Modeling and Optimizing Medium Composition for Shoot Regeneration of Chrysanthemum via Radial Basis Function-Non-dominated Sorting Genetic Algorithm-II (RBF-NSGAII)

The aim of the current study was modeling and optimizing medium compositions for shoot proliferation of chrysanthemum, as a case study, through radial basis function- non-dominated sorting genetic algorithm-II (RBF-NSGAII). RBF as one of the artificial neural networks (ANNs) was used for modeling four outputs including proliferation rate (PR), shoot number (SN), shoot length (SL), and basal callus weight (BCW) based on four variables including 6-benzylaminopurine (BAP), indole-3-butyric acid (IBA), phloroglucinol (PG), and sucrose. Afterward, models were linked to the optimization algorithm. Also, sensitivity analysis was applied for evaluating the importance of each input. The R2 correlation values of 0.88, 0.91, 0.97, and 0.76 between observed and predicted data were obtained for PR, SN, SL, and BCW, respectively. According to RBF-NSGAII, optimal PR (98.85%), SN (13.32), SL (4.83 cm), and BCW (0.08 g) can be obtained from a medium containing 2.16 µM BAP, 0.14 µM IBA, 0.29 mM PG, and 87.63 mM sucrose. The results of sensitivity analysis indicated that PR, SN, and SL were more sensitive to BAP, followed by sucrose, PG, and IBA. Finally, the performance of predicted and optimized medium compositions were tested, and results showed that the difference between the validation data and RBF-NSGAII predicted and optimized data were negligible. Generally, RBF-NSGAII can be considered as an efficient computational strategy for modeling and optimizing in vitro organogenesis.

providing an accurate answer of what levels of medium compositions may be applied to obtain the maximum PR, SN, and SL as well as the minimum BCW. Thus, the model linked to NSGAII for finding the maximum efficiency and the optimum medium compositions levels, which are essential for significant in vitro shoot regeneration.
The RBF provides adequate accuracy for interpolation but not for extrapolation 38 . Therefore, the upper bound and lower bound of input variables (Table 1) were set as constraints, and the point with the highest PR, SN, and SL as well as the lowest BCW was considered as the ideal point during the optimization process. According to RBF-NSGA-II (Table 3), optimal PR (98.85%), SN (13.32), SL (4.83 cm), and BCW (0.08 g) can be obtained from a medium containing 2.16 µM BAP, 0.14 µM IBA, 0.29 mM PG and 87.63 mM sucrose.
Sensitivity analysis of the models. The importance of each input was evaluated through the entire 729 data lines (training and testing) to determine the general VSR. The VSR achieved for the model output (PR, SN, SL, and BCW), with respect to medium compositions (Table 4). Sensitivity analysis showed that PR, SN, and SL were more sensitive to BAP, followed by sucrose, PG, and IBA (Table 4). In BCW model, the feed efficiency indicated more sensitivity for BAP, followed by IBA, PG, and sucrose (Table 4).

Validation experiment.
The results of the validation experiment (Table 5) showed that RBF-NSGAII could be able to propose the optimal level of medium compositions to achieve the most appropriate results of the  Table 5). The results indicated that the difference between the MLP predicted and validation data was negligible (Table 5).

