Thermodynamic and Kinetic Investigation on Aspergillus ficuum Tannase Immobilized in Calcium Alginate Beads and Magnetic Nanoparticles

Tannase from Aspergillus ficuum was immobilized by two different techniques for comparison of kinetic and thermodynamic parameters. Tannase was either entrapped in calcium alginate beads or covalently-immobilized onto magnetic diatomaceous earth nanoparticles. When immobilized on nanoparticles, tannase exhibited lower activation energy (15.1 kJ/mol) than when immobilized in alginate beads (31.3 kJ/mol). Surprisingly, the thermal treatment had a positive effect on tannase entrapped in alginate beads since the enzyme became more solvent exposed due to matrix leaching. Accordingly, the proposed mathematical model revealed a two-step inactivation process. In the former step the activity increased leading to activation energies of additional activity of 3.1 and 26.8 kJ/mol at 20–50 ◦C and 50–70 ◦C, respectively, while a slight decay occurred in the latter, resulting in the following thermodynamic parameters of denaturation: 14.3 kJ/mol activation energy as well as 5.6–9.7 kJ/mol standard Gibbs free energy, 15.6 kJ/mol standard enthalpy and 18.3–29.0 J/(K·mol) standard entropy variations. Conversely, tannase immobilized on nanoparticles displayed a typical linear decay trend with 43.8 kJ/mol activation energy, 99.2–103.1 kJ/mol Gibbs free energy, 41.1–41.3 kJ/mol enthalpy and −191.6/−191.0 J/(K·mol) entropy of denaturation. A 90-day shelf-life investigation revealed that tannase immobilized on nanoparticles was approximately twice more stable than the one immobilized in calcium alginate beads, which suggests its use and recycling in food industry clarification operations. To the best of our knowledge, this is the first comparative study on kinetic and thermodynamic parameters of a tannase produced by A. ficuum in its free and immobilized forms.


Introduction
The immobilization of enzymes, when combined with the fundamental principles of thermodynamics and kinetics, has proven to be a highly useful tool for the successful application of biocatalysis in industrial processes [1,2]. If, on the one hand, immobilization can enhance the stability and reusability of a biocatalyst, on the other hand, knowledge of kinetic and thermodynamic parameters-such as decimal reduction time, half-life, last two years up to the beginning of August 2023 has revealed that there has been an increase in the contributions related to "enzyme immobilization and thermodynamic and kinetic", with no less than 815 research articles (based on the Science Direct search engine: http://www.sciencedirect.com, accessed on 10 August 2023). However, there are a few investigations on the kinetic and thermodynamic parameters of immobilized or free tannases, especially those of the comparative type like this.
Based on this background, the present work aimed to determine the kinetic and thermodynamic parameters of activity and thermal stability of A. ficuum tannase immobilized either in calcium alginate beads or on magnetic nanoparticles. For this purpose, the performance of both preparations was compared for the first time to that of the free enzyme, providing valuable information for a better understanding of reaction mechanism and performance prediction in possible enzyme-based industrial applications.

Results and Discussion
The Aspergillus ficuum tannase was previously immobilized on two supports with different techniques, namely entrapment in calcium alginate beads [19] and covalent immobilization on magnetic nanoparticles composed of polyaniline-coated diatomaceous earth (mDE-PANI-tannase) [21]. In this work, both immobilized-enzyme preparations were subjected to kinetic and thermodynamic modeling to determine the parameters involved in both catalysis and protein thermal denaturation, whose related equations and definitions have been mostly reported for two different enzyme preparations [25,26] and partially summarized in Section 3.3. In addition, a comparative study of such parameters with those of the free enzyme was also performed. Figure 1A illustrates the straight lines obtained by plotting, according to Arrhenius, the experimental results of tannin hydrolysis using free tannase, tannase immobilized in calcium alginate beads and mDE-Pani-tannase. In particular, the straight lines on the right refer to the increase in activity, described by the Arrhenius equation, resulting from a temperature rise from 20 to 40 • C for both immobilized enzyme preparations and from 20 to 30 • C for the free enzyme. This 10 • C difference in the optimal temperature of catalysis (T opt ) can be ascribed to conformational changes that the enzyme underwent due to immobilization in different supports [21]. Free tannase exhibited the highest activation energy of reaction (E) (51 kJ/mol, R 2 = 0.98) followed by tannase immobilized in calcium alginate beads (31.3 kJ/mol, R 2 = 0.96) and by mDE-PANI-tannase (15.1 kJ/mol, R 2 = 0.99). The lowest value of E observed for the enzyme immobilized on magnetic nanoparticles could have been due to the covalent nature of bonds involved in such a type of immobilization, which may have allowed the enzyme to be present only on the support surface and therefore more available for contact with the substrate [14]. As reported by Fernandes et al. [27], lower activation energies are desirable for commercial enzymes, as they imply lower expenses during the process.

