plus C18 column (Agilent Technologies), of 4.6 mm x 250 mm and 5 μm particle diameter, was kept at 30 o C. The analysis methodology was performed using a gradient elution method, and the mobile phases used were acetonitrile and an aqueous solution of formic acid (0.05 vol.%) under a total flow rate of 0.500 mL min -1 .
The TPA, MHET, and BHET concentrations were determined based on standard curves.

Enzymatic characterization
The protein concentration of HiC commercial liquid preparation was determined according to the Bradford method [34], and the obtained result was 16.

PET hydrolysis
The PC-PET hydrolysis was carried out in a 10 mL reactor provided with magnetic stirring (EasyMax TM 102, Mettler Toledo), employing HiC as a biocatalyst diluted in phosphate buffer solution. All tests were carried out using 200 mmol L -1 phosphate buffer (pH 7.0), and HiC concentration of 1.0 mgprotein mL -1 . At appropriate intervals, samples of the reaction medium were withdrawn and diluted in methanol before chromatographic analysis. The sum of the TPA, MHET, and BHET concentrations was used to determine the initial reaction rates.
The TPA yield was defined as described in Equation 1, where [TPA] and [PET] 0 refer to the concentration of TPA and the initial concentration of PET on mass basis (g L -1 ), respectively. and are the molar mass of TPA (166.1 g mol -1 ) and the PET repeat unit (192.2 g mol −1 ), respectively [35]. (1)

Effect of substrate concentration on PET hydrolysis
The effect of PET concentration on the enzymatic hydrolysis was evaluated using different PET concentrations: 10, 35, 50, 80, 100, 120, 150, 200, 220, 250 g L -1 . For all assays, the reactor stirring speed was 800 rpm, and the reaction temperature was kept at 70 ºC.

Kinetic study of enzymatic PET hydrolysis and product inhibition
Barth and collaborators [30] proposed a model for the heterogeneous enzymatic PET hydrolysis. The following assumptions were made in order to simplify the mathematical representation [36]:  Considering that the catalytic site of HiC has the unique function of producing oligomers that can be represented as the product (P), the resulting system could be modeled analyzing only the hydrolysis and adsorption steps with no incorporation of an extra kinetic parameter.
 The substrate concentration is based on the surface concentration accessible to enzymes for adsorption and subsequent catalysis, which considers the effect of substrate particle size.
 The produced oligomers may inhibit the biocatalyst reversibly and competitively, forming an EP complex.
Based on these assumptions, the heterogeneous enzymatic PET hydrolysis can be described by the following reaction scheme: .
where i is the inhibition parameter that can be described for each product formed (P) as shown in Equation 9.
If the product did not interact with the enzyme, then  tends to 0; that is, no competitive inhibitory effect is evidenced, and therefore Equation 8 can be rewritten as described in Equation 10.
The kinetic model was adjusted for two different granulometric size ranges in order to test its validity to describe PET hydrolysis. Analysis of specific surface area of selected PET

Effect of temperature
The reactions were carried out at six temperatures (40, 50, 55, 60, The parameters of the non-linear kinetic model and all the activation parameters were obtained by numerical estimation using the Levenberg-Marquardt algorithm. The data were analyzed by Origin 8.1 software using the convergence criterion of chi-square minimization (tolerance of 10 -9 ) and the maximum number of iterations equal to 50.

Effect of PC-PET concentration on enzymatic hydrolysis
The enzymatic degradation of PET is a heterogeneous catalytic process, in which the reactant is not solvated in the bulk solution. The effectiveness of enzymes is sensitive by continuous enzyme-substrate-product interactions at the PET-water reaction interface, such as enzyme binding, desorption, inhibition, and diffusion [36], and all these phenomena can cause remarkable changes of substrate exposed area. Thus, the effect of substrate concentration (10 -250 g L -1 ) on the concentration of PET hydrolysis products was investigated, and Figure 1 represents the results obtained. The PET hydrolysis products (TPA, MHET, and BHET) concentration had the lowest value at 10 g L -1 , while the maximum product concentration (129.42 mmol L -1 ) was obtained with 220 g L -1 of PET after 96h. Moreover, a higher concentration of released products after 96 h was observed than reported for Carniel and collaborators [17] after 14 days of reaction using HiC at 60 ºC, and higher than measured for Gamerith and collaborators [38] after 21 days of enzymatic hydrolysis using Thermobifida cellulosilytica cutinase at 50 ºC, in almost the same enzyme concentration.

Figure 1
The results showed that higher substrate concentrations appear to have two positive effects, as expected, (i) faster initial reaction rate and (ii) higher product generation. However, the positive effects of high substrate concentration, concerning the products released, decreased for PET concentrations above 220 g L -1 . It seems that interactions between reactants and biocatalysts occur quite differently from their homogenous counterparts, indicating that the diffusion processes in two-or one-dimensional space occur differently [39]. Therefore, the mass transfer limitation was evaluated varying the reaction stirring speed. A more intensive mixing regime should facilitate a better mass transfer inside the reactor, being effective to reduce a potentially high local product concentration surrounding the enzyme active site [36].

