## 4 Kinetic Modelling of PPO Inactivation
### Thermal Inactivation Kinetics of PPO  
In terms of kinetic modelling, first-order kinetics adequately described the semi-logarithmic relationship between residual PPO activity and thermal treatment in different fruits, ranging from 60 to 85 \({}^{\circ}\)C. Table 2 shows that, according to the Arrhenius equation, the rate constants increased with temperature. Higher k was obtained for strawberry puree PPO, confirming that the resistance of the fruit enzyme is low. The D-values of PPO are crucial for a better understanding of enzyme stability. When PPO in different fruits was treated either in puree or in juice form, the D-values of PPO decreased as the temperature increased. When comparing the D-value of PPO in different fruits such as sapota (75 \({}^{\circ}\)C, 8 min), bayberry juice (75 \({}^{\circ}\)C, 6 min), pear puree (75 \({}^{\circ}\)C, 121 min), and apple puree (75 \({}^{\circ}\)C, 22 min), it could be concluded that the PPO in bayberry juice is more thermosensitive [28; 36; 39]. The minimum energy required for a chemical reaction is known as activation energy, and it varies depending on the reaction. The susceptibility to temperature changes is indicated by the activation energy (E\({}_{\text{a}}\)). High E\({}_{\text{a}}\) indicates greater changes in k as a function of temperature [28]. These differences in PPO inactivation kinetic parameters could be due to the different fruit sources or the difference between the form of the fruits, such as puree or juice.  
\begin{table}
\begin{tabular}{p{142.3 pt} p{142.3 pt}} \hline \hline
**Models** & **Equations** \\ \hline First-order kinetic model & \(\ln\left(\frac{\text{A}}{\text{A}_{0}}\right)=-\text{k}\,\text{t}\,\,\,\,\, \text{log}\left(\frac{\text{A}}{\text{A}_{0}}\right)=-\frac{t}{D}\) \\  & where A is the enzyme activity of sample at different treatment times (t), A\({}_{0}\) is the initial enzyme activity of the raw unprocessed sample, A/A\({}_{0}\) is the residual activity of the enzyme, k is the first-order inactivation rate constant at the operating conditions (min\({}^{-1}\)), and t is the treatment time (min). \\  & The decimal reduction time (D: the time required to reduce enzyme activity by 90\%) can be calculated from the inactivation rate constant (D = 2.303/k). The negative reciprocal slope of the regression line of log D as a function of T was used to calculate z, which is defined as the temperature increase required to reduce the D value by 90\%. \\  & \(\log\left(\frac{\text{D}_{1}}{\text{D}_{2}}\right)=\frac{\left(\text{T}_{2}- \text{T}_{1}\right)}{\text{z}}\) \\  & Furthermore, the temperature dependence of the PPO inactivation can be described using the linearized version of the Arrhenius equation as follows: \\  & where temperatures T\({}_{2}\) and T\({}_{1}\) correspond to decimal reduction times D\({}_{2}\) and D\({}_{1}\) or constants k\({}_{2}\) and k\({}_{1}\), respectively. E\({}_{\text{a}}\) is the activation energy (kJ/mol), and R is the universal gas constant (8.31 J/(mol-K)). Using linear regression analysis, the activation energy was calculated from the slope of the ln (k) versus 1/T plot. \\ \hline Biphasic model & \(\text{A}=\text{A}_{\text{s}}\exp\left(-\text{k}_{\text{s}}\text{t}\right)+ \text{A}_{\text{L}}\exp\left(-\text{k}_{\text{L}}\text{t}\right)\) \\  & For non-linear data, the biphasic model can be used, where A\({}_{\text{S}}\) and A\({}_{\text{L}}\) are activities of the stable and the labile fractions, respectively, and k\({}_{\text{S}}\) and k\({}_{\text{L}}\) are the inactivation rate constants of stable and labile fractions, respectively. \\ \hline Weibull model & \(\log\left(\frac{\text{A}}{\text{A}_{0}}\right)=-\text{bt}^{n}\) \\  & Weibull model is also suitable for non-linear, where b and n are scale and shape factors, respectively [61]. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Models for inactivation of PPO.