The HRs, obtained by genetic transformation and selection, were subcultured in the same conditions as described previously [12].

The use of the desirability function enabled the Jasmonic Acid concentration (JAC) and the exposure time (ET) to be optimized. The optimal JA concentration of 0.06 mM with an exposure time of 24 h was the optimal compromise. The elicitation treatment was initiated 24 h before harvesting the HRs line and was removed for analysis on the 28th day of culture. This length of time was identified as the best stage for elicitation for the highest hyoscyamine content [12].

The solution of Jasmonic Acid (JA) [(−)-jasmonic acid from Sigma-Aldrich] was prepared by dissolving Jasmonic Acid in an adequate volume of ethanol. The solution was filtered through membrane filter (pore size: 0.2 mm Nalgene) and completed with sterile distilled water. The HRs were elicited with this stock solution at 0.06 mM.

The controls were the same HRs lines in the same conditions of culture but without elicitation (ethanol-water solution without Jasmonic Acid). All cultures were conducted in Petri dishes containing 20 mL of B5 medium, in darkness at 26 ±1 °C.

Alkaloids were extracted using a method described by Amdoun et al. [12]. Hyoscyamine was analyzed using the GC-MS method previously reported by Kartal et al. [21].

The response surface methodology (RSM) consists of an adjustment of empirical models to the data obtained experimentally. Linear (1st degree) or quadratic (2nd degree) mathematical models are employed to describe the system to be optimized [22]. When several factors influence a particular system, it is impossible to screen and to control the contribution of each factor so only those factors which have a major influence must be considered. The screening design, such as the factorial designs 2^k, can be used to meet this objective. They are efficient and economical [15].

To evaluate the form of the true response, a second degree model is used. The factorial designs 2^k are used to determine the first-order effects but these are inefficient when additional effects, like the second-order effects, are so important. Experimental central points in the factorial design 2^k can be added to evaluate the shape. The polynomial function must contain other factors, which include the interaction between the various experimental variables. To determine a critical point (maximum, minimum or saddle), the polynomial function of second degree must contain quadratic factors.

The model of second degree is given by equation (1), where Y is the response, α ₀ is the intercept of the y axis, α_j , α_jj ,…, α_jl are the various coefficients of the model (linear and quadratic), X_j and X_l are the independent variables (factors) and ε is the error of model with ∀i, V(ε_i ) = σ ² (homoscedasticity) and ε ∼ N (0, σ ²) (normal distribution). (1) Y = α 0 + ∑ j = 1 k α j X j + ∑ j = 1 k α j j X j 2 + ∑ j < 1 ∑ l = 2 k α j L X j X l + ɛ

The equation system (1) can be resolved by the Method of Least Squares (MLS) and may be written in matrix form (2) [23], where y is the (n, 1) vector of measured responses, X is the (n, p) matrix of the mathematical model of factor levels, and α is the (p, 1) vector of unknown coefficients. (2) [ y 1 y 2 . . . y n ] ︸ y = [ 1 x 11 . . . x 1 k 1 x 21 . . . x 2 k . . . . . . . . . . . . . . . . . . 1 x n 1 x n 2 . . x n k ] ︸ X [ α 0 α 1 . . . α k ] ︸ α + [ ɛ 1 ɛ 2 . . . ɛ n ] ︸ ɛ

With n responses, and y and p coefficients in the model, there are n equations and n + p unknowns. To solve the system, the Method of Least Squares is used. After some calculations, the solution of system (2) is given by Equation (3): (3) a ^ = ( t X X ) − 1 t X Y

The low and high levels of variables (factors) are denoted by −1 and 1, respectively. The levels of X_i variable are coded and obtained from Equation (4), where X_i is the independent variable coded value, A_i is the independent variable real value, A ₀ is the independent variable real value on the center point and ΔA_i is the step change value.

The most used second-order symmetric designs for the RSM are: The general factorial design, the Box-Behnken design, the Central Composite Design and Doehlert’s design. These differ by the location of the experimental points in the studied region, the number of factor levels kept, the number of experiments and the blocks [14].

When the relationship between the variables and the response has been established by the modeling, predictions can be made. However, the mathematical model obtained after adjustments to experimental results, sometimes cannot describe the studied domain. It is necessary to analyze and examine the diagnostic of the obtained model to evaluate its pertinence to describe the studied phenomena. If the analysis and the diagnostic are satisfactory, the defined model can be used in predictions, but only if the conditions are identical and the standard error is present.

The global predicted capacity of a mathematical model is generally explained by the coefficient of determination R ² A large value of this coefficient does not necessarily suggest that the model is a good one. The R ² value improves if the variables are high even if they are not statistically significant [24]. Because R ² alone is not a measure of the model’s accuracy, it is also necessary to take into account the value of the Absolute Average Deviation (AAD) [13]. The AAD is a direct method which describes deviations, so its value must be as small as possible between the measured and the predicted data [13].