Thermodynamic Parameters of Tannase-Catalyzed Reaction
Unlike what occurred for E, the standard enthalpy variation of the enzyme unfolding equilibrium (∆H • u ) [25,26], which was estimated from the slopes of straight lines obtained at temperatures higher than T opt (on the left in Figure 1A), showed an opposite behavior, with the highest value being observed for tannase immobilized in calcium alginate beads (29.6 kJ/mol, R 2 = 0.97), followed by mDE-PANI-tannase (17.6 kJ/mol, R 2 = 0.90) and free tannase (17.6 kJ/mol, R 2 = 0.99). In this case, the higher this value, the greater the stability of the biocatalyst at high temperatures [4]. This outcome corroborates with what was expected from the technique used to immobilize tannase. The entrapment technique confined the enzyme within the matrix, which gave it greater protection against sudden variations or prolonged exposure to a given temperature. As explained better by Boudrant et al. [28], a proper immobilization protocol is important to achieve higher performance of the biocatalist and mainly to enhance its stability. Unlike what occurred for E, the standard enthalpy variation of the enzyme unfolding equilibrium (ΔH°u) [25,26], which was estimated from the slopes of straight lines obtained at temperatures higher than Topt (on the left in Figure 1A), showed an opposite behavior, with the highest value being observed for tannase immobilized in calcium alginate beads (29.6 kJ/mol, R² = 0.97), followed by mDE-PANI-tannase (17.6 kJ/mol, R² = 0.90) and free tannase (17.6 kJ/mol, R² = 0.99). In this case, the higher this value, the greater the stability of the biocatalyst at high temperatures [4]. This outcome corroborates with what was expected from the technique used to immobilize tannase. The entrapment technique confined the enzyme within the matrix, which gave it greater protection against sudden variations or prolonged exposure to a given temperature. As explained better by Boudrant et al. [28], a proper immobilization protocol is important to achieve higher performance of the biocatalist and mainly to enhance its stability.
The entrapment technique imprisons the enzyme molecules inside the support ( Figure 1B, part I), and the pore size influences directly the enzyme activity. Differently from adsorption ( Figure 1B, part II) and covalent immobilization, it does not use chemical bonds or electrostatic forces, and the enzyme is physically trapped in the matrix. On other hand, covalent immobilization ( Figure 1B, part III) resorts to chemical bonds. However, the enzyme structure cannot make a bond with polymer supports like alginate and polyaniline, for which a powerful crosslinking agent such as glutaraldehyde is needed. Glutaraldehyde, a bifunctional compound with an aldehyde group in each extremity, is able to make a chemical bond with amino groups from the side chains ( Figure 1B, part The entrapment technique imprisons the enzyme molecules inside the support ( Figure 1B, part I), and the pore size influences directly the enzyme activity. Differently from adsorption ( Figure 1B, part II) and covalent immobilization, it does not use chemical bonds or electrostatic forces, and the enzyme is physically trapped in the matrix. On other hand, covalent immobilization ( Figure 1B, part III) resorts to chemical bonds. However, the enzyme structure cannot make a bond with polymer supports like alginate and polyaniline, for which a powerful crosslinking agent such as glutaraldehyde is needed. Glutaraldehyde, a bifunctional compound with an aldehyde group in each extremity, is able to make a chemical bond with amino groups from the side chains ( Figure 1B, part III); so, the enzyme is immobilized on the support surface. Both techniques have their advantages depending mainly on the application [14].
It is noteworthy that the parameter ∆H • u refers to the enthalpic energy related to enzyme unfolding, which interferes with the formation of enzyme-substrate complex. As previously described by Abellanas-Perez et al. [29], the enzyme has a specific affinity for a particular substrate, which depends on its tridimensional structure; so, any change in structure leads to a loss in biological activity. As explained by Da Silva et al. [4], the tridimensional structures of enzymes are supported by non-covalent bonds such as hydrogen bonds, Van der Waals forces, dipole-dipole interactions, ion exchange, etc., which are easily broken by motion due to a temperature increase. At low temperatures there is little vibration, and enzyme molecules are present in rigid forms. At higher temperatures, due to some breaks of these bonds, enzyme molecules undergo structural modifications and acquire more malleable conformations, which allows the substrate to more easily fit in the active site and, as a consequence, the affinity to increase. However, when too many bonds are broken simultaneously, the enzyme cannot form its complex with the substrate, and thus a decline in the reaction rate is observed [30].

Thermodynamics of Tannase Thermal Inactivation
As explained earlier, thermodynamics of thermal inactivation of the enzyme is related to energy involved in the loss of protein structure. This occurs when the enzyme is subjected to prolonged exposure to a given temperature. Figure 2A shows the straight lines obtained by plotting the natural logarithm of the free enzyme residual activity (Ψ) as a function of time at different temperatures in order to determine the related constants of thermal inactivation (denaturation) (k d ). These k d values were later used to estimate the activation energy of free enzyme thermal inactivation (E d = 53.5 kJ/mol) through the Arrhenius equation ( Figure 2B). The straight lines showed high determination coefficients (R 2 ), and the values of kinetic and thermodynamic parameters of free tannase denaturation are listed in Table 1.

