Implications of additive x additive epistasis for common bean breeding

doi:10.21203/rs.3.rs-2608516/v1

The presence of epistasis in complex traits can be significant and affect the selection of segregating populations undesirably. The purpose of this study was to determine the epistatic genetic components that influence root and shoot traits of common bean and identify their effect on the performance of segregating populations. The field experiment consisted of 49 treatments (backcross progenies, parents and segregating populations in the F₂ and F₃ generations). The cross P₁-BAF53 (Andean) x P₂-IPR 88 Uirapuru (Mesoamerican) as reference. Six traits were taken into consideration: root distribution (%), first pod length (cm), number of grains (plot), plant height (cm), reproductive cycle (days) and number of basal branches (plot). Additive x additive epistasis significantly affected 50% of the above traits. For root distribution (RD), this genetic component deteriorated the mean performance of the progenies in relation to the parents, regardless of the presence of dominance deviations and additive x dominant and dominant x dominant epistasis. In addition, lower variation in the different generations (parents and F₂ and F₃ progenies) was found for RD than for the other traits. Rather, no additive x additive interaction component for plant height was detected, but predominance of genetic deviations of dominance and transgressive segregation events were identified. Thus, additive x additive epistasis can hamper the success of a breeding program, for being associated with low-variability traits. This fact may be related with the number of crosses and/or the choice of genetically more distant parents for the program, and possibly with the presence of linked genes.

Gene interactions

Gene pools

Genetic models

Phaseolus vulgaris L.

Root

Common bean (Phaseolus vulgaris L., 2n = 2x = 22) is doubtlessly the most important grain legume for direct human consumption (Jiménez, 2019). Wild ancestors of the crop that inhabit a relatively narrow ecological niche with soil and climate traits of its own (Central and South America) were domesticated. For this reason, the crop is limited by a range of biotic and abiotic restrictions in all cultivation environments (Assefa et al. 2019). Most likely, genetic bottlenecks caused by domestication further reduced the ability to respond to some stresses, such as specific pathogens and drought (Bitocchi et al. 2013; Schmutz et al. 2014). Traditionally, wherever in the world common bean has been cultivated, conditions were limiting and agricultural inputs minimal. With this in mind, in recent years, root-related traits have become a focus of investigation in common bean breeding programs (Burridge et al. 2016, 2019; Polania et al. 2017). If these traits are not taken seriously enough in common bean breeding, genotypes without phenotypic plasticity will be developed, specifically for highly favorable environments, which will result in crop failure under stress conditions.

Breeding for a quantitative trait such as root distribution is based on the study of its variation, which must be partitioned into components attributed to different causes (Falconer and Mackay 1996). Breeders pay primary attention to heritable or genetic variations, which are characterized by differences in size, shape, color or development, inherent to the genetic constitution of the plants and transmitted to the progenies (Poehlman 1979). This genotypic variation can be broken down into: i) Additive component: caused by differences in the homozygote performance at each locus; ii) Deviation caused by dominance: appears in case of dominance in at least some of the loci; and iii) Deviation caused by interaction: resulting from the interaction between non-allelic genes (Vencovsky and Barriga 1992; Falconer and Mackay 1996). Partitioning the genotypic variation is extremely important to understand the types of predominant gene action in the control of a trait. Additive and dominant gene actions are well known and their role in crop breeding has been widely discussed (Bernardo 2002). On the other hand, the implications of gene interaction effects and possible applications in breeding are not yet fully understood.

Epistasis is only briefly mentioned in breeding books (Allard 1960; Fehr 1987), as the interaction of alleles at different loci (Graner 1950; Holland 2001). The term has caused some confusion between the definitions proposed by classical geneticists and breeders, and was understood in this paper as “any form of non-allelic interaction” (Phillips 1998). Although epistasis usually represents only a minor fraction of the genotypic value, it can, if significant: i) be converted into additive variance (under genetic drift or inbreeding); ii) contribute to a temporary response to selection, especially in allogamous species and iii) give rise to populations with specific aptitudes for certain locations or regions of cultivation (Holland 2001). In addition, theoretical bases and empirical evidence have proven the contribution of epistasis to inbreeding depression (Fu and Ritland 1996) and heterosis (Hill 1982). These findings and their implications are still fragmentary for common bean.

In general, the progress of breeding programs in selecting progenies with a satisfactory root development has been slow, since other, shoot-related traits are prioritized (Tracy et al. 2020). For common bean, the presence of transgressive segregants has been reported as an exception rather than the rule (Toaldo et al. 2013; de Melo et al. 2016; Grigolo et al. 2021). Of the few progenies detected, a significant portion was found to have an inferior performance to the parents (Cerutti et al. 2020; Grigolo et al. 2021). A possible explanation for this pattern could be the presence of epistatic components that influence the trait. Thus, this study hypothesized that: i) the trait root distribution in common bean may be controlled by genes that interact additively (additive x additive) and ii) significant epistatic effects can be detected equally for root and shoot-related traits. The specific objective was to: determine the epistatic components that control the expression of root and shoot traits in common bean, associating their presence or absence with the performance of segregating populations.

Analysis of variance is the most important information synthesis of a controlled experiment. By this tool, the variation in observed values of the response variable caused by each experimental factor can be discriminated to draw conclusions about the experimental factors and each unit factor that classifies observation units determined by relevant extraneous variables (soil chemical variations, soil moisture, seed size etc). For this reason, the first steps in mating schemes already allow an estimation of the genetic components involved in inheritance. For example, in different generations derived from a given cross (F₂, F₃, backcrosses) different phenotypic responses can be expressed. In this regard, one must know how much of these responses are due to the treatment (genetic variation) and how much to chance (error). In an incomplete block design, the sources of variation are treatment effects and unit factor effects. In this study, only the treatment effect was needed, for which the results of analysis of variance are displayed (Table 1).

