*3.1. Calculation of E*ff*ective Treatment Temperature (T*eff*)*

In this study, *T*eff for HPP treatments on α-Lac and β-Lg was calculated using Equation (1) and tabulated in Table 1. In all cases, *T*eff was significantly lower than the maximum temperature in HPP (*T*max), which further establishes the application of the concept of *T*eff in denaturation studies.


**Table 1.** HPP conditions at different temperatures and the corresponding parameters.

*3.2. Kinetics of* α*-Lac and* β*-Lg Denaturation during HPP Treatments*

Activation energy (*E*a) for α-Lac and β-Lg denaturation was calculated using the results obtained from HPP treated samples following Section 2.6. The reaction orders for HPP-induced denaturation of α-Lac and β-Lg were determined to compare the rate constants (*k*) at different temperatures and pressures, and to calculate the *E*a. Experimental data obtained in this work and their graphical representation in Figures 2 and 3 yielded the reaction order (*n*) as 1 and 2 for α-Lac and β-Lg, respectively. These orders consistently produced reasonably straight lines with good correlation of coefficients (*r*<sup>2</sup> > 0.98) and agree well with previous studies [38–40].

**Figure 2.** High-pressure denaturation of alpha-lactalbumin (α-Lac), HPP applied at ~20 ◦C (**A**), ~30 ◦C (**B**), and ~40 ◦C (**C**). Temperature at all HPP conditions (*T*eff) is presented in Table 1. The concentration of α-Lac is expressed as *c*t/*c*o, where *c*<sup>t</sup> = α-Lac concentration after HPP and *c*<sup>o</sup> = initial α-Lac concentration before HPP.

**Figure 3.** High-pressure denaturation of beta-lactoglobulin (β-Lg), HPP applied at ~20 ◦C (**A**), ~30 ◦C (**B**), and ~40 ◦C (**C**). Temperature at all HPP conditions (*Te*ff) is presented in Table 1. The concentration of β-Lg is expressed as *c*t/*c*o, where *c*t= β-Lg concentration after HPP and *c*<sup>o</sup> = initial β-Lg concentration before HPP.

Furthermore, Figures 4 and 5 represent the effect of HPP on the rate constant (*k*) for the denaturation of α-Lac and β-Lg, respectively. *E*<sup>a</sup> was calculated from the gradients of respective lines from Figure 4 for α-Lac and from Figure 5 for β-Lg. *E*<sup>a</sup> indicates the energy barrier that a protein is required to overcome to take part in a reaction. Values of *E*<sup>a</sup> during HPP are presented in Table 2 where we observed a distinctive higher *E*<sup>a</sup> in β-Lg denaturation compared to α-Lac. These results correspond well with the previous studies investigated by Huppertz et al. [31] for whole milk and Mazri et al. [32] for skim milk.

**Figure 4.** Effect of HPP on the rate of constant (*k*) for denaturation of α-Lac. Temperature at all HPP conditions (*T*eff) is presented in Table 1.

**Figure 5.** Effect of HPP on the rate of constant (*k*) for denaturation of β-Lg. Temperature at all HPP conditions (*T*eff) is presented in Table 1.


**Table 2.** Activation energy (*E*a) after HPP of 300–600 MPa.

However, *E*<sup>a</sup> values obtained in this work for β-Lg denaturation differ to some extent from those reported earlier by Anema et al. [39] from HPP treatments of 200–600 MPa at 10–40 ◦C up to 60 min. They reported *E*<sup>a</sup> as 103.8 and 114.35 kJ/mol for 500 and 600 MPa, respectively; twice higher than the reported values of this work.

IMF is a complex food containing a variety of ingredients. Therefore, this difference in *E*a could be attributed to the dissimilarity in treatment media, treatment duration, and the estimation of treatment temperature, since this study considered the temperature attained during the pressure come-up time.