During the analysis, the statistical global significance of the model is determined by variance analysis (ANOVA, Fisher’s Test: F-value). The lack of fit test is also applied to estimate the model (fitting). The statistical significance of the model terms is determined by the Student test (t-value). The model is adjusted to the experimental data when it is significant and when the lack of fit value of the model is not significant. The truth of the hypothesis related to the homoscedasticity and the normal distribution of variations is very important during the diagnostic. The visual analysis of graphical plots of deviations gives some information about the robustness of the model. Concerning the influential observations on the predictions of the model, the main statistics are the leverage, Cook’s distance and the DFFITS.

Depending on the object, the optimal point can be characterized as the maximum, the minimum or the saddle. The value of the maximal point can be calculated from the first derivative of the model’s equation. The positive values of the explanatory variables, where the first derivative is equal to zero, correspond to the optimal values. The accuracy of the values can be verified by comparing the predicted values obtained with the mathematical model, and the measured values obtained after the experiments with the same conditions.

The aim of this paper is to optimize the composition of the medium, which influences the response to elicitation [12,25,26]. This work succeeds a precedent paper [12], where we applied the screening analysis to the influence of minerals on the hyoscyamine content of elicited HRs.

As has been reported in [12], nitrate [NO₃ ⁻], calcium [Ca²⁺], the combination [NO₃ ⁻ × Ca²⁺] and phosphorus [H₂PO₄ ⁻], are the most significant factors in the intensity of the response to elicitation for hyoscyamine production by HRs. Although phosphorus [H₂PO₄ ⁻] is statistically efficient, only [NO₃ ⁻] and [Ca²⁺] are used for optimization by RSM. Sucrose is also included for optimization due to its effect on biomass and alkaloid production [6,8]. The minimal and maximal values of nitrate and calcium have been selected from the precedent results of Amdoun et al. [12]. The real values of the concentrations have been codified by the Equation (4) (see Section 2.4).

The Central Composite Design (CCD) is used to obtain the measured responses which will be useful for the mathematical model. The CCD was presented by Box and Wilson in the 1950s. It consists of the following two parts:

A factorial design with at least one experimental point located in the center of the experimental area;

It is necessary to carry out the experiments corresponding to the star design’s points and to calculate whether the results are explained by the linear model. Figure 1 shows the studied experimental region, the range and the levels of the studied variables. Five levels have been considered for each of the three studied factors (NO₃ ⁻, Ca²⁺ and sucrose). So, the application of CCD consists of carrying out 20 experiments combining three factors, six of which are located in the center of the studied region. The optimal criterion is orthogonality, in this case α calculated is equal to ±1.52 [27]. Each point (R₁…R₂₀) corresponds to one experiment. The points R₁ to R₈ represent the factorial design 2^k. The points R₉ to R₁₄ are the star design. The points R₁₅ to R₂₀ represent the experiment done in the central experimental design.

For all experiments, the HRs cultures were performed in Petri dishes containing 20 mL of B5 medium [28]. In order to avoid other modifications of the medium due to the addition of other counterions, each element was chosen and then HNO₃ for [NO₃ ⁻] and Ca(OH)₂ for [Ca²⁺] were added to the medium before adjustment of the pH to 5.8 [29]. The quantity of inoculum for each dish was 0.3 g fresh weight from the root tips of HRs of Datura stramonium. The measured response was the hyoscyamine level (mg/L). The averages were calculated from the three replicates. The control corresponds to the same HRs line; in the same B5 medium containing 25 mM [NO₃ ⁻], 1.0 mM [Ca²⁺] and 3% sucrose; and the same conditions of temperature and darkness. Dry weight was obtained by oven drying the HRs for 48 h at 40 °C. The final dry biomass is expressed by the average of the three values with a precision balance.

The coefficient of determination R ² measures the variability explained by the factors and their interactions in the observed responses [30]. It is 0.97 for the model, from which it can be concluded that 97% of the HS level of elicited HRs is attributed to independent variables and only 3% of the total variability is not explained by the model. The AAD value is low (1.2%) and the value of deviation is σ = 0.9. Thus, the model explains the global variability; it is globally predictive.

The F-value and the value of probability show that the model is statistically significant. The lack of fit test is not significant (Table 1), which indicates that the model fits well with the experimental data.

All the linear terms related to [NO₃ ⁻], [Ca²⁺], [NO₃ ⁻/Ca²⁺] interaction and the sucrose effect are significantly positive (Table 2). The linear effect of the latter is almost identical to that of [Ca²⁺]. As the concentrations of these three elements increase in the B5 medium, the response to elicitation of HRs becomes more significant up to a certain limit where the response is reduced.