Significant treatment effects were observed for all traits, except for the reproductive cycle (p = 0.2816) (Table 1). This suggests that differences were expressed in the genotypes derived from the cross P₁-BAF53 (Andean) x P₂-IPR 88 Uirapuru (Mesoamerican), which may lead to an erroneous decision in approximately one of every 10,000 repetitions of this experiment, i.e., the reliability was very high. This result could be expected since the parents were chosen for their differences, with potential for genetic variation. As the analysis attributed the errors from the experimental units (plots) and the observation units (plants) differently – a rather common situation in the early stages of breeding programs – any detected variation can only be attributed to the primary treatment effects. In addition, variation was highest for the treatment effect for plant height (mean square), exponentiating the phenotypic differences between these genotypes with regard to plant architecture (Table 1).

The detected variation can be partitioned into each genetic component in order to understand the possible epistasis effects affecting each trait. The treatment factor has an orthogonal structure in which previously established comparisons analyze the mean values of each treatment. To obtain the additive, dominant and epistatic genetic components, predicted functions with fixed and random effects were generated. These functions are suitable to recover inter-block information (Littell et al. 2006). Since common bean is an authentic autogamous species, the genetic effects of additive nature are of primary interest (Table 2).

The trait root distribution had a significant effect for both the additive genetic component (4.13) and additive x additive epistasis (1.12). These findings were similar for the traits first pod length and number of grains (Table 2). Therefore, a single additive-dominant model is not reliable to generate data for these three traits. Usually, the value of the additive component is higher than that of the additive x additive component. However, this was not confirmed for first pod length, which can be attributed to an inaccurate estimate of the additive component under epistasis. On the contrary, no significant effects of additive x additive epistasis were observed on the traits plant height (p = 0.2339), reproductive cycle (p = 0.3484) and number of basal branches (p = 0.6280) (Table 2). For these traits, only allelic interaction was relevant. These results may be associated with the variation of each trait within the fixed and segregating populations.

Among all traits, a considerable range of variation was observed for plant height and number of basal branches. For these characters, the coefficient of variation oscillated from 21 to 79% and 22 to 29%, respectively (Table S2), and variance was also considerable in the fixed and consequently the segregating populations. For plant height for example, the variances of parents P₁-BAF53 and P₂-IPR 88 Uirapuru were 465.75 and 322.16, respectively. These were the highest and lowest observed values for this trait and resulted in F₂ and F₃ progenies with equally high variance. The identified variation was partly due to the components of non-additive inheritance (dominance, additive x dominant and dominant x dominant epistasis). On the contrary, for the traits for which the additive x additive epistatic component was significant, the coefficient of variation was only 12%, and variance was lower compared to the previous values. For root distribution, the magnitude of variance of parents and progenies was the same. The genetic constitutions P₁, P₂ and F₂ were part of the mating scheme of this study, which helps to explain the presence or absence of additive x additive epistasis for the different studied traits. In this sense, the results showed that for the traits for which additive x additive epistatic effects were identified, variation was lower in the parents and F₂ and F₃ progenies compared to the other traits.

In addition to the variation resulting from the additive components of genetic variation, the remaining part of the genotypic value (dominance deviations and additive x dominant and dominant x dominant epistatic deviations) must be considered (Table 3). Deviations result from the property dominance between the alleles of a locus (Falconer and Mackay 1996). Of the genotypic dominance deviations estimated for each progeny, only two, with negative means (-0.5 and − 1), were significant for root distribution, while two other deviations bordered the significance level (data not shown). The predicted function by which this component can be estimated for every i^− th progeny (L1_i - L2_i), computes a negative genetic difference between P₁-BAF53 and P₂-IPR 88 Uirapuru; this may be related to alleles favorable for the trait expression of the Mesoamerican parent. In addition to dominance deviations, the number of significant genetic deviations for additive x dominant and dominant x dominant epistasis was highest of all traits for root distribution (Table 3). Possibly, these deviations were related to the differentiated performance of segregating progenies for this trait.

For the other traits, different results were observed regarding the remaining components of the genotypic value. A greater expression of positive genetic deviations was identified, as well as higher additive x dominant and dominant x dominant epistasis deviations in the traits first pod length and number of grains, compared to the dominance deviations. On the contrary, for the reproductive cycle, number of basal branches and mainly for plant height, more dominance deviations than non-additive epistasis deviations were identified (Table 3).

The main genetic phenomena involved in plant breeding are generally determined by the extent and magnitude of these remaining components of genotypic value. A direct association of these components with heterosis or inbreeding is not an easy task. Otherwise, the genetic causes of heterosis could be elucidated by an overall unifying theory, for example. Some reasons for this difficulty: i) a high number of genes is involved in the control of quantitative traits; ii) for a given genotype involving several loci in heterozygosis, some can be positive and others negative, i.e., they are favorable or unfavorable in relation to the trait and, last but not least, iii) the impossibility of representing the entire genetic variation of the segregating populations, especially in field experiments. Therefore, any explanatory attempts of these effects are mere speculations that must be interpreted with caution. Due to the fact that the fixed and segregating populations were cultivated in the same experiment as the backcross progenies, predicted functions were designated that would allow genetic comparisons between progenies (F₂ and F₃) and parents (P₁ and P₂) (Table 3).

