*3.1. Single-Factor Test Results*

Quinoa protein extraction rate increased sharply with prolongation of ultrasonic time and reached a maximum value of 69.07 ± 0.89% at 90 min, and then decreased gradually (Figure 1A). When the ultrasonic time was between 30 and 90 min, cavitation and thermal effects of the ultrasonic wave caused the solvent molecules to rapidly enter the solid matter, resulting in quick dissolution of QPI and increase in extraction rate. When the ultrasonic time exceeded 90 min, the protein structure was destroyed and degraded due to longterm cavitation and thermal effects of the ultrasound. This resulted in protein folding and aggregation due to excessive increase in hydrophobicity, leading to a decline in the dissolution rate of QPI [33]. Therefore, the optimum ultrasonic time was 60–120 min.

With an increase in the amount of solvent, quinoa protein extraction rate rapidly increased initially, reached a maximum value of 71.44 ± 1.49% when the solid-liquid ratio was 1:15, and then decreased slowly (Figure 1B). This trend is observed because the quinoa particles do not fully collide with each other under the action of ultrasonic cavitation when the concentration of quinoa is low. Appropriate solid-liquid ratio enables the quinoa particles to fully accept the force of emptiness and collide strongly with each other, which increases the quinoa protein dissolution and extraction rate [34]. However, excessive quinoa concentration will affect the mass and heat transfer effect of the material, thereby affecting the cavitation effect and decreasing the quinoa protein extraction rate. Therefore, the optimum solid-liquid ratio was 1:10–1:20.

As the ultrasonic temperature increased, quinoa protein extraction rate gradually increased, reached a maximum value of 69.07 ± 0.89% at 45 ◦C, and then decreased rapidly (Figure 1C). This occurs because within a certain range, the increase in temperature is conducive to the acceleration of thermal motion inside the molecule, which increases the protein dissolution and extraction rate. However, very high temperatures damage the integrity of the protein structure and reduce the dissolution and extraction rate [35]. Therefore, 35–55 ◦C was considered as the optimum temperature range.

With increase in pH, quinoa protein extraction rate gradually increased, reached a maximum value of 71.10 ± 0.58% at pH 11.0, and then decreased rapidly (Figure 1D). It was observed that pH changes significantly affected the extraction rate of quinoa protein (*p* < 0.05). This is mostly likely due to the loosening of tight cell structure and destruction of the protein structure in an alkaline environment, thereby increasing the amount of protein eluted [36]. As the pH increased above 11.0, the extraction rate of quinoa protein decreased, which might be due to the destruction of secondary bonds in the protein by strong bases, resulting in denaturation of the protein secondary and tertiary structure [37]. However, previous studies have shown that the higher the pH when extracting QPI, the worse the thermal stability, and the purity of QPI is not high and the color is dark at this time [1]. Therefore, the pH value of the solvent was set to 10.

**Figure 1.** The effects of ultrasonic time (**A**), solid-liquid ratio (**B**), ultrasonic temperature (**C**), pH (**D**) and the response surface methodology and contour plots for the effects of various factors (**E**–**G**) on the extraction rate of QPI. Different letters (a–e) represent significant differences (*p* < 0.05).


Response surface test design and results are shown in Table S2.

#### 3.2.2. Regression Equation Fitting and Analysis of Variance (ANOVA)

A mathematical model with the regression equation was established by statistical analysis of the experimental data: extraction rate of QPI (%) = 73.48 + 1.61A + 2.55B − 0.16C + 3.66AB + 0.66AC +2.09BC − 9.49A<sup>2</sup> − 1.05B<sup>2</sup> − 4.61C2

ANOVA was performed for the regression model (Table S3). *p* < 0.01 indicated that the model equation was highly significant and the model's lack of fit was insignificant. Therefore, the selected model was appropriate. The correlation coefficients were R2 = 0.9895 and RAdj<sup>2</sup> = 0.9759, indicating that the equation fitted the test well, and can effectively reflect the relationship between each factor and its response value. Therefore, it was feasible to use this model equation to predict the process parameters of ultrasound-assisted quinoa protein extraction.

ANOVA of regression equation coefficients showed that the effect of ultrasonic time, the solid:liquid ratio on quinoa protein extraction rate, the interaction terms AB, and the interaction terms BC, were extremely significant; the effect of ultrasonic temperature on the quinoa protein extraction rate and the interaction terms AC was insignificant. The F value represents the degree of influence of ultrasonic time, solid-liquid ratio, and ultrasonic temperature on the extraction rate of QPI. A higher F value indicated greater influence. The order of influence of the three influencing factors on quinoa protein extraction rate, as inferred from the magnitude of F values, was solid-liquid ratio > ultrasonic time > ultrasonic temperature.

#### 3.2.3. Response Surface Analysis of Interaction of Various Factors

Response surface diagram and contour diagram of the interaction among ultrasonic time (A), solid-liquid ratio (B), and ultrasonic temperature (C) were prepared according to the regression equation (Figure 1E–G). A very steep curve was obtained for interactions between ultrasonic time and solid-liquid ratio, and solid-liquid ratio and ultrasonic temperature, indicating that these interactions had extremely significant effects on the extraction rate of QPI. A gentle curve obtained for the interaction between ultrasonic time and temperature indicated that these interactions did not have significant effects on the protein extraction rate. This was consistent with the results of ANOVA of regression equation.

The optimal process parameters of ultrasound-assisted quinoa protein extraction, determined by the mathematical model, were as follows: ultrasonic time 98.51 min, solidliquid ratio 1:20, ultrasonic temperature 46.82 ◦C, and pH 10; the predicted extraction rate of QPI under these conditions was 75.87%. For actual operation, the following conditions were considered: ultrasonic time 99 min, solid-liquid ratio 1:20 g·mL−1, ultrasonic temperature 47 ◦C, and pH 10; under these conditions, the extraction rate could reach 74.67 ± 1.08% and the purity of QPI obtained was 87.17 ± 0.58%. The time taken by the ultrasound-assisted method for quinoa protein extraction was significantly lower than the time taken by the traditional alkaline extraction method. This is probably because during the ultrasound-assisted extraction process, the rapidly formed cavitation bubbles have mechanical, chemical, and thermal effects on the medium, which significantly increases the extraction rate of QPI [38].
