Number of experiments necessary to more accurately differentiate common bean genotypes for grain physical traits and minerals in cluster analysis

ABSTRACT The number of experiments that provides greater detail in the differentiation of common bean genotypes for grain physical traits and minerals in cluster analysis is not known. This study was undertaken to determine the number of experiments necessary to more accurately differentiate common bean genotypes for grain physical traits and minerals in cluster analyses. Seven traits of grain physical quality and the concentration of six minerals were evaluated in 17 common bean genotypes with carioca (9) and black (8) grains. Statistical analyses were performed in data obtained from one, two, three and four experiments. A significant genotype × experiment interaction occurred for all traits, except for the potassium concentration. Tocher’s and the unweighted pair group method with arithmetic mean (UPGMA) cluster analyses were efficient in differentiating common bean genotypes by grain type when the data obtained from one experiment were considered. However, the use of data obtained from four experiments made it possible to recognize differences regarding grain lightness and brightness as well as the other traits. Four experiments are need for the Tocher’s and the UPGMA cluster analyses to more accurately differentiate carioca and black bean genotypes for grain physical traits and minerals.


INTRODUCTION
Common bean (Phaseolus vulgaris L.) has an important roles in food security, being a source of carbohydrates, proteins and minerals in the human diet (Câmara et al., 2013).Because common bean grains have a great diversity of colors and sizes, their consumption has become popular among consumers from several countries who accept different grain types in their diet.However, 90% of the common bean produced in Brazil is restricted to two grain types: carioca (beige seed coat with brown streaks) and black (Lemos et al., 2015).These grain types are highly appreciated by consumers and, for this reason, breeding programs have put greater efforts into the development of carioca and black bean cultivars with high grain physical quality and mineral concentration.This is because characteristics such as color, size, cooking time and nutritional value of beans are analyzed by consumers who are mindful of its quality.

Carioca and black bean cultivars growing at Brazil
show high genetic similarity because in the process of developing of these cultivars were performed crosses between parents of both grain types, which difficulty the separating the genotypes by grain type (Veloso et al., 2015).For this Number of experiments necessary to more accurately differentiate common bean genotypes for grain physical traits and minerals in cluster analysis reason, cluster analyses are shown to be potential tools to identify promising parents for use in crosses as well as recognize very similar, i.e., duplicate, genotypes.Several clustering methods have been described in the literature (Cruz & Carneiro, 2014), but Tocher's optimization and the UPGMA hierarchical cluster analyses are widely employed in common-bean breeding programs.
Tocher's method is widely used in differentiating common bean genotypes regarding agronomic traits evaluated in one experiment (Lima et al., 2012;Gonçalves et al., 2016;Santos et al., 2019).However, when data of agronomic traits obtained individually in two (Kumar et al., 2009;Coelho et al., 2010) or four (Ceolin et al., 2007) experiments were used, there was variation in the number of groups and in the composition of the groups formed by Tocher's method.Most agronomic traits determined in common bean genotypes showed a significant genotype × experiment interaction effect, therefore the use of data obtained from six experiments were suggested by Carg nelutti Filho et al. (2009) to identify divergent common bean cultivars by Tocher's method.For common bean genotypes no previous work was found with the definition of the necessary number of experiments to be recommended in Tocher's cluster analysis for traits of grain physical quality and minerals.
The UPGMA method has been efficient in differentiating common bean genotypes based on different morphological and/or agronomic traits evaluated in a single experiment (Kloster et al., 2011;Grahic et al., 2013;Bertoldo et al., 2014;Hegay et al., 2014;Veloso et al., 2015;Gonçalves et al., 2016;Canci et al., 2019;Santos et al., 2019;Long et al., 2020) or on the average of two (Guidoti et al., 2018;Arteaga et al., 2019;Savic et al., 2019) or three (Cabral et al., 2011) experiments.None of these studies described the criterion used to define the number of experiments employed in the UPGMA cluster analysis.For agronomic and grain physical quality traits and minerals, no recommendations were found in the literature as to the necessary number of experiments to be conducted to achieve greater accuracy in differentiating common bean genotypes using the UPGMA method.
Defining the number of experiments that provides greater detail in the differentiation of common bean genotypes based on traits of grain physical quality and minerals in cluster analyses will allow greater efficiency in identifying promising genotypes for use in controlled crosses.This information is unprecedented and constitutes an important innovation for common-bean breeding programs.Therefore, this study was conducted to determine the number of experiments necessary to more accurately differentiate common bean genotypes for grain physical traits and minerals in cluster analyses.The soil in the experimental area is a typic alitic Argisol, Hapludalf, which was prepared by the conventional cultivation system.All experiments were conducted observing the minimum requirements for determining the Value for Cultivation and Use (VCU) of common bean cultivars (Brasil, 2006).The management practices were implemented following the guidelines of the technical recommendations for the cultivation of common bean in the southern region of Brazil (CTSBF, 2012).

