#### 3.1. P–B Experimental Design and Results

The result of the P–B design are shown in

Table 3; Y

_{1} (%) and Y

_{2} (%) are the hydroxyl free radical scavenging rate and DPPH radical scavenging rate, respectively.

X1, X5, X6, X7, X8, X9, X10 represent casein, casein peptone, glucose, soybean peptone, inulin, calcium lactate, and cysteine, respectively, while X2, X3, and X4 are three error terms.

The effects of nutrients were different during fermentation, therefore the antioxidant activity changed under different conditions. As shown in

Figure 1, three variables, i.e., casein peptone (X5), glucose (X6), and calcium lactate (X9), accounted for a large proportion of the percent sum of squares on the Pareto chart for both hydroxyl free radical scavenging rate and DPPH radical scavenging rate. This indicated that the three variables had significant positive effects on the antioxidant activity (

Figure 2). According to the results of the P–B design and principal factor analysis, casein peptone, glucose, and calcium lactate were selected as the main factors for further analysis by the steepest ascent experiment; Y1 (%) and Y2 (%) are the hydroxyl free radical scavenging rate and DPPH radical scavenging rate, respectively.

#### 3.2. The Experimental Design and Results of the Steepest Ascent Experiment

The steepest ascent experiment was preformed to determine the central points of the RSM.

Table 4 lists the design and results of the steepest ascent experiment for nutrients promoting peptide production. Y

_{1} (%) and Y

_{2} (%) represent the hydroxyl free radical scavenging rate and DPPH radical scavenging rate, respectively.

As can be seen from

Table 4, both hydroxyl radical scavenging rate and DPPH radical scavenging rate reached the maximum in the fifth step during the steepest ascent experiments. Therefore, the levels of each factor in step five, which were 1.0% (

w/

v) for calcium lactate, 0.2% (

w/

v) for glucose, and 0.3% (

w/

v) for casein peptone, were used as the center points of the subsequent RSM.

#### 3.3. B–B Experimental Design and Results

B–B experimental design and results are shown in

Table 5. The hydroxyl free radical scavenging rate is represented by Y

_{1} (%), and the DPPH radical scavenging rate is represented by Y

_{2} (%).

The data were analyzed to get a quadratic regression model by using Design Export based on the data from the B–B experiment. A multiple regression equation correlating the response function with the independent variables could be obtained as:

where Y

_{1} and Y

_{2} are the corresponding expected values of the hydroxyl free radical scavenging rate and the DPPH radical scavenging rate, and A, B, and C are the coded values of the independent variables calcium lactate, glucose, and casein peptone, respectively.

Analysis of variance (ANOVA) was used to verify the validity of the model and its parameters according to the significance, as shown in

Table 6. Y

_{1} (%) and Y

_{2} (%) represent the hydroxyl free radical scavenging rate and DPPH radical scavenging rate, respectively.

As shown in

Table 6, a significant value for the hydroxyl free radical scavenging rate (

p = 0.0045 < 0.01) and an insignificant value of the lack of fit (

p = 0.1835 > 0.05) revealed the effectiveness of the regression analysis, which suggested that the regression model could be used to fit the effect of the three factors on hydroxyl free radical scavenging rate. As the ratio of the explained variation to the total variation, the coefficient of determination (R

^{2}) can be used to measure the degree of fit. The value of R

^{2} was 96.3%. This suggested that 96.3% of the response to the hydroxyl free radical scavenging rate was caused by changing the concentration of A, B, and C and by their interactions. Two-dimensional contours revealed that hydroxyl free radical scavenging rate changed with changes in the temperature, whey powder, and calcium lactate concentration, and their corresponding three-dimensional response surface were generated to better determine the interaction of the three variables with the corresponding variables (AB, BC, AC,

Figure 3). The contour plots seemed to be elliptical or nearly circular. This implied that AB, AC, and BC had mutual interactions affecting the hydroxyl free radical scavenging rate, while AB and BC were weak interactors in this respect (

p_{AB} = 0.3442 > 0.05,

p_{BC} = 0.6430 > 0.05). In addition, there was not a simple linear correlation between the variables, calcium lactate, casein peptone, and hydroxyl free radical scavenging rate (

p_{A}^{2} = 0.0012 < 0.01,

p_{C}^{2} = 0.0046 < 0.01).

The ANOVA showed that the model for the DPPH radical scavenging rate had a

p-value <0.01, which was statistically significant (

Table 6). The model equation was corroborated to be a suitable model to describe the value of the DPPH radical scavenging rate, while the lack of fit (

p = 0.0854 > 0.05) was insignificant. The value of R

^{2}, which was 97.96%, indicated that only 2.02% of the variability on the DPPH radical scavenging rate could not be explained by the predicted equation of model. The

p-value of BC was 0.2759, which suggested that A and B had a strong mutual interaction affecting the DPPH radical scavenging rate (

Figure 4).

The maximum responses values of the hydroxyl free radical scavenging rate (88.36%) and DPPH radical scavenging rate (63.79%) were obtained at additive amounts of 0.99% (w/v) calcium lactate, 0.21% (w/v) glucose, and 0.29% (w/v) casein peptone as predicted. The design of the verification experiment was dependent on the optimization results (A = 0.99%, B = 0.21%, C = 0.29%). The results demonstrated that the hydroxyl free radical scavenging rate and DPPH radical scavenging rate were 88.01 ± 0.69% and 63.48 ± 1.22% under the optimum conditions (n = 3). There was no significant difference from the predicted values, indicating the model was appropriate. The control had a scavenging rate of hydroxyl free radical of 56.50 ± 0.57% and a scavenging rate of DPPH radical of 41.97 ± 0.72% without addition of calcium lactate, glucose, and were peptone (n = 3). The hydroxyl free radical scavenging rate and DPPH radical scavenging rate was increased by 31.51% and 21.51%, respectively, after optimization by RSM.