Comparisons of the parent means with F₂ progenies detected no significance for root distribution. The comparison with F₃ progenies however found significant evidence, although the result of this predicted function was a negative value, indicating a poorer performance of the progenies than the parents. The same performance pattern was observed for first pod length, with a 2 cm longer mean value of the parent than F₃ progenies. From another angle, progenies with transgressive segregation were observed for the traits plant height and reproductive cycle; in other words, under predominant additive effects, certain progenies with specific allelic combinations may have phenotypes that are superior to the parents. For example, the F₂ progenies were 21 cm taller than their parents. These results can also be explained by the significant dominance deviations observed for these traits (Table 3).

The mating schemes that can detect gene interactions do not include genotypes to estimate transgressive segregation (parents and F₂ and F₃ progenies). Therefore, to illustrate the effect of genetic components on this phenomenon of transgressive segregation, models were developed for two traits of this study (root distribution and plant height), for which the results were contrasting (Table 4). By least square models, two matrices are established: the first (X) defines the genetic parameters that control the model and the second (Y) the means of each treatment. Based on the results of Triple Test Cross analysis for plant height, only the basic parameters of the additive-dominant model were considered (mean – m; homozygote deviation from the mean - a and heterozygote deviation from the mean - d). For root distribution, on the other hand, these three parameters were added plus the additive x additive epistatic component - aa.

The genetic model proposed for the trait root distribution was based on the classical genetic definitions for two genes with two alleles, with some reservations. The contrast or genetic distance between the parents was not assumed to be 100%. Consequently, in the F₂ generation, the contribution of heterozygous loci (parameter d) was only 0.25 instead of 0.5 (50% reduction in heterozygosity). Finally, the presence of an additive x additive epistatic component in the backcross progenies and parents was suggested. In this way, the following model for root distribution was established: 50.79–3.75a + 16.88d + 22.04aa. This revealed that the additive x additive epistatic component (estimated at 22.04) in fact contributed to the phenotype of root distribution (Table 4).

On the other hand, the genetic model proposed for plant height took all basic premises into consideration, from the total contrast between parents to the composition of segregating populations with positive contributions from heterozygote deviations (d). The estimated model (55.34 + 22.46a + 30.15d) revealed significant contributions from the additive (22.46) as well as the dominance component (30.15), while the deviations from the observed and estimated means were only due to chance (non-significant deviation). In view of the magnitude of the estimated dominance parameter and the absence of additive x additive epistasis effects, transgressive segregation effects on this trait seem likely.

For the cross used in this study, potential for genetic variation, at different magnitudes according to the trait considered, was detected. For root distribution, first pod length and number of grains, the additive and additive x additive components were significant. On the contrary, for plant height, reproductive cycle and number of basal branches, no interaction component was observed. These results can be explained by the magnitude of variance determined for these traits. Our findings suggested that additive x additive epistasis had a negative influence on the mean performance of segregating progenies, because even for traits for which genetic deviations of dominance and non-additive epistasis were observed, the mean of the progenies was not superior to that of the parents. Contrariwise, transgressive segregation effects on plant height were observed as a function of genetic dominance deviation. Using the genetic models proposed for root distribution and plant height, the genetic performance of the progenies derived from this cross could be validated.

The model of genetic components can be improved at a progressive rate, as segregating plants are included, regardless of the mating scheme. However, hybridization between Andean and Mesoamerican genotypes usually results in phenotypic abnormalities among the descendants, e.g., infertility of F₁ hybrids or partial sterility in segregating generations (Gepts and Bliss 1985). This peculiar feature substantially reduces the amount of seeds produced per cross. In addition, the Andean genotype (BAF53) was the female parent in this study, further reducing the expectations of seed production (due to the lower number of seeds per pod) in both fixed (F₁) and segregating (F₂) generations. Not only was the number of established progenies low, but they were also highly sensitive to various environmental factors when grown in the field. This led to the death of a significant number of plants, thus disturbing the balance of the experimental data. In this research, due to the insufficient amount of seeds, the F₁ populations were not planted either. However, this progeny had been evaluated for the same traits in the field in the previous two years (Cerutti et al. 2020; Grigolo et al. 2021). In those experiments, 30 different F₁ hybrid combinations as well as the F₂ progenies and parents had been evaluated.

The trait root distribution has a certain phenotypic plasticity that can be altered according to the environmental conditions (Lanna et al. 2016). Among the main root functions, water and nutrient uptake are particularly important. Thus, under water stress, the genotype tends to strike its roots deeper into the soil than when the soil moisture conditions are optimal, for example. The 49 treatments completed the growth and development cycle irrigated by only 124.20 mm of cumulative rainfall in the four months of the experiment (Figure S1). This level is almost three times lower than the ideal amount for the crop (300–400 mm). Due to the trait expression (in terms of root percentage) reported for this environmental condition, the partitioning of the genetic components may have been more refined than under favorable environment conditions (well-distributed rainfall in the field or a greenhouse experiment).

Previous experiments with autogamous plants have shown the predominance of additive effects on inheritance of root traits (Kashiwagi et al. 2008; de Melo et al. 2016, 2018) and shoot traits (Toledo et al. 2000). Likewise, an additive x additive component was detected in the genetic control of root length and root dry weight for chickpea (Kashiwagi et al. 2008), as well as for five root traits of common bean, in the cross of the genotypes ICA Pijao x Ouro Negro (Araújo et al. 2005). Although both genotypes of the latter study belong to the Mesoamerican group, they come from different regions (Ouro Negro - Brazil, ICA Pijao - Colombia).