Description of the experiments
Nerinéia Dalfollo Ribeiro et al.

Evaluation of grain physical quality and mineral concentration
The physical quality of the common bean grains was analyzed based on seven traits, namely, L*, a*, b*, absorption, normal grains, cooking time and mass of 100 grains.
The values of L*, a* and b* characterized grain color and were evaluated using a portable colorimeter.The L* parameter measures the variation from black (0) to white (100); a* quantifies the intensity of the colors from green (-60) to red (+60); and b* indicates the color spectrum between blue (-60) and yellow (+60).
The traits of absorption, normal grains and cooking time were determined in a sample of 25 grains after soaking in 50 mL of distilled water for 8 h, at room temperature (20 ± 2 ºC).Absorption was obtained by the difference in weight of the grains after and before soaking relative to the initial weight of the grains, and expressed in %.Normal grains differed from hard grains in their ability to absorb water, and were expressed in %.Cooking time was evaluated using a Mattson cooker, by following the methodology described in Ribeiro et al. (2021).Mass of 100 grains was quantified in three random 100-grain samples, whose moisture was standardized to 13%, and expressed in g.
The concentration of six minerals (potassium, phosphorus, calcium, magnesium, iron and copper) was analyzed according to the methodology described by Miyazawa et al. (2009).Reading of these minerals were carried out in an atomic absorption spectrophotometer, except for potassium, which was obtained on a flame photometer, and phosphorus, which was quantified using an optical emission spectrophotometer.

Statistical analyses
The obtained data were subjected to analysis of variance adjusted for traits with heterogeneous residual variances (Cruz, 2016).
Multicollinearity diagnostics was based in the combined analysis of variance of four experiments, using the phenotypic correlation matrix for the 13 traits evaluated.
The collinearity classes were evaluated based on the criteria proposed by Montgomery et al. (2012).
The cluster analyses were implemented for data obtained from one, two, three and four experiments.For this, the residual variance and covariance matrices obtained in the analyses of variance of these experiments were used to generate matrices of genetic dissimilarity using Mahalanobis' generalized distance with standardized means (D 2 ).The traits that most contributed to genetic divergence were identified in the Mahalanobis' generalized distance analysis.
Cluster analyses were performed using two methods: Tocher and UPGMA.The cophenetic correlation coefficient (CCC) was established from Pearson's linear correlation between the elements of the cophenetic matrix and the elements of the dissimilarity matrix to measure the consistency of the clustering pattern.All statistical analyses were carried out using Genes software (Cruz, 2016).