Free Tannase Thermal Inactivation
structure leads to a loss in biological activity. As explained by Da Silva et al. [4], the tridimensional structures of enzymes are supported by non-covalent bonds such as hydrogen bonds, Van der Waals forces, dipole-dipole interactions, ion exchange, etc., which are easily broken by motion due to a temperature increase. At low temperatures there is little vibration, and enzyme molecules are present in rigid forms. At higher temperatures, due to some breaks of these bonds, enzyme molecules undergo structural modifications and acquire more malleable conformations, which allows the substrate to more easily fit in the active site and, as a consequence, the affinity to increase. However, when too many bonds are broken simultaneously, the enzyme cannot form its complex with the substrate, and thus a decline in the reaction rate is observed [30].

Thermodynamics of Tannase Thermal Inactivation
As explained earlier, thermodynamics of thermal inactivation of the enzyme is related to energy involved in the loss of protein structure. This occurs when the enzyme is subjected to prolonged exposure to a given temperature. Figure 2A shows the straight lines obtained by plotting the natural logarithm of the free enzyme residual activity (Ψ) as a function of time at different temperatures in order to determine the related constants of thermal inactivation (denaturation) (kd). These kd values were later used to estimate the activation energy of free enzyme thermal inactivation (Ed = 53.5 kJ/mol) through the Arrhenius equation ( Figure 2B). The straight lines showed high determination coefficients (R²), and the values of kinetic and thermodynamic parameters of free tannase denaturation are listed in Table 1.    The activation Gibbs free energy of enzyme denaturation (∆G d ), which reflects the amount of remaining energy in the protein structure after exposure to a certain temperature [31], increased from 100.15 kJ/mol at 30 • C to 103.54 kJ/mol at 50 • C, thereby highlighting a certain thermal stability. This result suggests that, within the tested temperature range and exposure time (120 min), free tannase denaturation was likely to be reversible, as better explained in Section 3.3. Such an assumption is corroborated by the negative values of activation entropy of free tannase denaturation listed in Table 1 (∆S d ).

Free Tannase Thermal Inactivation
In fact, according to Wahba et al. [23], negative entropy values suggest greater enzyme stability and that the irreversible denaturation process is a further step forward involving higher temperatures and longer times.
Consistently, the enthalpy of denaturation (∆H d ), which reflects the amount of broken non-covalent bonds, showed only a slight variation. In fact, considering that the energy to break a non-covalent bond can be estimated at about 5.4 kJ/mol [30], in the temperature range under investigation only 9 non-covalent bonds were likely to be broken in free tannase, a number that does not seem to have been sufficient for the denaturation process to become irreversible. In this respect, it is important to notice that enzymes with thermostability characteristics in their free form are more suitable to be immobilized in supports, as they can be recovered for later applications.
Other kinetic parameters evaluated for free tannase were the half-life (t 1/2 ) and the decimal reduction time (D-value), which characterize the thermal inactivation behavior at each temperature to which an enzyme is exposed [24]. Both decreased by increasing temperature, showing the temperature influence in reducing the overall activity due to thermal inactivation. The thermal resistance constant (Z-value), a parameter of sensitivity to temperature variation, was then estimated from the slope of thermal-death-time plot of logD-value versus temperature. It revealed that an increase of about 19 • C is required for D-value to be reduced by one log cycle, i.e., by 90% [14].

Thermal Inactivation of Tannase Immobilized on Magnetic Nanoparticles
The kinetic and thermodynamic parameters of thermal inactivation of mDE-PANItannase were close to those found for the free enzyme. As can be seen in Figure 3, the semi-log plots used to determine the values of k d ( Figure 3A) and E d ( Figure 3B) showed high R 2 values ( Table 2), demonstrating that mDE-PANI-tannase behavior can be described by the equations outlined in Section 3.3. However, this tannase immobilization technique reduced the value of E d to 43.9 kJ/mol, i.e., by approximately 10 kJ/mol compared to the free enzyme. Considering that the higher this parameter, the greater the stability of a biocatalyst, such a reduction suggests that the enzyme was slightly more sensitive to temperature variations [14].
(ΔSd). In fact, according to Wahba et al. [23], negative entropy values suggest greater enzyme stability and that the irreversible denaturation process is a further step forward involving higher temperatures and longer times.
Consistently, the enthalpy of denaturation (ΔHd), which reflects the amount of broken non-covalent bonds, showed only a slight variation. In fact, considering that the energy to break a non-covalent bond can be estimated at about 5.4 kJ/mol [30], in the temperature range under investigation only 9 non-covalent bonds were likely to be broken in free tannase, a number that does not seem to have been sufficient for the denaturation process to become irreversible. In this respect, it is important to notice that enzymes with thermostability characteristics in their free form are more suitable to be immobilized in supports, as they can be recovered for later applications.
Other kinetic parameters evaluated for free tannase were the half-life (t1/2) and the decimal reduction time (D-value), which characterize the thermal inactivation behavior at each temperature to which an enzyme is exposed [24]. Both decreased by increasing temperature, showing the temperature influence in reducing the overall activity due to thermal inactivation. The thermal resistance constant (Z-value), a parameter of sensitivity to temperature variation, was then estimated from the slope of thermal-death-time plot of logD-value versus temperature. It revealed that an increase of about 19 °C is required for D-value to be reduced by one log cycle, i.e., by 90% [14].