The beneficial effect of [NO₃ ⁻] is twofold: it improves the biomass [7,8,12] and the HS production by HRs simultaneously [8,12]. It is one of the crucial elements of the medium to improve the level of HS in elicited HRs. Effectively, its deficiency during the first week of culture, negatively affects the response to elicitation of HRs [12]. Moreover, its excess in the medium (from 95 mM) shows a negative effect on the biomass [8,12,31]. The [Ca²⁺] is in second position for increasing the HS level after elicitation [12]. The [Ca²⁺] ion activates Putrescine Methyl Transferase (PMT), which is one of the enzymes involved in the biosynthesis of tropane alkaloids [5,9]. It is a secondary messenger and participates in signal transduction and cellular regulation. Its intracellular flux is activated by a stimulus (such as elicitation) and also induces the defense reactions [26,32]. For sucrose, Saenz-Carbonell and Loyola-Vargas [6] showed that it improves the specific production of hyoscyamine. In our case, sucrose particularly increases the biomass.

The graphical representation of the standard error (StdErr) function (Figure 4A) is symmetric and its highest value (>1) is obtained in the four corners of the experimental area. Nevertheless, its value is 0.3 inside the central area. This low value is related to a good model quality, so the model Equation 6 can be used to evaluate the values of the Ŷ_HS response in the global experimental domain.

The RS is the graphical representation of the mathematical model in the experimental domain. Figure 4B shows the response surface for the HS level from elicited HRs as a function of [NO₃ ⁻] (X ₁) and [Ca²⁺] (X ₂). The sucrose value has been fixed to its coded value 0.0 (equivalent to 40 mg/L). From this figure, for different combinations of [NO₃ ⁻] and [Ca²⁺], the responses can be predicted. This is an effective tool to see the simultaneous effect of [NO₃ ⁻] and [Ca²⁺] on the HS level from elicited HRs. The improvement in the HS level is correlated with the increasing [NO₃ ⁻, Ca²⁺] concentrations until limits where the response decreases, beyond the central zone of the experimental domain (Figure 4B). The beneficial effect of the combination has been described by Amdoun et al. [12]. In the case of a low response with low [NO₃ ⁻, Ca²⁺] values, the cells are in a critical condition for plant defense reactions [12]. The high [NO₃ ⁻, Ca²⁺] values have a negative impact on the biomass and also on the HS level. Our results are identical with those obtained by Sikuli and Demeyer [7] for the biomass. These authors found low values of biomass and HS for HRs of Datura stramonium cultivated in B5 medium with high [NO₃ ⁻, Ca²⁺] concentrations. This effect is greater when the [Ca²⁺] value is high. We also note that the nitrate reductase activity measured in the HRs is high in this medium [7].

All the combinations [NO₃ ⁻, Ca²⁺] located in the red area (Figure 4B) illustrate the optimal conditions. Nevertheless, the local optimum concentrations correspond to positive values of variables where the first derivatives (δY_HS /x ₁; δY_HS /x ₂; δY_HS /x ₃) of the model Equation 6 cancel each other out. The optimal concentrations in coded values are 0.37 for [NO₃ ⁻], 0.26 for [Ca²⁺] and 0.14 for sucrose. This is equivalent to 79.1 mM, 11.4 mM and 42.9 g/L respectively in real values and corresponds to the optimized B5 medium (B5-OP). These optimal calculated concentrations are located in the area where the standard error function is lower and where the predicted accuracy of the model Equation 6 is confirmed.

According to the simulation of the HS level from elicited HRs, cultured in B5-OP medium, the predicted value is 107.90 mg/L. The calculated prediction interval, at 95%, is PI = [106.17; 109.72]. The measured level for the same concentrations and in the same conditions is 110.3 ±1.4 mg/L. This value is in the 95% PI range and validates the optimal concentrations of [NO₃ ⁻], [Ca²⁺] and sucrose.

Figures 5A and 5B show a culture of HRs in B5-OP medium in a Petri dish or Erlenmeyer. It displays a high biomass in comparison with HRs cultivated in B5 control (Figure 5C).

There is a significant improvement in biomass (51.2%) with the B5-OP medium in comparison with the HRs control (Table 4). A significant difference was observed for the specific production of HS (mg/g DW) in HRs with or without elicitation. The optimization was determined as 81% and 101.2% for elicited or non-elicited HRs respectively in B5-OP medium.

Finally, the improvement in the level of HS (mg/L) was 173.6% for non-elicited HRs, and 212% for elicited HRs cultivated in B5-OP, in comparison with the control medium.