Individual and combined analyses of variance
The genotype effect was significant for 57.69% of the traits evaluated in the different analyses of variance (Table 1), which shows that the common bean genotypes differed for most traits of grain physical quality and minerals.Previous works also found expressive variation for the traits of grain physical quality (Rivera et al., 2016;Herrera-Hernández et al., 2018;Dias et al., 2021) and mineral concentration (McClean et al., 2017;Steckling et al., 2017;Yeken et al., 2019;Delfini et al., 2020;Jan et al., 2021)  However, a significant genotype × experiment (environment) interaction effect for 34 of the 39 tested combinations (13 traits × 3 combined analyses of variance) was obtained.Therefore, the common bean genotypes exhibited variation for all traits of grain physical quality and five minerals when cultivated in different environment,

Tocher's cluster analysis
The L* value was the trait that most contributed to the differentiation between common bean genotypes, according to the results obtained from Mahalanobis' generalized distance (Table 2).The percentage contribution of the L* value to the discrimination of genotypes ranged from 90.11% (experiment I) to 96.78% (experiments I, II, III and IV), i.e., this trait had a major participation in the formation of the different groups in cluster analyses.The visual assessment of grain color was also efficient to distinguish different groups of common bean landraces evaluated in Spain (Arteaga et al., 2019) and in Turkey (Canci et al., 2019).However, no previous studies were found including the L* value in cluster analyses of common bean genotypes.The present study is the first record that grain color, as quantified by the L* value, is an efficient descriptor to differentiate carioca and black bean genotypes, suggesting great potential for use in cluster analyses.The number of groups formed and the genotypes belonging to each clustering generated by Tocher's method were different if we consider data obtained from one, two, three or four experiments (Table 3).Similar results were described for agronomic traits evaluated in two consecutive years (Kumar et al., 2009;Coelho et al., 2010) and in four experiments (Ceolin et al., 2007), and for nine nutritional traits analyzed in two years (Pereira et al., 2011).In all these papers were described differences regarding the clustering pattern obtained by Tocher's method when were used data of individually experiments.The most evaluated traits in the present study showed a significant genotype × experiment interaction effect (Table 1).Therefore, the formation of groups of common bean genotypes obtained by Tocher's method will be specific for each environment for traits of grain physical quality and minerals.The use of data obtained from individual experiments result in low coincidence in the stratification of groups of common bean genotypes based on genetic dissimilarity.
The use of data from one experiment resulted in the formation of three groups by Tocher's method (Table 3).
Group 1 comprised the nine genotypes of carioca beans.
The black bean genotypes were clustered into groups 2 (opaque grains) and 3 (grains of intermediate brightness

Number of experiments necessary to more accurately differentiate common
bean genotypes for grain physical traits and minerals in cluster analysis and bright).None previous study was found of genetic dissimilarity analysis for traits of grain physical quality and minerals evaluated in common bean genotypes using Tocher's method.For agronomic traits, the Tocher's method was applied with efficient to differentiate common bean genotypes based on data obtained from a single experiment (Lima et al., 2012;Veloso et al., 2015;Gonçalves et al., 2016;Santos et al., 2019).In the present study, Tocher's method allowed differentiate the genotypes of carioca and black beans, based on analysis performed with data obtained from a single experiment.Nevertheless, not was efficient for the identification of differences with respect to grain lightness and brightness in common bean genotypes of the same grain type.A similar response was observed when the Tocher's method was applied to data from two or three experiments.Tocher's clustering has also been performed based on the average data from two (Maziero et al., 2017) or several experiments (Pereira et al., 2019) for agronomic traits.Cargnelutti Filho et al. (2009) suggested the use of data of six experiments so that common bean genotypes can be more accurately differentiated in the Tocher's method based on agronomic traits.For traits of grain physical quality and minerals, no definition was found on the minimum number of experiments to achieve greater accuracy in the differentiation of common bean genotypes in the Tocher's cluster analysis.
In the present study, five groups were formed when Tocher's method was analyzed using the data obtained from four experiments.Group 1 contained the carioca bean genotypes with the highest grain lightness (L* = 56.20)and the highest absorption (94.67%), namely, Pérola, CNFC 15 097, Carioca, BRS MG Uai, LP 09-33, LEC 02-16, LEC 01-16 and GEN 45-2F-293P (Tables 3 and 4).Group 2 included the black bean genotypes of  The results of data obtained from four experiments enabled the differentiation of common bean genotypes by grain type (carioca and black), based on the L* value, with greater detail in terms of grain lightness and brightness as well as other traits of grain physical quality and minerals.
Thus, the use of data obtained from four experiments was more discriminative in the formation of groups by Tocher's method for traits of grain physical quality and minerals evaluated in common bean genotypes.

