## 2 Materials and Methods
### Experimental Design for Bread Formulation  
The response surface methodology (RSM) is a collection of techniques that allow for the inspection of a response that can be displayed as a surface when the experiments investigate the effect of varying quantitative factors on the values that a dependent variable or response takes [23]. Accordingly, the bread formulation was first optimized through an RSM in which the effect of independent variables (proportion of gluten-free flows, X1; proportion of hydrocolloids, X2; and hydration percentage, X3) was evaluated to determine the most appropriate ratio of ingredients to obtain a gluten-free bread with desirable physical characteristics. For this purpose, a Box-Behnken Design (B-BD) was carried out. The Box-Behnken design is a response surface which designs place points on the midpoints of the edges of the cubical design region, as well as points at the center, circumscribed on a sphere. This design can be applied for the optimization of various chemical and physical processes, where the number of experiments is determined according to the requirements of the process [23]. Thus, experiments, performed in triplicate, were conducted using B-BD (Table 1) with three factors (X1, X2, and X3) at three levels (low (\(-\)1), medium (0) and high (+1)), as follows: X1: 60% (\(-\)1), 80% (0), 100% (+1); X2: 0% (\(-\)1), 50% (0), 100% (+1); X3: 70% (\(-\)1), 95% (0), and 120% (+1). The dependent variables studied in the 13 experiments were the specific volume and several instrumental parameters (hardness, adhesiveness, springiness, cohesiveness, chewiness, resilience, and color). To prepare the bread for these experiments, the flour weight was used as a reference, considering this weight as 100%. Accordingly, a 1.5% (based on flour weight) of both yeast and salt was set, while the percentages of flour, hydrocolloids, and water varied depending on the experiment since they were the factors evaluated in the experimental design.  
A multi-response optimization was accomplished using a desirability function provided by XLSTAT Statistical Software for Microsoft Excel version 2021.1 (Addinsoft, Paris, France). Desirability analysis was employed to assess if a combination of variables satisfied the goal that was defined for the response, using a scale ranging from 0.0 (undesirable) to 1.0 (highly desirable), and finally, the design space of responses was predicted using the regression model equation [24; 25]. Experimental data were fitted to the quadratic model using a second-order polynomial model. Coefficients (linear, quadratic, and interaction) were determined by least squares regression. Analysis of variance (ANOVA) was used to determine the significance and interactions of the factors (\(p<0.05\)). The experimental data obtained were optimized considering the higher specific volume and springiness and the lower hardness, as these are key parameters influencing the quality of bread [26].  
A second-order polynomial model provided by the Box-Behnken design relating the responses to factors X1, X2, and X3 was used to optimize the responses, as follows:  
\[\text{Y}=\text{a}_{0}+\text{a}_{1}\text{X}_{1}+\text{a}_{2}\text{X}_{2}+\text {a}_{3}\text{X}_{3}+\text{a}_{1,2}\text{X}_{1}\text{X}_{2}+\text{a}_{1,3} \text{X}_{1}\text{X}_{3}+\text{a}_{2,3}\text{X}_{2}\text{X}_{3}+\text{a}_{1,1} \text{X}_{1}{}^{2}+\text{a}_{2,2}\text{X}_{2}{}^{2}+\text{a}_{3,3}\text{X}_{3} {}^{2} \tag{1}\]  
where Y = response value, \(a=\) constant coefficients, and X = independent variable.  
The variables individually considered to evaluate the best formulation of gluten-free bread were used as the response (Yi): Y1, specific volume; Y2, springiness; and Y3, hardness. The experimental conditions that independently maximized Y1 and Y2, and minimized Y3 were obtained from the fitted models.