Thermal Inactivation of Tannase Immobilized on Magnetic Nanoparticles
The kinetic and thermodynamic parameters of thermal inactivation of mDE-PANItannase were close to those found for the free enzyme. As can be seen in Figure 3, the semi-log plots used to determine the values of kd ( Figure 3A) and Ed ( Figure 3B) showed high R² values ( Table 2), demonstrating that mDE-PANI-tannase behavior can be described by the equations outlined in Section 3.3. However, this tannase immobilization technique reduced the value of Ed to 43.9 kJ/mol, i.e., by approximately 10 kJ/mol compared to the free enzyme. Considering that the higher this parameter, the greater the stability of a biocatalyst, such a reduction suggests that the enzyme was slightly more sensitive to temperature variations [14].   This behavior is corroborated by the decrease of Z-value from 19 to 16 • C. The values of the other kinetic denaturation parameters, namely, D-value and t 1/2 ( Table 2), were also smaller than those of free tannase (Table 1), confirming the lower thermostability of this enzyme preparation.
∆G d increased from 99.25 to 103.08 kJ/mol with increasing temperature from 30 to 50 • C ( Table 2), suggesting that tannase maintained its reversible profile of thermal denaturation after immobilization on magnetic nanoparticles. This occurrence is consistent with the results of de Lima et al. [21], who reported that tannase immobilized on magnetic diatomaceous earth nanoparticles coated with polyaniline could be reused up to 10 times losing only 34% of its initial activity. ∆S d also had negative values (−191.58/−191.04 J/(K·mol)), which are characteristic of reversible inactivation. Finally, ∆H d (41.17-41.33 kJ/mol) was reduced by almost 10 kJ/mol compared to the free enzyme, which corresponds to the breakdown of approximately eight non-covalent bonds.

Thermal Inactivation of Tannase Immobilized in Calcium Alginate Beads
Unlike what was expected from a typical thermal stability study, A. ficuum tannase entrapped in calcium alginate beads exhibited an unusual behavior throughout the whole tested temperature range (20-70 • C). While the free form of tannase and the mDE-PANItannase showed the characteristic decay profile of protein denaturation (Figures 2A and 3A), the activity of tannase entrapped in alginate beads, after an initial increase until a maximum value after about 30 min, progressively decreased still keeping above its starting value (Figure 4). This behavior is not commonly observed. Only Rodríguez-Lopez et al. [32] reported the same behavior for mushroom polyphenol oxidase upon inactivation using an 80 • C hot water bath and 22.6 W/cm 3 microwave power irradiation. However, there is no mathematical model available in the literature on other biocatalysts that acted in the same or similar way. enzyme preparation.
ΔGd increased from 99.25 to 103.08 kJ/mol with increasing temperature from 30 to 50 °C ( Table 2), suggesting that tannase maintained its reversible profile of thermal denaturation after immobilization on magnetic nanoparticles. This occurrence is consistent with the results of de Lima et al. [21], who reported that tannase immobilized on magnetic diatomaceous earth nanoparticles coated with polyaniline could be reused up to 10 times losing only 34% of its initial activity. ΔSd also had negative values (−191.58/−191.04 J/(K•mol)), which are characteristic of reversible inactivation. Finally, ΔHd (41.17-41.33 kJ/mol) was reduced by almost 10 kJ/mol compared to the free enzyme, which corresponds to the breakdown of approximately eight non-covalent bonds.