Other studies also reported epistasis in crosses between parents from different gene pools. Similarly to this study, in a cross of Andean x Mesoamerican parents (BRS Radiante x Carioca-MG), the interactions “additive x dominant and dominant x dominant” were significant for the traits number of pods per plant and number of grains per plant. For grain weight per plant, all types of epistasis were significant, but the additive x additive component was more relevant than the others (larger mean square) (Moreto et al. 2012). Moreover, a previous study on the same genetic constitution (BRS Radiante x Carioca-MG) highlighted that the F₂:F₃ and F₂:F₄ progeny means were reduced by 44.91% (grain yield, in g plot^− 1) and 61.61% % (100-grain weight, in g) in relation to the parent mean, when they contributed equally to the mating (frequency = 0.50) (Baldoni et al. 2008). In other words, the epistatic component (especially additive x additive) was detected in crosses involving parents from different gene pools, as reported in this research. This component seems to be involved in the progeny performance over the selfing generations, with direct influence on plant selection in breeding programs.

The occurrence of epistasis has been more frequent in crosses between than within the gene pool (Borel et al., 2016). In the above study, shoot traits (yield and 100-grain weight) were evaluated, and mainly additive x dominant and dominant x dominant epistasis was detected. Bearing in mind that the Triple Test Cross (TTC) design was developed to detect non-allelic interactions in genes for which the parents differ (Kearsey and Jinks 1968), the identification of epistatic components is to be expected in hybridizations between different pools.

Why does additive x additive epistasis occur in certain traits but not in others?

The presence of additive and non-additive epistasis is related to the genetic background in a given cross (Mather and Jinks 1971). If both parents maintain alleles in their genome that interact predominantly additively, additive x additive genetic interactions may be identified in the progeny resulting from this cross. Possibly, the alleles that control root distribution, first pod length and the number of grains of the genotypes of this study interact additively. In addition, a proportion of genes involved in genetic control may or may not be contrasting between parents, as for example A¹A¹ vs A¹A¹ or A²A² vs A²A².

The descriptive measures estimated for each genotype (segregating and fixed) showed that for root distribution, first pod length and number of grains, the variation was relatively low compared to plant height, for example. This suggests that no additive x additive epistasis affected the traits with lower variation (both between parents and between progenies). This hypothesis would however not explain the result for the reproductive cycle, for which no additive x additive epistasis was detected, but considerable variation between genotypes. In this situation, this was the only trait for which no significant variation between treatments was observed (Table 1). In addition, the climatic conditions during the experiment, above all precipitation, may have influenced the reduction in days from flowering to harvest in all genotypes (Figure S1).

The genetic model test confirmed the presence of the additive x additive component for root distribution. Furthermore, the model detected overestimation of the dominance component (16.88) to the detriment of the additive component (-3.75). When the least square method is used, additive effects are expected to predominate, except in the case of epistasis, genetic linkage or overdominance (Bernardo 2002). As common bean is an authentic autogamous plant, overdominance as a possible cause can be ruled out. On the other hand, in traits controlled by many genes, the occurrence of genetic linkage is to be expected. In this case, especially if linkage occurs in the repulsion phase, the dominance component tends to be overestimated (Kearsey and Pooni 1996). Thus, the presence of epistasis and a possible linkage disequilibrium may have caused the dominance deviations detected for root distribution.

Is there any relationship between additive x additive epistasis and the mean progeny performance?

Commonly, the mean performance of progenies derived from crosses is compared to the mean parent performance. When additive inheritance is predominant for a trait, transgressive performance of these superior progenies is expected. On the other hand, progenies may also have a superior performance in the case of dominance deviations and non-additive epistasis. These deviations were observed for root distribution, first pod length and number of grains. In this way, the appearance of superior progenies could be explained. However, no improved F₂ and F₃ progenies were found that would allow selection by the breeder. This fact can be explained by the presence of the genetic component of interaction.

Most likely, in addition to the epistatic components in the progenies, the parents must have genes interacting additively across their genome. These interactions play an important role for the adaptation of each genotype in its native region (Andes or Mesoamerica). In progenies derived from these parents, the coadapted gene complexes are undone, resulting in an inferior progeny performance (Johnson and Gepts 2002). This pattern is probably highly peculiar for the trait root distribution. It is known that the ecological niches of each gene pool are relatively specialized. For example, the Mesoamerican genotypes are well-adapted to dry spells in the middle of the season and to fertile soils of volcanic origin in an almost equatorial geographic range, whereas the Andean population is better adapted to cold regions (Bitocchi et al. 2013; Ariani et al. 2016). These factors may have triggered the development of the gene interactions, according to the cultivation environment.

What is the real effect of epistasis or how can this interaction be properly exploited in common bean breeding programs?

More often than not, the probability of identifying superior plants in a breeding program can be compared to the odds of winning the lottery (Van Ginkel and Ortiz 2018). Numerous factors affect the development of this search, and the chances of breeding an ideal genotype could be improved by refining the genetic knowledge about a given trait (Allard 1960). Although apparently unimportant, epistasis was significant for the traits analyzed in this research and affected the performance of segregating progenies negatively, in particular with regard to root distribution. The phenotypic plasticity of this trait is high, so that the presence of additive x additive interaction may be related to the adaptability of a genotype to the environment of cultivation, to ensure species survival and perpetuation. Thus, in more restrictive environments, this genetic component may become more important than in environments under intense fertilization, input application and irrigation. Classical breeding programs recommend that the selection of segregating populations should be carried out in the environments where the genotypes will be cultivated (Ceccarelli 2015). Most likely, breeders will therefore detect the said genetic component, which can obscure the results of plant selection, or rather, the selection of an ideal genotype for root distribution is hampered.