Effect of stirring speed
The enzymatic PET hydrolysis system is composed of a liquid phase, containing enzymatic extract, and a solid phase consisted of the PET particles. An adequate interaction of the enzyme with the substrate at the solid-liquid interface is essential for efficient enzymatic PET hydrolysis [40]. Therefore, the rate of PET hydrolysis could be affected by diffusion limitations. According to Gan and collaborators [36], the heterogeneous hydrolytic reaction rate is determined by three events in a stirred batch reactor. The first one is the mass enzyme transfer through the stagnant liquid film layer adjacent to the solid substrate, the second event refers to the enzyme adsorption at the substrate surface, and finally, the third one is the catalysis. Thus, with the progress of the enzymatic PET hydrolysis, the amount of available substrate decreases, and the overall reaction rate depends on the enzyme penetration and diffusion inside the solid substrate.
The effect of stirring speed on the PET hydrolysis reaction was investigated with the varying initial concentration of PC-PET (10 to 250 g L -1 ) and stirring reaction speed (100, 400, and 800 rpm).

Figure 2
According to the results shown in Figure  For the three series of experiments, the initial product formation rate goes through a maximum, corroborating the limited enzyme diffusion into substrate surface for PC-PET concentrations above this maximum. As the reactor stirring speed decreases, a thicker boundary layer around solid particle surfaces increases, resulting in a higher mass transfer resistance in the external film of the solid [41]. Such behavior was observed in the PET enzymatic hydrolysis conducted at 100 rpm, in which lower initial rates were noted for all substrate concentrations.
The PET enzymatic hydrolysis has similar aspects to the hydrolysis of other polymers as cellulose [24,30]. Ingesson and collaborators [42] studied the enzymatic hydrolysis of cellulose and they reported that an increase of 7.5 % on the initial substrate concentration resulted in the reduction of the initial hydrolysis rate and the conversion yield for different shaking regimes (reaction flasks were shaken continuously or intermittently). The authors attributed this behavior to end-product inhibition and mass transfer limitations within the reaction mixture due to the high viscosity of the slurry (reaction medium).
Thus, considering that the best results were obtained for reactions carried out at 800 rpm stirring speed, further experiments were conducted with the same stirring speed to minimize the diffusion restriction for the kinetic study and to evaluate the enzyme inhibition by hydrolysis products.

Effect of PC-PET concentration: Kinetic study
The enzymatic hydrolysis of PET is influenced by product inhibition [43].  approach could reduce the cost and the complexity of reactor design or operation [44], once multi-enzyme systems could be limited concerning different optimal activity ranges and deactivation kinetics [45].

Figure 3
The kinetics of PC-PET hydrolysis catalyzed by HiC was studied by employing a heterogeneous model based on enzyme-limiting conditions proposed by Barth and collaborators . Even though cutinases from Thermobifida sp. present a competitive inhibition effect by reaction products [30,38,43], the cutinase from Humicola insolens seems not to be inhibited at 70 o C by any of PET products, which suggests that the enzyme could be used in its maximum activity. Thus, by adjusting the model, the inhibitor contribution term () was considered equal to 0, since there was no inhibition contribution, and then, it was possible to determine the reaction constants through Equation 10.
The reaction rates were determined from the sum of released hydrolysis product concentrations in the reactions with different initial PET concentrations, and it was plotted as a function of substrate concentration for two PC-PET granulometric size ranges (Figure 4).
high determination coefficient (R 2 ) was obtained in the fitting to the kinetic model given by Equation 10, which was superior to 0.978 for both ranges, thereby confirming the model validity. The kinetic parameters were obtained by nonlinear regression analysis, and the results are presented in Table 1.

Figure 4
Table 1 The PC-PET fraction with the smaller particle size range (F1) presented a larger specific area than F2 (Table 1), which means a higher substrate surface area available.
Therefore, the reaction carried out with the smallest PET particles showed a 2.4-fold higher hydrolysis rate constant (k) than the larger particle size range (F2). This is expected, as the enzymatic PET hydrolysis occurs in a heterogeneous interface, and PET erosion depends on substrate surface availability. Similar behavior was also observed in other studies [46,47], in which the decreasing particle size led to higher PET depolymerization due to increased surface area. Besides, comparable adsorption equilibrium constants (K) were obtained for both PC-PET particle size ranges by analyzing them based on initial substrate concentration. On the other hand, the value of K (0.0051 mL cm -2 ) estimated for the F1 range, based on the PC-PET surface area, was 6.6-fold lower than that obtained for F2 (0.0339 mL cm -²), which is almost equal to the specific area ratio between F1 and F2. Therefore, it can be suggested that the accessibility of the degradable ester bonds for different substrate particle sizes was not significantly affected by enzyme affinity to PET, differently from what was seen when PET samples with different crystallinities were evaluated [40]. The proposed kinetic model was able to describe PC-PET erosion since this reaction depends on the substrate surface availability. Thus, different temperatures were also evaluated using this model to investigate this effect on enzymatic PET hydrolysis, which is relevant to take into account in bioreactor design and further scaling-up strategies to implement the enzymatic PET hydrolysis route.