Thermal Inactivation of Tannase Immobilized in Calcium Alginate Beads
Unlike what was expected from a typical thermal stability study, A. ficuum tannase entrapped in calcium alginate beads exhibited an unusual behavior throughout the whole tested temperature range (20-70 °C). While the free form of tannase and the mDE-PANItannase showed the characteristic decay profile of protein denaturation (Figures 2A and  3A), the activity of tannase entrapped in alginate beads, after an initial increase until a maximum value after about 30 min, progressively decreased still keeping above its starting value (Figure 4). This behavior is not commonly observed. Only Rodríguez-Lopez et al. [32] reported the same behavior for mushroom polyphenol oxidase upon inactivation using an 80 °C hot water bath and 22.6 W/cm 3 microwave power irradiation However, there is no mathematical model available in the literature on other biocatalysts that acted in the same or similar way.  Therefore, a new mathematical approach to the kinetic and thermodynamic parameters of the biocatalyst entrapped in calcium alginate beads was proposed in the present study. Since the calcium alginate beads were porous polymeric spheres where tannase had been entrapped, they may have suffered leaching due to exposition to temperatures higher than the optimum one, which may have partially degraded their polymeric structure and increased pore size ( Figure 5, part II). Obviously, a similar mechanism can be proposed for the action of other external agents such as suboptimal pH, chemical agents and so on. Therefore, the contact between enzyme and outer environment, i.e., the solvent in the case of residual activity tests performed in this study at temperatures higher than that optimal one or reaction medium in industrial applications, became progressively more effective along the starting 30 minutes. However, when the enzyme was exposed for a longer period, its structure began to be denatured, and there was a decrease in its activity ( Figure 5, Part III). Nonetheless, the residual activity still remained above the initial one. Calcium alginate probably conferred extra protection to tannase, so that an activity lower than the initial one would have required exposition longer than 100 min.
in the case of residual activity tests performed in this study at temperatures higher than that optimal one or reaction medium in industrial applications, became progressively more effective along the starting 30 minutes. However, when the enzyme was exposed for a longer period, its structure began to be denatured, and there was a decrease in its activity ( Figure 5, Part III). Nonetheless, the residual activity still remained above the initial one. Calcium alginate probably conferred extra protection to tannase, so that an activity lower than the initial one would have required exposition longer than 100 min. Figure 5. Mechanism proposed in this study for the degradation of calcium alginate beads exposed to temperatures higher than the optimum one.
As mathematically described by Equation (5) onwards in Section 3.3, the kinetic constant of the resultant of these two phenomena is the sum of both positive (kL) and negative (kd) contributions in terms of residual activity. The values of these constants are gathered in Table 3 along with the respective determination coefficients. It is possible to observe in Figure 4 that the straight lines describing the variations of lnΨ over time did not obey the traditional one-step decreasing profile as the temperature was raised from 20 to 70 °C. Since these constants are related to the activation energies Figure 5. Mechanism proposed in this study for the degradation of calcium alginate beads exposed to temperatures higher than the optimum one.
As mathematically described by Equation (5) onwards in Section 3.3, the kinetic constant of the resultant of these two phenomena is the sum of both positive (k L ) and negative (k d ) contributions in terms of residual activity. The values of these constants are gathered in Table 3 along with the respective determination coefficients. Table 3. Kinetic constants describing the activity increase due to matrix leaching (k L ) and activity decrease due to subsequent denaturation (k d ) of tannase entrapped in calcium alginate beads. It is possible to observe in Figure 4 that the straight lines describing the variations of lnΨ over time did not obey the traditional one-step decreasing profile as the temperature was raised from 20 to 70 • C. Since these constants are related to the activation energies involved in the two proposed phenomena, it is possible to identify in Figure 6A two different regions, the former between 20 and 50 • C (straight line on the right) and the latter between 50 and 70 • C (straight line on the left). Thus, it was possible to estimate two activation energies linked to the increase in tannase activity resulting from matrix leaching (E L1 = 3.1 kJ/mol and E L2 = 26.8 kJ/mol, respectively) ( Figure 6A) and an activation energy linked to the decrease in activity due to denaturation under prolonged exposure (E d = 14.3 kJ/mol) ( Figure 6B).
As is well known, the Gibbs free energy variation is a parameter that measures the spontaneity of any process, in that, positive values are characteristic of a spontaneous process, negative values of a non-spontaneous process and a null value of a process in thermodynamic equilibrium. As can be observed in Table 4, the standard Gibbs energy variations related to both the activity increase due to leaching of calcium alginate beads (∆G • L ) and to the activity decrease due to denaturation of the entrapped enzyme after prolonged exposure (∆G • d ) were negative, demonstrating that the increase in biocatalyst activity was a spontaneous phenomenon characteristic of the support outwear. However, the values of ∆G • that referred to the combination of the two phenomena (∆G • RT ) were positive, suggesting that the increase in activity was a non-spontaneous phenomenon forced by the leaching process.
involved in the two proposed phenomena, it is possible to identify in Figure 6A two different regions, the former between 20 and 50 °C (straight line on the right) and the latter between 50 and 70 °C (straight line on the left). Thus, it was possible to estimate two activation energies linked to the increase in tannase activity resulting from matrix leaching (EL1 = 3.1 kJ/mol and EL2 = 26.8 kJ/mol, respectively) ( Figure 6A) and an activation energy linked to the decrease in activity due to denaturation under prolonged exposure (Ed = 14.3 kJ/mol) ( Figure 6B). Figure 6. Semi-log plots used to estimate the activation energies linked to the increase in activity resulting from matrix leaching (EL1 and EL2) (A) and to the decrease in activity due to enzyme denaturation under prolonged exposure (Ed) (B). Enzyme system: tannase entrapped in calcium alginate beads.
As is well known, the Gibbs free energy variation is a parameter that measures the spontaneity of any process, in that, positive values are characteristic of a spontaneous process, negative values of a non-spontaneous process and a null value of a process in thermodynamic equilibrium. As can be observed in Table 4, the standard Gibbs energy variations related to both the activity increase due to leaching of calcium alginate beads (ΔG°L) and to the activity decrease due to denaturation of the entrapped enzyme after prolonged exposure (ΔG°d) were negative, demonstrating that the increase in biocatalyst activity was a spontaneous phenomenon characteristic of the support outwear. However, the values of ∆G° that referred to the combination of the two phenomena (∆G°RT) were positive, suggesting that the increase in activity was a non-spontaneous phenomenon forced by the leaching process.
Standard enthalpy variations related to increased activity due to support leaching (∆H°L) and decreased activity due to denaturation after prolonged exposure (∆H°d) did not vary with rising temperature, being 29.9 and 14.3 kJ/mol, respectively, while the one that referred to the combination of the two phenomena (∆H°RT) was 15.6 kJ/mol in the entire temperature range studied. This value corresponds approximately to the breaking of only three non-covalent bonds after 90 min of biocatalyst exposure. Finally, the fact that the related entropy variations were positive and very small (18.2 ≤ ΔS°RT ≤ 29.0 J/(K•mol)) confirms that, despite representing an irreversible denaturation process, such thermal inactivation was poorly significant and that the biocatalyst had excellent thermostability.

