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Traditionally, optimization in biological analyses has been carried out by monitoring the influence of one factor at a time; this technique is called
Hairy roots (HRs) are an efficient system for the production of secondary metabolites [
In general, the experimental procedure of optimization is achieved by studying a single factor at any one time. While this factor is modified to find the optimal response, others are kept at a constant level. This is known as the
During recent years, the RSM has been used extensively for optimization in many areas of industrial research and process development in chemistry and biochemistry. Although Calam [
Its use for the optimization and analysis of biotechnological processes with microorganisms and enzymatic engineering has given good results. For example, the work of Maddox and Richert [
This study follows the work of Amdoun
 First, to use the RSM method for the optimization of nutrients (nitrate, calcium and sucrose) in the culture medium B5 [Gamborg, 1968] to improve hyoscyamine production in elicited HRs;
 Second, to show the value of the RSM method in the optimization of several responses in plant material cultures.
The HRs, obtained by genetic transformation and selection, were subcultured in the same conditions as described previously [
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 [
The solution of Jasmonic Acid (JA) [(−)jasmonic acid from SigmaAldrich] 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 (ethanolwater 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
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 [
To evaluate the form of the true response, a second degree model is used. The factorial designs 2^{k} are used to determine the firstorder effects but these are inefficient when additional effects, like the secondorder 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
The equation system (1) can be resolved by the Method of Least Squares (MLS) and may be written in matrix form (2) [
With
The low and high levels of variables (factors) are denoted by −1 and 1, respectively. The levels of
The most used secondorder symmetric designs for the RSM are: The general factorial design, the BoxBehnken 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 [
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
During the analysis, the statistical global significance of the model is determined by variance analysis (ANOVA, Fisher’s Test:
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 [
As has been reported in [
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;
A star design whose axial points
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.
For all experiments, the HRs cultures were performed in Petri dishes containing 20 mL of B5 medium [
The coefficient of determination
The
All the linear terms related to [NO_{3}
^{−}], [Ca^{2+}], [NO_{3}
^{−}/Ca^{2+}] interaction and the sucrose effect are significantly positive (
The beneficial effect of [NO_{3}
^{−}] is twofold: it improves the biomass [
The validity of a model can be evaluated by the residual analysis. We defined as residual, the difference between the observed values and the calculated values obtained by the model. It is the part which is not explained by the equation of the model. This analysis can also detect some outliers among the total data. We use principally graphical methods for the residual analysis, such as the graphical presentation of the residuals as a function of the estimated values [
If the value of an observation diverges from the supposed form of the distribution of the total observations, it is called an outlier. If this form is modified, the observation can converge with the new model [
We called the leverage
After this operation, only the observations R_{4}, R_{8} and R_{14} presented main DFFITS values (
The removal of the observations R_{4} and R_{14} improves the accuracy of the estimated coefficients. The CV% goes from 2.0 to 0.9, in this case the AAD value was calculated as 0.4%. More precisely, it is the model
The graphical representation of the standard error (StdErr) function (
The RS is the graphical representation of the mathematical model in the experimental domain.
All the combinations [NO_{3}
^{−}, Ca^{2+}] located in the red area (
According to the simulation of the HS level from elicited HRs, cultured in B5OP medium, the predicted value is 107.90 mg/L. The calculated prediction interval, at 95%, is
There is a significant improvement in biomass (51.2%) with the B5OP medium in comparison with the HRs control (
Finally, the improvement in the level of HS (mg/L) was 173.6% for nonelicited HRs, and 212% for elicited HRs cultivated in B5OP, in comparison with the control medium.
The selection of significant statistical variables enables the accuracy of the model to be improved. After the diagnostics, some outliers related to the model with the global terms are eliminated after the screening. This selection of outliers must be carried out carefully; their removal is implemented when they are due to pure error. In our case, only two observations were removed after the study of the screening of variables.
Optimization of the B5 medium improves the response to elicitation. Under optimal conditions, the optimization of the HS level is 212.7% for elicited
The optimal concentrations for the selected line and in our conditions are: 79.1 mM, 11.4 mM and 42.9 g/L for [NO_{3} ^{−}], [Ca^{2+}] and sucrose, respectively. These values correspond to the solutions obtained with the first derivatives of the model. Nevertheless, all the concentrations included in the optimal area of the surface response can be used.
This new approach to the optimization of B5 medium could be employed to maximize the response to elicitation. The B5OP medium significantly influenced the biomass and alkaloid production. This was possible by the simulation and the predictability of the quadratic model used. Indeed, unlike classical methods, this mathematical model is not time consuming and a large number of experiments are not needed to optimize the parameters. Mathematical modeling is a powerful tool for biological studies. In this work, by applying the RSM, only 20 experiments were required to optimize the B5 medium composition in terms of [NO_{3} ^{−}], [Ca^{2+}] and sucrose. This quadratic model led to a demonstration of the effects of these nutrients on the HS level of HRs elicited by Jasmonic Acid (JA).
However, the HS level of the elicited HRs, cultured in B5OP medium, can still be improved by an irregular deficiency in [NO_{3}
^{−}] [
Experimental region and levels of each of the three factors of the CCD (
Diagnostic plot: (
BoxCox plot for power transformations (
Response Surface plots showing effects of [NO_{3}
^{−}] and [Ca^{2+}] as a function of standard error (StdErr) (
Appearance of HRs on the 28th day of culture in B5OP (
ANOVA table for the quadratic model (results in bold are significant).