Discussion
Being successful in plant tissue culture depends on various factors such as gelling agents, the composition of the medium, use of specific combinations of PGRs, and light and temperature conditions 2 . Adjusting medium compositions for increasing plant growth and development is the most common method in plant tissue culture 2,9 . In the current study, RBF-NSGAII model was used to achieve a comprehensive understanding of the effect of different levels of BAP, IBA, PG and sucrose on shoot proliferation of chrysanthemum, and to obtain new insights into improving chrysanthemum organogenesis. According to the best of our knowledge, this study is the first report of using RBF-NSGAII for modeling and optimizing medium compositions for shoot proliferation of this ornamental plant.
High coefficient of determination between observed and predicted values for both training and testing process indicated good performance of the models for the studied parameters. The high efficiency of ANN in plant tissue culture has been shown by several studies 26,27,30 . Recently, several studies used ANN-GA for modeling and optimizing in vitro organogenesis 26,27 . GA is mainly used for optimizing different in vitro conditions using a single objective function. However, plant tissue culture problems have to satisfy various objective functions  www.nature.com/scientificreports www.nature.com/scientificreports/ by considering different constraints and GA, as a single-objective algorithm, cannot optimize multi-objective simultaneously 37 . Therefore, the use of GA would not be the applicable solution to optimize multiple-objective functions. Thus, there is a dire need for applying the multi-objective algorithm for the optimization process. In this study, we have introduced NSGAII as a multiple-objective algorithm. BCW has a negative effect on micropropagation due to the somaclonal variation and restriction of vascular system. Therefore, the ultimate aim of this study was to analyze RBF model to provide an accurate answer of what levels of medium compositions may be applied to obtain the maximum PR, SN, and SL as well as the minimum BCW.    www.nature.com/scientificreports www.nature.com/scientificreports/ Shoot proliferation usually produces true clones of an explant. In some plants such as chrysanthemum, it is not feasible to achieve micro-shoots by using PGRs or manipulation culture-room conditions 2,9 . Adjusting medium composition as an alternative for the promotion of shoot proliferation would benefit in vitro culture 19 . Several studies 15,[39][40][41] reported the successfulness of shoot proliferation of chrysanthemum via single node explants. However, those studies focused on the effects of different hormonal combinations on shoot proliferation. Therefore, there is a lack of a comprehensive study on the effect of medium composition in shoot proliferation. The results of this study showed the necessity of balancing cytokinin/auxin (BAP/IBA) for shoot regeneration. Similarly to our results, Iizuka, et al. 42 showed the maximum shoot regeneration in MS medium supplemented with 8.88 µM BAP and 0.1 µM IBA. Also, Lu, et al. 41 reported that the combination of 8.88 µM BAP and 1.07 µM NAA resulted in 100% PR in chrysanthemum. It is well known that in vitro organogenesis significantly depends on the ratio between cytokinins and auxins 12,13 . The previous studies indicated that the high ratio of cytokinin/ auxin promote shoot induction 9,13,14,41,42 .
In our experiment higher PR, SN and SL were achieved on media with supplemented PG, however PG concentrations over 0.3 mM had an inhibitory effect on these parameters. A recent study 23 reported that PG could promote shoot regeneration in Vitex negundo. Additionally several studies 19,23,24 associated PG with plant growth and development. Shoot proliferation of Minuartia valentina was improved on MS medium with PG in combination with BAP 43 . Sarkar and Naik 22 indicated PG × sucrose interaction plays an important role in shoot regeneration. Similarly, our results showed that 0.29 mM PG and 87.63 mM sucrose had a striking impact on shoot proliferation.
According to our results, 87.64 mM sucrose was better during the shoot proliferation stage, which was consistent with the findings of Lu, et al. 41 , da Silva 2 , Iizuka, et al. 42 , and Arun, et al. 44 who suggested that 87.64 mM sucrose should be used in all culture stages for the in vitro multiplication of chrysanthemum.
Finally, according to the validation experiment, RBF-NSGAII can be considered as a new computational algorithm in analyzing data derived from in vitro culture parameters for predicting optimized levels of medium compositions required in the shoot proliferation stage.