Inactivation of Immobilized Enzyme under Storage Conditions
Tannase immobilized either on magnetic nanoparticles or in calcium alginate beads was finally subjected to a shelf-life study to determine the kinetic parameters often used to characterize the stability of enzyme preparations to be used in industrial applications, i.e., the half-life (t1/2) and the decimal reaction time (D-value).  Table 4. Thermodynamic parameters referred to the activity of tannase entrapped in calcium alginate beads. Parameters referred to the increase in activity resulting from matrix leaching are denoted by the subscript "L", those referred to the decrease in activity due to enzyme denaturation under prolonged exposure by the subscript "d" and those referred to combination of both phenomena by the subscript "RT". Standard variations of Gibbs free energy (∆G • ), enthalpy (∆H • ) and entropy (∆S • ). 1 ∆H • L was calculated as the sum (E L ) of both activation energies linked to the activity increase resulting from matrix leaching (E L1 ) and activity decrease due to denaturation after prolonged exposure (E L2 ) ( Figure 6A and Equation (21)).

Temperature ( • C)
Standard enthalpy variations related to increased activity due to support leaching (∆H • L ) and decreased activity due to denaturation after prolonged exposure (∆H • d ) did not vary with rising temperature, being 29.9 and 14.3 kJ/mol, respectively, while the one that referred to the combination of the two phenomena (∆H • RT ) was 15.6 kJ/mol in the entire temperature range studied. This value corresponds approximately to the breaking of only three non-covalent bonds after 90 min of biocatalyst exposure. Finally, the fact that the related entropy variations were positive and very small (18.2 ≤ ∆S • RT ≤ 29.0 J/(K·mol)) confirms that, despite representing an irreversible denaturation process, such thermal inactivation was poorly significant and that the biocatalyst had excellent thermostability.

Inactivation of Immobilized Enzyme under Storage Conditions
Tannase immobilized either on magnetic nanoparticles or in calcium alginate beads was finally subjected to a shelf-life study to determine the kinetic parameters often used to characterize the stability of enzyme preparations to be used in industrial applications, i.e., the half-life (t 1/2 ) and the decimal reaction time (D-value).

Materials and Methods
Commercial tannase (300 U/g) from Aspergillus ficuum used in this study was purchased from Sangherb Bio-Tech (Xi'an, China). It was immobilized either in calcium alginate beads or on magnetic nanoparticles.

Immobilization in Calcium Alginate Beads
Tannase was entrapped in calcium alginate beads as described by de Lima et al. [19]. Briefly, 10 mL of A. ficuum tannase solution (6 mg/mL, corresponding to enzyme activity of 127.5 U/mL) in 0.2 M acetate buffer, pH 5.0, were mixed with 30 mL of 3.0% (w/v) sodium alginate solution and dropped in 60 mL of 2.0% (w/v) CaCl2 solution at 4 °C under stirring for beads formation. The resulting tannase-loaded beads, with approximately 0.4 mm mean diameter, were left to cure in the same CaCl2 solution for 4.0 h, collected, washed twice with distilled water, suspended in the above working buffer and finally stored at 4 °C for further use.

Materials and Methods
Commercial tannase (300 U/g) from Aspergillus ficuum used in this study was purchased from Sangherb Bio-Tech (Xi'an, China). It was immobilized either in calcium alginate beads or on magnetic nanoparticles.

Immobilization in Calcium Alginate Beads
Tannase was entrapped in calcium alginate beads as described by de Lima et al. [19]. Briefly, 10 mL of A. ficuum tannase solution (6 mg/mL, corresponding to enzyme activity of 127.5 U/mL) in 0.2 M acetate buffer, pH 5.0, were mixed with 30 mL of 3.0% (w/v) sodium alginate solution and dropped in 60 mL of 2.0% (w/v) CaCl 2 solution at 4 • C under stirring for beads formation. The resulting tannase-loaded beads, with approximately 0.4 mm mean diameter, were left to cure in the same CaCl 2 solution for 4.0 h, collected, washed twice with distilled water, suspended in the above working buffer and finally stored at 4 • C for further use.