Whether the presence of epistasis is advantageous or disadvantageous is defined by a series of factors, e.g., which stage the breeding program is in, namely: i) choice of parents or ii) early selection stages or iii) final selection stages. In phase i, breeders who work with a high number of crosses between a selected group of parents will be less surprised by negative epistatic and gene linkage effects. A higher number of crosses will allow greater inter- and intra-chromosomal recombination, releasing greater genetic variation. This variation will be reflected in a spectrum of genotypes that includes plants with good performance for the target trait. In autogamous breeding for example, preferably three or four generations of crosses should be established, using a sufficiently high number of parents to consolidate the improved genotype (Johnson and Bernard 1962; Vernetti 1983). Extra care must also be taken with regard to the genetic distance between the parents, since genetically similar parents will not convey new characteristics to their progenies. In the early selection stages (ii), if additive x additive gene action influences the trait, selection should be less rigorous in the first selfing generations (Vernetti 1983). The reason is that epistasis can alter the performance of the progenies until they become lines. In the final stages of a breeding program (iii), the additive x additive interaction can be converted into additive variance, contributing to the selection of superior plants.

Genotypes and assessment site

The trait root distribution has been extensively evaluated in the field since more than a decade (more precisely since 2009), by a methodology that measures root distribution in the soil profile as response variable (Rocha et al. 2010; Toaldo et al. 2013; de Melo et al. 2016, 2018; Velho et al. 2018; Grigolo et al. 2021). Based on the published results, the cross P₁-BAF53 x P₂-IPR 88 Uirapuru was defined as study object to detect the presence of epistasis for the said trait, as well as for other agronomically characteristics relevant for common bean breeding. The genetic constitution represented in BAF53, an accession of the Active Germplasm Bank of the University of the State of Santa Catarina (UDESC), was originally identified in the county of Cunha Porã, Santa Catarina. This accession with sulfur-colored beans belongs to the Andean gene pool (phaseolin type T), has a determinate growth habit and a mean root distribution of 25%, while cultivar IPR 88 Uirapuru was developed by the breeding program of the Agronomic Institute of Paraná (IAPAR) and is widely cultivated in Brazil. It belongs to the black bean commercial group, of the Mesoamerican gene pool (phaseolin type S), has an indeterminate growth habit and mean root distribution of 35%. In previous experiments, broad-sense heritability for root distribution in the segregating population F₂ of P₁-BAF53 x P₂-IPR 88 Uirapuru was estimated at 23%.

The F₁ progenies (P₁-BAF53 x P₂-IPR 88 Uirapuru) were obtained by flower bud emasculation and artificial pollination. Subsequently, the F₁ progenies were selfed to establish the F₂ progenies. Once the P₁, P₂, F₁ and F₂ populations were available, they were crossed according to the half-sib backcross model (Triple Test Cross - TTC). This mating scheme is an extension of the method proposed by Comstock and Robinson, which includes, in addition to the additivity and dominance estimates, an unbiased epistasis test (Comstock and Robinson 1952; Kearsey and Jinks 1968). The design involves backcrosses of each F₂ plant with three testers: P₁ x F₂ (backcross L₁), P₂ x F₂ (backcross L₂) and F₁ x F₂ (backcross L₃). Apart from the backcross progenies (45 treatments), the genotypes P₁, P₂, F₂ and F₃ were integrated in the experiment, i.e., a total of 49 treatments. With the inclusion of these genetic constitutions, the segregating could be compared with the fixed populations.

The experiment was carried out in the field in the county of Lages (Santa Catarina, Brazil) from December 2019 to March 2020 (Figure S1). The treatments were randomized in a partially balanced incomplete block design (7 x 7), with two replications. The experimental units consisted of two 1-m rows. Two seeds per row were sown, i.e., four plants per plot. The plant density (0.50 x 0.50m spacing between plants and rows) allowed maximum root development of each bean plant, without physical obstacles or barriers. In this way, the treatments were comparable and the genetic components could be determined.

Evaluated traits

Root distribution (%) was evaluated in the field during full flowering (stage R₆), by the excavation method (Böhm 1979), with adaptations. The original method is based on digging a profile in the soil beside the plants, perpendicular to the sowing line, leaving the plants in the soil. After opening the profile, the inside is cleaned by removing excess soil between the roots. The proposed modifications consisted of: i) evaluation of two rather than one plant per plot, and ii) after cleaning the roots, insertion of a grid template of the same size as the profile (0.50m long, 0.30m deep), subdivided in 60 grid squares (0.05 x 0.05m). The entire grid frame was photographed and each of the 60 squares was evaluated for presence or absence of roots (1 or 0, respectively), indicating the percentage of total roots per plant in the template space (\(\left(\sum \text{1/60}\right)\text{*100}\)).