UPGMA cluster analysis
The CCC ranged from 0.9772 (experiment I) to 0.9880 (experiments I and II) (Figure 1), with significance detected at 1% probability by the t test.These values were higher than the CCC obtained in UPGMA cluster analysis, considering agronomic traits evaluated in one (Veloso et al., 2015;Gonçalves et al., 2016), two (Arteaga et al., 2019) and three (Cabral et al., 2011) experiments with different Number of experiments necessary to more accurately differentiate common bean genotypes for grain physical traits and minerals in cluster analysis common bean genotypes.The CCC values obtained indicate a high adjustment between the cophenetic matrix and the dissimilarity matrix based on Mahalanobis' generalized distance (Cabral et al., 2011).In the present study, CCC values ≥ 0.9772 are associated with greater reliability in the clustering pattern provided by the UPGMA method for traits of grain physical quality and minerals analyzed in one, two, three and four experiments.Nerinéia Dalfollo Ribeiro et al.
In the four dendrograms obtained by the UPGMA method, two groups were formed: 1 -comprising the carioca bean genotypes; and 2 -containing all black bean genotypes, adopting 90% similarity as a criterion for the definition of the groups.Similarly, common bean genotypes were clustered into different groups according to grain type (Kloster et al., 2011) and to grain color (Canci et al., 2019) by the UPGMA method when were evaluated different morphological descriptors.The UPGMA method has been efficient in differentiating common bean genotypes based on morphological (Kloster et al., 2011;Grahic et al., 2013;Hegay et al., 2014;Canci et al., 2019) and agronomic (Bertoldo et al., 2014;Gonçalves et al., 2016;Santos et al., 2019;Long et al., 2020) traits evaluated in single experiment.However, differentiating common bean genotypes using the UPGMA method based in grain physical quality traits and minerals is unprecedented in common bean.
It was only possible to differentiate the line SM 0312 (black-streak carioca) from the other carioca common bean genotypes in group 1 when data obtained from four experiments were used.For the black common bean genotypes, cultivar Guapo Brilhante (bright grains) and line TB 02-19 (lighter black grains) clearly stood out among the genotypes present in group 2. Therefore, the use of data obtained from four experiments was more discriminative in the stratification of groups by the UPGMA method, similar to the observations with Tocher's method for traits of grain physical quality and minerals.Previous studies have shown that the UPGMA method has also been used to differentiate common bean genotypes based on morphological and/or agronomic traits evaluated in two (Guidoti et al., 2018;Arteaga et al., 2019;Savic et al., 2019) and three (Cabral et al., 2011) experiments.None of these studies defined the number of experiments that should be used in cluster analysis.The number of experiments to be employed in UPGMA cluster analysis to provide a better differentiation between common bean genotypes for traits of grain physical quality and minerals was not found in literature.
In the present study was observed that the most traits of grain physical quality and minerals showed a significant genotype × experiment interaction effect (Table 1).This result shows that the effects of years and seasons in the same growing location need to be considered in cluster analyses.The use of data from four experiments in the Tocher's and the UPGMA cluster analyses always greater efficiency in the differentiation between carioca and black bean genotypes, especially for recognition differences in grain lightness and brightness (L* value) and other traits of grain physical quality and minerals (Table 3 and Figure 1).Thus, the use of four experiments in the Tocher's and the UPGMA cluster analyses is recommended for common-bean breeding programs with an emphasis on traits of grain physical quality and minerals.This strategy provides greater detail about the differences between common bean genotypes, thereby allowing a more accurate identification of promising parents for use in controlled crosses as well as of duplicate accessions.

CONCLUSIONS
Tocher's and the UPGMA cluster analyses are efficient in differentiating common bean genotypes by grain type using data obtained from one experiment.
Four experiments are need for the Tocher's and the UPGMA cluster analyses to more accurately differentiate carioca and black bean genotypes for grain physical traits and minerals.