Immobilization on mDE-PANI Nanoparticles
Diatomaceous earth (DE) particles were magnetized (mDE) according to Cabrera et al. [20]. A black precipitate was collected and coated with polyaniline (PANI) as follows. The precipitate was submitted to treatment with 0.1 M KMnO 4 solution at 25 • C for 60 min, and nanoparticles were washed with distilled water and immersed into 0.25 M aniline solution prepared in 2.0 M HCl at 4 • C for 30 min. Then, they were washed with distilled water and 1.0 M HCl to remove residual aniline using an external 0.6-T magnetic field (Ciba Corning Medical Diagnostics, Walpole, MA, USA). Finally, mDE-PANI nanoparticles were washed several times, dried at 50 • C and stored at 25 • C for later use. mDE-PANI nanoparticles were activated with glutaraldehyde (2.5% (v/v)) by stirring at 25 • C for 2 h and washed several times with distilled water for eliminating unreacted glutaraldehyde. Tannase from A. ficuum prepared in 0.2 M sodium acetate buffer (1 mL of 3 mg/mL solution, corresponding to 342 U/mg specific activity), pH 5.0, was incubated with mDE-PANI nanoparticles (0.05 g) at 4 • C under mild stirring for 20 h. Then, tannase immobilized on mDE-PANI nanoparticles (mDE-PANI-tannase) was collected by an external 0.6-T magnetic field, and the supernatants were used for protein determination according to the Bradford method [33] using bovine serum albumin as a standard. The immobilized derivatives were stored in sodium acetate buffer at 4 • C until use.

Determination of Free and Immobilized Tannase Activity
Tannase activity was measured according to Pinto et al. [34]. Briefly, 0.1 mL of the free enzyme solution in 0.2 M acetate buffer, pH 5.0, or 15 calcium alginate beads with immobilized tannase or 100 µg of tannase immobilized on 0.05 g mDE-PANI were added to 2.0 mL of 0.05% (w/v) tannic acid solution. After homogenization for 7 min at 30 • C, 100 µL of this reaction mixture were added to 150 µL of ethanolic rhodanine solution (0.667% w/v) and allowed to react for 5 min. Then, 100 µL of 0.5 M KCl and, after 2.5 min, 2.15 mL of distilled water were added to the mixture. The formation of a complex with maximum absorbance at 520 nm was followed by means of a UV-Vis spectrophotometer, model Lambda 25 (Perkin Elmer, Wellesley, MA, USA). One unit of tannase was defined as the amount of enzyme necessary to obtain 1 µmol of gallic acid per minute under the assay conditions (30 • C for 7 min). Control experiments were also performed using immobilized beads without tannase [19].