Effect of temperature
It is well known that temperature significantly affects the hydrolysis of PC-PET [20,38,48]. Cutinases must be thermally stable (at temperatures above 60 ºC) for the hydrolysis of semi-crystalline PET [33,[49][50][51]. Such behavior is necessary because the mobility of the polymer chain increases above the glass transition temperature (Tg), which for PC-PET occurs at 78.5 ºC [52] but it may be lowered by approximately 10 °C in aqueous solutions [53,54].
The temperature effect on the hydrolysis rate was addressed in this work through Arrhenius law in order to seek the best temperature range to use the HiC. Thus, several PC-PET concentrations were evaluated at each temperature (40,50,55,60,70, and 80 ºC), and the heterogeneous kinetic model was adjusted, as shown in Figure 5. An increase in the initial rate was observed with the rise of temperature, except at 80 ºC. The highest hydrolysis rate was observed at 70 ºC, closer to the Tg of PET, where chain mobility of the polymer is higher.
Probably, the temperature of 80 ºC led to enzymatic denaturation, even though the literature reports that the optimum temperature range is between 75 and 80 ºC for the hydrolysis of PET using HiC [20,32].

Figure 5
It can be seen from Figure 5 and Table 2  temperatures. These results show that the reaction rate constant (k) has a stronger influence on the overall reaction rate than the adsorption equilibrium constant (K), as reported by Basu and collaborators [55].

Table 2
The availability of thermodynamic reaction parameters is of interest in investigating the structure-reactivity relationship [37]. The enzymatic reaction starts with the initial binding of a substrate by an enzyme, and then, it proceeds by increasing their mutual affinity in water, which leads to a rate enhancement [56]. The enzyme binding interactions include fixation of the reacting groups of the substrate at the active site in the correct position and a destabilization of the ground state that brings a considerable loss of entropy, thus decreasing the activation energy of the reaction [57]. As the enzyme conformation changes during the catalytic event, it would be useful to study structures approaching the transition state of the biocatalyzed reaction.
The transition state itself is too short-lived, but an indirect approach, such as Pauling's transition state stabilization theory, may provide a partial solution to this problem [58]. This theory proposes that enzymes are developed to bind tightly to the transition state rather than the reactant or product. The high affinity of the enzyme active site by transition state results in lowering the activation state.
An exponential approach was adopted to determine the activation parameters, and the results are shown in Figure 6 and Table 3 kJ mol -1 and 10.8 J ± 9.8 mol -1 K -1 , respectively. Wolfenden and collaborators [65] observed that ΔS # is low for hydrolytic reactions, and on the other hand, ΔH # is high and positive for slow reactions that are catalyzed by enzymes. Besides, the entropy contribution (-TΔS # ) implies that the degree of randomness decreases with the adsorption [55] of the enzyme. As both phenomena occur simultaneously, the adsorption step is followed by a considerable release of enthalpy, which results from the increase in substrate affinity by the enzyme [56].
The Circe effect hypothesis postulates that enzymes spend some part of the binding free energy on binding tightly to the substrate [57]. Thus, the reaction progress, that is, the ES complex shifting from the ground state to the transition state, is mainly enthalpic in its origin [56]. It was reported that enthalpy tends to dominate the enzyme thermodynamics on reactions involving single substrates and hydrolytic reactions in which water is present in abundance [65]. Therefore, the activation barrier can be climbed without any entropy loss, producing an increase in the enzymatic reaction rate [66].

Figure 6
Table 3 Thus, the estimation of thermodynamic activation parameters for PC-PET hydrolysis allowed the understanding of the origin of enthalpic contributions to enzymatic rate enhancement. This comprehension will be useful for rationalizing further enzymological experiments on PC-PET hydrolysis and enzyme (HiC) engineering and design.

Conclusion
The

Acknowledgments
The authors thank UFRJ, UERJ, IFRJ, and Petrobras for funding. The authors thank the Professor Marcos Lopes Dias of the Federal University of Rio de Janeiro for kindly providing PC-PET in powder form.        concentration of 1.0 mgprotein mL -1 , and PC-PET with a particle size range 0.075-0.250 mm.