Four
experiments were established in two consecutive years -2016 and 2017 -and in the two seasons recommended for the cultivation of common bean in the southern region of Brazil: rainy and dry, which correspond to sowing carried out in the months of October and February, respectively.The experimental area is located on the Universidade Federal de Santa Maria (UFSM), in Santa Maria -RS, Brazil (29º42´S latitude, 53º49´W longitude and 95 m altitude).The region is characterized by its humid subtropical climate, with hot summers and no clearly defined dry season.A randomized block design with three replicates was used in all experiments.Each experimental plot consisted of 4-m four rows, spaced 0.5 m apart and a usable area of 4 m 2 .A total of 17 common bean genotypes (lines and cultivars) were evaluated: SM 0312, BRS MG Uai, CNFC 15 097, LEC 02-16, GEN 45-2F-293P, LP 09-33, LEC 01-16, Pérola, Carioca, IAC Netuno, LP 11-117, TB 02-19, CHP 04-239-52, CHP 01-182-48, TB 03-11, BRS Valente and Guapo Brilhante.All genotypes have grains of the Mesoamerican gene pool and were developed by different public research institutions that release new carioca and black bean cultivars for cultivation in the southern region of Brazil.
for the following number of experiments: one: experiment I (2016 rainy season crop); two: experiments I and II (2016 rainy and 2017 dry season crops); three: experiments I, II and III (2016 rainy, 2017 dry and 2017 rainy season crops); and four: experiments I, II, III and IV (2016 rainy, 2017 dry, 2017 rainy and 2018 dry season crops).In combined analyses of variance, all effects were considered random, except for the genotype effect, which was analyzed as fixed.The homogeneity of residual variances was analyzed by Hartley's maximum F-test.The degrees of freedom of the error and of the genotype × experiment interaction were confirming previous results described bySteckling et al. (2017),Ribeiro & Kläsener (2020),Dias et al. (2021) andRibeiro et al. (2021).In this case, the formation of groups of common bean genotypes based in genetic dissimilarity will be specific for each environment and the selection of common bean parents for use in controlled crosses will be different for each environment.Thus, the environmental variability between growing years and seasons is very important in cluster analyses in the evaluation of the genetic dissimilarity of the common bean germplasm.With regard to the potassium concentration, the F test was significant for the genotype effect only in the combined analysis variance of three experiments (I, II and III).For this reason, potassium concentration was excluded of the cluster analyses.However, multicollinearity diagnostics showed a condition number (CN) equal to 5,698.14, indicating severe collinearity, by the classes proposed byMontgomery et al. (2012).Thus, highly correlated traits and with a greater weight in the last eigenvectors had to be discarded before cluster analyses were performed.The exclusion of the traits of b*, a* and normal grains resulted in a CN = 16.82, that is, weak multicollinearity.Cruz & Carneiro (2014) recommended deletion the multicollinear variables to prevent that these variables implicitly receiving greater weights in the cluster analyses, which allows the proper interpretation of the results obtained in the analyses.Nerinéia DalfolloRibeiro et al.

Figure 1 :
Figure 1: Dendrograms and cophenetic correlation coefficient (CCC) of the unweighted pair group method with arithmetic mean (UP-GMA), from Mahalanobis' generalized distance, obtained in 17 common bean genotypes evaluated in experiments I (2016 rainy season crop), I and II (2016 rainy and 2017 dry season crops), I, II and III (2016 rainy, 2017 dry and 2017 rainy season crops) and I, II, III and IV (2016 rainy, 2017 dry, 2017 rainy and 2018 dry season crops).

Table 3 :
Common bean genotypes classified in each group obtained by Tocher's optimization method, from Mahalanobis' generalized distance, in experiments I (2016 rainy season crop), I and II (2016 rainy and 2017 dry season crops), I, II and III (2016 rainy, 2017 dry and 2017 rainy season crops) and I, II, III and IV (2016 rainy, 2017 dry, 2017 rainy and 2018 dry season crops)