conclusion
In vitro culture issues have to satisfy different opposite objective functions; so, there is a dire need of using the multi-objective algorithms such as NSGA-II for optimizing the process. RBF-NSGAII has been introduced as an applicable algorithm for modeling and optimizing shoot proliferation of chrysanthemum. Based on the results of this study, the interaction effects of medium compositions can be precisely identified via RBF-NSGAII. Generally, RBF-NSGAII can be considered as a powerful and applicable model for applying in various areas of plant tissue culture.
Methods plant materials. The single-node explants of chrysanthemum 'Hornbill Dark' were cut from indoor mother plants. After that, the explants were washed for 30 min under tap water, followed by washing after cleaning by a liquid soap solution. Additional surface sterilization was implemented under a laminar airflow chamber. The explants were sterilized by 70% aqueous ethanol for 40 s, dipped 15 min in 1.5% (v/v) NaOCl solution, and three times washed by sterilized distilled water. Then, the nodal segments (0.5 cm) were vertically inoculated on 200-ml glass flasks consisting of 40 ml basal medium.  Table 4. Importance of medium composition for proliferation rate (PR), shoot number (SN), shoot length (SL), and basal callus weight (BCW) of chrysanthemum according to sensitivity analysis on the developed RBF model to rank the importance of inputs.
BAP, IBA, PG and sucrose were used as inputs while PR, SN, SL, and BCW were considered as outputs (Fig. 5).
The algorithm chosen to model these relationships was RBF-ANN. RBF is a statistical neural network used for regression-based problems. The input of the transfer function for each neuron in such a network is the Euclidean distance between the input and the center of that neuron 46 . RBF was applied for obtaining the maximum rate of PR, SN, and SL as well as the minimum rate of BCW. Prior to modeling the data was split into a 75% training and 25% testing set and checked to confirm that the ranges of the train and test data are overlapping. The different values for the significant model's parameters were examined based on a trial and error analysis for determining and improving the overall performance of the best-constructed model. For trial and error analysis, we used MSE as the default criteria for tuning ANN models. In addition, we used K-Fold Cross Validation (K = 5) for our training set. Also, we trained four RBF models for each outputs including PR, SN, SL, and BCW. In the end, the best-resulted output with the minimum estimation error was determined for each individual model based on Mean Bias Error (MBE), Root Mean Square Error (RMSE), and the coefficient of determination (R2) as follows: where y i is the value of predicted datasets, ŷ i is the value of observed datasets, and n is the number of data. Best fit can be achieved when R 2 values closer to 1 and RMSE values closer to 0. The MBE value stands for negative and positive calculation error, indicating the similarity of the predicted values with observational values.
Popular transfer function in RBF is the Gaussian function, the Gaussian function uses the following relation 46 : where X r : input with unknown output, X b : observed inputs in time b, and h: spread. The output of the function close to 0 when − X X r b approaches a large value, and close to 1 when − X X r b approaches 0. Eventually, the dependent variable (Yr) by predictor X r is calculated as follows: where w j is the weight of connections from the b th hidden layer to the output layer and w 0 : bias. optimization process (nSGAii). NSGA-II is an evolutionary optimization algorithm that is used in multi-objective problems. This algorithm starts by generating a set of random solutions; the objective function value is then calculated for each solution, and the process of refining the solutions begins. At this step, the solutions are elected for crossover using the binary tournament operator based on two criteria: non-dominated www.nature.com/scientificreports www.nature.com/scientificreports/ sorting and crowding distance. The algorithm can be kept from getting stuck in the local optimum by applying a mutation operator. The objective function values are calculated once again after the refining solutions are determined. This process is repeated until one of the stopping criteria is satisfied. In each generation, non-dominated solutions in objective space constitute a pareto front; any point on this front can be an optimal solution of the problem (Fig. 6). In this study, 200 initial population, 1000 generation, 0.7 crossover rate, 0.05 mutation rate, the uniform of mutation function, two-point crossover function, and a binary tournament selection function were considered. Also, PR, SN, SL, and BCW were considered as four objective functions to determine the optimum values of inputs based on the results of RBF algorithm. The ideal point of pareto was chosen such that PR, SN, and SL were maximized, while BCW was minimized. In other words, a point in the pareto front was considered as the solution such that was minimal; where m, n, and o are the maximum PR, SN, and SL respectively, and p is the minimum BCW in observed data. Before applying Eq. 6, objective function values were scaled between 0 and 1.

Sensitivity analyses.
The sensitivity PR, SN, SL, and BCW against the investigated medium compositions, was assessed by considering the following criterion; Sensitivity error (VSE) value: overall performance of the developed RBF model if the certain independent variable is not available.
Variable sensitivity ratio (VSR) value: indicates the correlation between the VSE and the error of the RBF model in the case that all variables are available.
High rate of VSR is paramount. Therefore, all input variables should be ranked based on their VSR rate (importance). The mathematical code for constructing and assessing models and optimization analysis was written conveniently for Matlab (version 9.5) software.
Validation experiment. During the validation experiment, the levels of medium compositions optimized by RBF-NSGAII were evaluated for confirming the efficiency of RBF-NSGAII to model and optimize the medium composition for shoot proliferation parameters (i.e., PR, SN, SL, and BCW).