In addition to root distribution (RD), the following shoot traits were evaluated: i) First pod length (FPL): at maturity, the longitudinal section of the first pod above the soil was measured (in cm; total plants per plot); ii) Number of grains (NG): counting the total number of grains of each plant at maturity (total plants per plot); iii) Plant height (PH): at maturity, each plant was measured from the soil to the tip of the vine, when present (in cm; total plants per plot); iv) Reproductive cycle (RC): number of total days from flowering to maturity (two plants per plot) and v) Number of basal branches (NBB): number of (vegetative and reproductive) branches within a standard distance (0.10m) from the ground to the plant apex (total plants per plot).

Statistical analysis

Because of the experimental design of incomplete blocks (each block represents only part of the treatments), a combined intra-block analysis with recovery of inter-block information became necessary (Littell et al., 2006). In addition, certain agronomic traits such as root distribution, number of grains and reproductive cycle were not normally distributed. Considering these two aspects, these traits were analyzed by generalized mixed linear models, based on the following statistical model: \(\text{E}\left[\text{Y|}\text{γ}\right]\text{=}{\text{g}}^{\text{-1}}\left(\text{X}\text{β}\text{+Z}\text{γ}\right)\). In this model are expressed Y: is the observation vector; \(\text{γ}\): the random effects vector; \({\text{g}}^{\text{-1}}\): inverse link function, for each distribution used; X: the known incidence matrix of the fixed effects vector (repetition and treatment); β: fixed effects vector; Z: incidence matrix of random effects (blocks). The linear predictor also considered the error vectors between (experimental error) and within (sampling error). The concept of two errors was linked to the presence of the experimental unit (plot) and observation unit (plant). This allows the correct characterization of the experimental error to ensure valid (unbiased) inferences. For the estimation of random effects, the technique known as restricted pseudo-likelihood (RPL) was used, with an expansion around the estimates of the best unbiased linear predictors of random effects - RSPL. The degrees of freedom were determined by the Containment method, suitable for moderately unbalanced experiments.

The Triple Test Cross (TTC) analysis consists of two parts: one related to the test for additive x additive epistasis (aa) and the other to the testing and estimation of the other genetic components (additive - a, dominant - d and epistatic ad + dd). To this end, the treatment factor is partitioned into genetic effects by means of inference spaces (McLean et al. 1991). The (narrow) inference spaces are derived for specific levels of the random effect, in view of the need to recover inter-block information. As previously mentioned, three backcross types are established (L₁, L₂ and L₃). Initially, additive x additive epistasis is tested according to the inference space L₁ + L₂ − 2L₃ for all progenies evaluated (1 degree of freedom). In the absence of additive x additive epistasis, the comparison of means shows no significant differences. This scale test to determine additive x additive epistasis is the key difference between the TTC mating scheme and others used to study trait inheritance. According to the genetic basis of this test, for the i^− th plant sampled from population F₂, the additive (a) and dominant (d) terms neutralize each other, leaving (if any) only the epistatic effects. This applies to any number of loci, and does not depend on the inbreeding degree, gene frequency or genetic correlations (Kearsey and Jinks 1968). The effects of additive and dominant genetic components are estimated similarly as for the design proposed by Comstock and Robinson, except for the inclusion of F₁. Thus, the additive effect corresponds to the linear combination L₁ + L₂ + L₃ in the mean of the progenies (1 degree of freedom), while the dominance effects are calculated by the expression L1_i - L2_i for every i^− th progeny (14 degrees of freedom). Finally, the additive x dominant and dominant x dominant epistasis effects are estimated simultaneously by the independent scale test L1_i + L2_i − 2L3_i for every i^− th progeny (14 degrees of freedom).

To associate the detected genetic components and possible effects of transgressive segregation, the segregating (F₂ and F₃) were compared with the fixed populations (P₁ and P₂). Additionally, genetic models were tested by the least-squares method, whose parameters are estimated by solving the matrix equation: \(\widehat{\text{B}}\text{=}{\left({\text{X}}^{{\prime }}\text{X}\right)}^{\text{-1}}{\text{X}}^{{\prime }}\text{Y}\), where: \(\widehat{\text{B}}\) is the vector of the estimates of the genetic parameters and Y the matrix of the observed values of each fixed or segregating population.

All analyses were performed using SAS software (SAS OnDemand for Academics) and the GLIMMIX and IML procedures. GLIMMIX was used to analyze variance and partition the treatment effects according to the mating scheme (Table S1). By this procedure, the distribution of each response variable was specified with the function DIST and the tests and estimates of each genetic component were computed using function ESTIMATE. In addition, the means, standard errors and other descriptive measures for each treatment were estimated by function LSMEANS with the specification of confidence intervals (option CL). Function IML was used to test the proposed genetic models. Vector Y (observed mean values) was constructed from the original scale or based on the means, defined by the linkage function, according to the respective trait and its distribution.

Table 1

Analysis of variance for the fixed treatment effect of the traits: root distribution (RD), first pod length (FPL), number of grains (NG), plant height (PH), reproductive cycle (RC) and number of basal branches (NBB).
Trait	Distribution	Treatment effect
Trait	Distribution	NDF ^a	DDF ^b	MS ^c	F value	P > F
RD	Poisson	44	82	0.05	3.65	0.0001
FPL	Normal	43	146	8.90	3.66	0.0001
NG	Lognormal	43	146	0.91	3.19	0.0001
PH	Normal	43	146	1494.67	3.15	0.0001
RC	Lognormal	43	82	0.02	1.16	0.2816
NBB	Normal	43	146	6.56	3.12	0.0001

^a Numerator degrees of freedom. ^b Denominator degrees of freedom. ^c Mean square.

