2.6. Optimization Using Central Composite Design (CCD) and Response Surface Modeling (RSM)

To determine the optimum concentration of the most effective factors identified by the OVAT (one-variable-at-a-time) analysis, and to study their interactions, a response surface methodology (RSM) approach, in terms of a central composite design (CCD), was applied. Design-Expert^® Software (Version 9.0.3.1, Stat-Ease Inc., Minneapolis, MN, USA) was used in this study to construct the experimental design and statistically analyze the experimental data. Based on the OVAT results, the basic chemical composition of the medium was fixed as follows (g/L): 15, sucrose; 0.2, FeSO₄·7H₂O; 0.25, KCl added to different concentrations of yeast extract, KH₂PO₄, and MgSO₄·7H₂O. Then, the pre-defined key variables were denoted as follows: yeast extract [X₁], KH₂PO₄ [X₂], and MgSO₄·7H₂O [X₃]. Each variable in the design was studied at five different levels, with all variables taken at a central coded value of zero (Table 1).

Experimental range and levels of the three independent variables for the central composite design (CCD) matrix in terms of coded and actual factors.

* α: axial point.

A factorial experimental design, with an axial point (α = 1.68) and six replicates at the center point, was employed, with a total of 20 experiments (Table 2). The responses Y1, Y2, Y3, refer to the orange pigment yield (OPY), red pigment yield (RPY), and dry biomass concentration (DBW), respectively. A multiple regression analysis of the data was carried out to draw out an empirical model linking the response measured (Y) to the independent variables (X1, X2, X3). The following second-order polynomial equation describes the relationship between the independent variables and allows the calculation of the expected responses (Equation (4)):

where Y is the response; x_i and x_j are variables (i and j range from 1 to k); β₀ is the model intercept coefficient; β_j, β_jj, and β_ij are interaction coefficients of the linear, quadratic and second-order terms, respectively; k is the number of independent parameters (k = 3 in this study); and e_i is the error [74,75].

Central composite experimental design matrix with observed responses over process parameters (OPY, RPY, DBW) in T. albobiverticillius 30548 SmF, using the three independent variables: yeast extract, KH₂PO₄, MgSO_4.7H₂O.

¹: OPY (orange pigment yield at 470 nm in terms of g/L quinizarin equivalent); ²: RPY (red pigment yield at 500 nm in terms of g/L red yeast rice pigment equivalent); ³: DBW (g/L dry biomass concentration in the culture broth).