Thermodynamic Modeling
Tannase, either free or immobilized in calcium alginate beads or magnetic nanoparticles, was incubated for 2 h in 0.2 M sodium acetate buffer, pH 5.0, at various temperatures (from 20 to 90 • C), and the residual enzyme activities were determined as described in Section 3.2.
Activation energy of the enzyme-catalyzed reaction (E) and standard enthalpy variation of the enzyme unfolding equilibrium (∆H • u ) were estimated from semi-log plots of the starting enzyme activity (A i ) versus the reciprocal temperature (1/T) in the temperature range of 20-70 • C. In particular, as described elsewhere [25,26], the linearized log form of the Arrenhius equation was used to estimate E in the temperature ranges of 20-40 • C for immobilized enzyme preparations and 20-30 • C for free tannase: while ∆H • u was estimated from the slope of the left straight line of the Arrhenius-type plot of lnA i at temperatures higher than the optimal ones (30 and 40 • C, respectively).
Thermal inactivation of most free or covalently immobilized enzymes can be described by the following general deactivation model [35]: where N is the native biomolecule in its totally active state, while U and D the biomolecule in its reversibly unfolded and ultimate irreversibly denatured states, while k' 1 , k' 2 and k' 3 are the rate constants of enzyme unfolding, folding and denaturation, respectively. This was the case of both free tannase and tannase immobilized on magnetic nanoparticles, for which the traditional kinetic and thermodynamic approach proposed for other enzyme systems was successful [25,26].
However, after immobilization some enzymes may become more thermostable due to variations in their tertiary structure induced by crosslinking agents or the matrix itself. Since the entrapment of enzymes in calcium alginate beads implies their immobilization within the matrix during the gelation process ( Figure 5, part I), the access of substrate takes place through the polymer pores. Organic polymers like this may be subject to degradation when exposed to chemical, physical or microbiological agents. In this case, their pores may increase in size and number, leaving the immobilized enzyme, i.e., tannase in this study, more exposed to the reaction medium and allowing the substrate to enter the pores more easily ( Figure 5, part II). Equation (2) can then be rewritten in order to take into account such an increase in enzyme activity due to exposure of the polymeric matrix to these agents: where I and A are the immobilized-enzyme preparations at the beginning and after exposure, k o is the rate constant of this phenomenon, while k 1 , k 2 and k 3 are the new unfolding, folding and denaturation constants.
Assuming that polymer degradation is an irreversible and first-order process, enzyme activity tends to increase over time: where k is the overall kinetic constant and A the enzyme activity. Because enzymes are molecules with biological activity, they progressively lose their activity due to the denaturation process. In fact, when exposed for long time to agents such as high temperature in the present case, non-covalent bonds can be broken, and their three-dimensional structure becomes predominantly unfolded ( Figure 5, part III). Normally, as is the case of free tannase and tannase immobilized on magnetic nanoparticles, this decay shows a linear trend and can be treated as an irreversible first-order reaction. On the other hand, in the case of tannase entrapped in calcium alginate beads, k is influenced by these opposite contributions, i.e., the activity increase resulting from polymer degradation, described by the degradation constant (k L ), and the activity reduction due to enzyme denaturation described by the denaturation constant (k d ): The contribution of the former phenomenon is more pronounced than that of the latter when k > 0, and vice versa when k < 0. To better understand the meaning of this constant, it is important to remember that the enzyme is in a thermodynamic folding/unfolding equilibrium, which is governed by k 1 and k 2 , while the formation of final product is governed by k 3 . The overall kinetic constant can then be written as follows [36]: By equaling Equations (5) and (6), we obtain the Equation: that simplifies to Equation (8) when k 3 k 2 : where K eq is the constant of the folding/unfolding equilibrium. Considering that k 3 = k B T h [37], one can write: where T is the absolute temperature, h the Planck's constant, and k B the Boltzmann's constant.
To describe the thermal inactivation kinetics, Ortega et al. [38] proposed a multifraction inactivation model, which supposes the existence of multiple enzyme fractions, each of which can be independently analyzed with first-order kinetics. Considering the well-known relationship between the standard variation of Gibbs free energy (∆G • ) and K eq : and applying the approach of Ortega et al. [38] to the above two contributions governed by k L and k d , we can calculate this parameter for the tannase system under investigation (∆G • TR ) as: Considering that: under the equilibrium conditions described above, ∆G • TR can also be described by the equation: By equaling the right members of Equations (11) and (13), we obtain: RT ln ln By differentiating Equation (16) with respect to T, we can write: Applying the Arrhenius equation to the two above-mentioned contributions governed by k L and k d , we can write the equation: where E L and E d are their respective activation energies, while A o and B o are the corresponding pre-exponential factors. The linearized version of this equation: can be differentiated with respect to T to omit the constant term, thus leading to the equation: Equaling the right terms of Equations (17) and (20), the enthalpy of the reaction catalyzed by tannase entrapped in calcium alginate beads can be calculated as the difference of E d and E L : Considering Equation (12), one can calculate the entropy of tannase-catalyzed reaction for this enzyme preparation by the equation:

Shelf Life Stability of the Immobilized Enzyme Preparations
Storage stability of both free and immobilized tannase, with starting activity of about 170 U/mL, was checked by determining the residual activity after long-time storage (90 days) at 4 • C. Free tannase was stored as solution in 0.2 M acetate buffer, pH 5.0, while the immobilized one was stored in wet form. Enzyme activity was determined at regular time intervals (15 days). The kinetic shelf-life parameters of tannase immobilized either in mDE-PANI or in calcium alginate beads were evaluated for 90 days to 4 • C. The enzyme activity was performed every 15 days using tannic acid as a substrate as described in Section 3.2. k d , t 1/2 and D-value were determined as described by Silva et al. [14].

Conclusions
The Aspergillus ficuum tannase, which was immobilized either by entrapment in alginate calcium beads or covalently on magnetic nanoparticles, showed good stability when subjected to different temperatures and exposed to the solvent for a long period. The thermodynamic analysis of the reaction revealed that tannase immobilized on nanoparticles had the lowest activation energy and, therefore, would be the most appropriate enzyme preparation to conduct low-cost industrial tannin degradation treatments. However, thermodynamics and kinetics of biocatalyst denaturation showed that, although the enzyme either in its free or immobilized form is subject to a reversible denaturation mechanism, calcium alginate immobilization ensured greater stability for longer. Using this entrapment technique, tannase hydrolytic activity was increased due to leaching of support accompanied by pore enlargement, which, in addition to allowing greater biocatalyst contact with the substrate, provided greater protection against thermal inactivation after 90 min of incubation. Finally, shelf-life tests performed on immobilized biocatalysts at 4 • C for 90 days revealed that the enzyme immobilized on magnetic nanoparticles kept its activity for almost twice the time as the enzyme entrapped in calcium alginate beads. Such findings suggest that this tannase in both immobilized forms, which showed great potential for tannin degradation and thermal stability, could be profitably exploited for applications in the food industry.