Table 2

Predicted functions (BLUP) with partial representation of the coefficients to determine the additive effect (L1 + L2 + L3) and additive x additive effect (L1 + L2-2L3), based on a model with fixed effects: mean (µ), repetition (R_i, with two levels), treatment (T_j, with 45 levels) and the random effect: repetition(block) (R_kB_l, with 14 levels), for the traits root distribution (RD), first pod length (FPL), number of grains (NG), plant height (PH), reproductive cycle (RC) and number of basal branches (NBB).
		Fixed effect											Random effect
^a	Function	µ	R₁	R₂	T₁	T₂	T₃	T_j ^b	T₄₆	T₄₇	T₄₈	T₄₉	R₁B₁	R₂B₁	R_kB_l ^c	R₁B₇	R₂B₇		BLUP	P > \|t\|
RD	L₁ + L₂ + L₃	(33	16.5	16.5	1	1	1	...	0	0	0	0	1	1	...	1	1)	/33	4.13	0.0001
RD	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	1.12	0.0135
FPL	L₁ + L₂ + L₃	(30	15	15	1	1	1	...	0	0	0	0	1	1	...	1	1)	/30	9.99	0.0001
FPL	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	13.20	0.0001
NG	L₁ + L₂ + L₃	(30	15	15	1	1	1	...	0	0	0	0	1	1	...	1	1)	/30	4.67	0.0001
NG	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	4.55	0.0002
PH	L₁ + L₂ + L₃	(30	15	15	1	1	1	...	0	0	0	0	1	1	...	1	1)	/30	71.53	0.0001
PH	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	59.38	0.2339
RC	L₁ + L₂ + L₃	(30	15	15	1	1	1	...	0	0	0	0	1	1	...	1	1)	/30	3.66	0.0001
RC	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	0.35	0.3484
NBB	L₁ + L₂ + L₃	(30	15	15	1	1	1	...	0	0	0	0	1	1	...	1	1)	/30	4.89	0.0001
NBB	L₁ + L₂-2L₃	(0	0	0	1	1	-2	...	0	0	0	0	1	1	...	1	1)	/2	-1.66	0.6280

^a Distribution considered: RD (Poisson), FPL (Normal), NG (Lognormal), PH (Normal), RC (Lognormal) and NBB (Normal). ^b Other treatment means, continued. ^c Other replication(block) means, continued.

Table 3

Genetic dominance effects (dominance deviations) and additive x dominant + dominant x dominant epistasis (ad + dd deviations) for the traits root distribution (RD), first pod length (FPL), number of grains (NG), plant height (PH), reproductive cycle (RC) and number of basal branches (NBB). Comparisons between segregating populations in the F₂ and F₃ generations *versus* the parent mean are indicated (P₁-BAF53 + P₂-IPR 88 Uirapuru).
Trait ^a	Dominance deviations ^b		ad + dd deviations ^b		Comparison ^c
Trait ^a	Number	Magnitude	Number	Magnitude	F₂ x parents		F₃ x parents
RD	2	(- -)	4	(+ + + -)	0.1341	^NS	-0.3106	^**
FPL	1	(+)	3	(+ + +)	-0.5417	^NS	-1.5790	^**
NG	0	N/A^††††	2	(+ +)	0.0516	^NS	0.0565	^NS
PH	2	(+ +)	1	(+)	21.1497	^*	6.2409	^NS
RC	1	(+)	0	N/A ^¶	0.1854	^*	0.1653	^NS
NBB	1	(+)	2	(+ -)	-0.7919	^NS	0.1819	^NS

^a Distribution considered: RD (Poisson), FPL (Normal), NG (Lognormal), PH (Normal), RC (Lognormal) and NBB (Normal). ^b Significant deviations by inference spaces, according to the TTC mating scheme. ^c Positive values indicate progeny superiority and negative values indicate parent superiority. ^¶ Not applicable. ^*, ^** significant at 5% and 1% probability by the t test. ^NS not significant.

Table 4

Analysis of variance (least-squares method) for the traits root distribution (RD) and plant height (PH) for significance tests of fixed (P₁ and P₂) and segregating genotypes (F₂, F₃, L₁, L₂ and L₃) of the cross P₁-BAF53 x P₂-IPR 88 Uirapuru.
Trait	Source of variation	DF ^a	SS ^b	MS ^c	F value	P > F
RD	Model	3	943.27	314.42	2.95	0.0374
	Deviation	41	6174.45	150.59	1.41	0.0923
	Error	82	8735.99	106.53	-	-
PH	Model	2	4644.96	2322.48	4.88	0.0088
	Deviation	41	9605.46	234.27	0.49	0.9951
	Error	146	69419.83	475.47	-	-

^a Degrees of freedom. ^b Sum of squares. ^c Mean square.

Funding

This work was supported by Brazilian Council for Scientific and Technological Development (CNPq), Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES) and Foundation for the Support of the Scientific and Technological Research of Santa Catarina (FAPESC).

Competing Interests

The authors have no relevant financial or non-financial interests to disclose.

Author Contributions

Each author contributed individually and significantly to the development of this study. Conceptualization: Melo, R.C.; Coimbra, J.L.M.; Silva, J.G.C. Data curation: Melo, R.C.; Carbonari, L.T.S.; Cerutti, P.H. Formal analysis: Melo, R.C.; Coimbra, J.L.M. Supervision: Coimbra, J.L.M.; Silva, J.G.C. Writing – original draft: Melo, R.C.; Coimbra, J.L.M.; Guidolin, A.F.; Silva, J.G.C.; Carbonari, L.T.S.; Cerutti, P.H. Writing – review & editing: Melo, R.C.; Carbonari, L.T.S.; Cerutti, P.H.