Model  9537.1  10  953.7  414.1 

Residual  20.7  9  2.3  


Lack of fit  16.1  4  4.0  4.3  0.1 
Pure Error  4.7  5  0.9  


Total  9557.8  19 
Analysis terms for the quadratic model (results in bold are significant).






104.8  172.2   

10.5  24.6 


5.5  12.9 


4.2  9.8 


3.5  6.5 


1.0  1.9  0.1 

1.0  1.9  0.1 

−16.4  −35.4 


−14.4  −31.1 


−14.5  −31.3 


0.7  1.3  0.2 
Diagnostics for influential observations (Results in bold are outliers).


















R_{1}  −1  −1  −1  43.8  43.9  −0.1  0.8  0.0  −0.4 
R_{2}  1  −1  −1  56.8  57.5  −0.7  0.8  0.5 

R_{3}  −1  1  −1  46.8  47.5  −0.7  0.8  0.5 

R_{4}  1  1  −1  70.8  72.0  −1.3  0.8 


R_{5}  −1  −1  1  51.2  49.8  1.4  0.8 


R_{6}  1  −1  1  65.2  64.4  0.8  0.8  0.7 

R_{7}  −1  1  1  55.2  54.4  0.8  0.8  0.7 

R_{8}  1  1  1  86.2  86.0  0.2  0.8  0.1  0.7 
R_{9}  −1.52  0  0  50.0  50.9  −0.9  0.6  0.1 

R_{10}  1.52  0  0  83.6  83.0  0.7  0.6  0.1  0.8 
R_{11}  0  −1.52  0  62.3  63.1  −0.8  0.6  0.1  −0.9 
R_{12}  0  1.52  0  80.6  79.9  0.7  0.6  0.1  0.7 
R_{13}  0  0  −1.52  66.7  64.8  1.8  0.6  0.4 

R_{14}  0  0  1.52  75.5  77.5  −2.0  0.6  0.5 

R_{15}  0  0  0  103.8  104. 8  −1.0  0.2  0.0  −0.3 
R_{16}  0  0  0  104.7  104.8  −0.0  0.2  0.0  0.0 
R_{17}  0  0  0  105.1  104.8  0.3  0.2  0.0  0.1 
R_{18}  0  0  0  106.4  104.8  1.6  0.2  0.0  0.5 
R_{19}  0  0  0  103.8  104.8  −1.0  0.2  0.0  −0.3 
R_{20}  0  0  0  104.9  104.8  0.1  0.2  0.0  0.0 






R_{4}  1  1  −1  70.8  74.8  −4.0  0.6  0.9 

R_{8}  1  1  1  86.2  83.2  2.3  0.5  0.5 

R_{14}  0  0  1.52  75.5  77.5  −2.0  0.6  0.5 

Biomass and HS production of HRs cultivated in B5 control medium or B5OP medium after the 28th day of culture
















B5 
8.4 ± 0.6  2.1 ± 0.1  4.2 ± 0.6  17.6 ± 1.6  35.3 ± 2.0  
B5OP 
12.7 ± 0.2  3.8 ± 0.1  8.5 ± 0.3  48.3 ± 2.3  110.3 ± 1.4  
Optimization  51.2%  81%  101.2%  173.6%  212.7%  


LSD test  differences  −4.3  −1.7  −4.3  −30.6  −75.0 


±limits  0.9  0.2  0.8  4.4  13.8  


significance  significant  significant  significant  significant  significant 
B5 control (25 mM NO_{3} ^{−}, 1.0 mM Ca^{2+} and 3% sucrose);
B5OP optimized (79.1 mM NO_{3} ^{−}, 11.4 mM Ca^{2+} and 42.9% sucrose).