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Acknowledgments

The authors are indebted to the State University of Santa Catarina (UDESC), Brazilian Council for Scientific and Technological Development (CNPq), Brazilian Federal Agency for Support and Evaluation of Graduate Education (CAPES) and Foundation for the Support of the Scientific and Technological Research of Santa Catarina (FAPESC) for granting scholarships, funding and support for the development of this publication.

Allard RW (1960) Principles of plant breeding. John Wiley, New York
Araújo AP, Antunes IF, Teixeira MG (2005) Inheritance of root traits and phosphorus uptake in common bean (Phaseolus vulgaris L.) under limited soil phosphorus supply. Euphytica 145:33–40. https://doi.org/10.1007/s10681-005-8772-1
Ariani A, Berny Mier y Teran JC, Gepts P (2016) Genome-wide identification of SNPs and copy number variation in common bean (Phaseolus vulgaris L.) using genotyping-by-sequencing (GBS). Mol Breed 36:. https://doi.org/10.1007/s11032-016-0512-9
Assefa T, Assibi Mahama A, Brown A V., et al (2019) A review of breeding objectives, genomic resources, and marker-assisted methods in common bean (Phaseolus vulgaris L.). Mol Breed 39:. https://doi.org/10.1007/s11032-018-0920-0
Baldoni AB, Patto Ramalho MA, De Abreu Â de FB (2008) Allele frequency and selection efficiency in cross populations of Andean x Mesoamerican common beans (Phaseolus vulgaris L. Fabales, Fabaceae). Genet Mol Biol 31:914–919. https://doi.org/10.1590/S1415-47572008000500019
Bernardo R (2002) Breeding for quantitative traits in plants. Woodbury, Minnesota
Bitocchi E, Bellucci E, Giardini A, et al (2013) Molecular analysis of the parallel domestication of the common bean (Phaseolus vulgaris) in Mesoamerica and the Andes. New Phytol 197:300–313. https://doi.org/10.1111/j.1469-8137.2012.04377.x
Böhm W (1979) Methods of Studying Root Systems. Springer-Verlag, New York
Burridge JD, Findeis JL, Jochua CN, et al (2019) A case study on the efficacy of root phenotypic selection for edaphic stress tolerance in low-input agriculture: Common bean breeding in Mozambique. F Crop Res 244:. https://doi.org/10.1016/j.fcr.2019.107612
Burridge JD, Jochua CN, Bucksch A, Lynch JP (2016) Legume shovelomics: High-Throughput phenotyping of common bean (Phaseolus vulgaris L.) and cowpea (Vigna unguiculata subsp, unguiculata) root architecture in the field. F Crop Res 192:21–32. https://doi.org/10.1016/j.fcr.2016.04.008
Ceccarelli S (2015) Efficiency of Plant Breeding. Crop Sci 55:87–97. https://doi.org/10.2135/cropsci2014.02.0158
Cerutti PH, Grigolo S, de Melo RC, et al (2020) Combining ability between common bean gene groups for root distribution trait. Cienc e Agrotecnologia 44:. https://doi.org/10.1590/1413-7054202044011520
Comstock RE, Robinson HF (1952) Estimation of the average dominance of genes. In: Heterosis. Iowa State College, Ames, pp 494–512
de Melo RC, Schmit R, Cerutti PH, et al (2016) Genetic variation in the trait root distribution over segregating generations of common bean. Euphytica 207:665–674. https://doi.org/10.1007/s10681-015-1568-z
de Melo RC, Trevisani N, Corrêa SC, et al (2018) Inheritance of root distribution in common bean and selection strategy. Crop Breed Appl Biotechnol 18:373–381. https://doi.org/10.1590/1984-70332018V18N4A56
Falconer DS, Mackay TFC (1996) Introduction to Quantitative Genetics. Longman
Fehr WR (1987) Principles of cultivar development. Macmillan Publishers Company, New York
Fu YB, Ritland K (1996) Marker-based inferences about epistasis for genes influencing inbreeding depression. Genetics 144:339–348. https://doi.org/10.1093/genetics/144.1.339
Gepts P, Bliss FA (1985) F1 hybrid weakness in the common bean. J Hered 76:447–450. https://doi.org/10.1093/oxfordjournals.jhered.a110142
Graner EA (1950) Elementos de genética: Bases para o melhoramento de plantas e animais, 4th edn. Edições Melhoramentos
Grigolo S, de Melo RC, Fioreze AC da CL, et al (2021) Heterosis for the root distribution trait in common bean. Acta Sci - Agron 43:1–8. https://doi.org/10.4025/actasciagron.v43i1.46712
Hill WG (1982) Dominance and epistasis as components of heterosis. Zeitschrift für Tierzüchtung und Züchtungsbiologie 99:161–168. https://doi.org/10.1111/j.1439-0388.1982.tb00375.x
Holland JB (2001) Epistasis and Plant Breeding. Plant Breed Rev 21:27–92
Jiménez OR (2019) Common Bean (Phaseolus vulgaris L.) Breeding. In: Al-Khayri J, Jain S, Johnson D (eds) Advances in Plant Breeding Strategies: Legumes. Springer, pp 151–200
Johnson HW, Bernard RL (1962) Soybean Genetics And Breeding. Adv Agron 14:149–221. https://doi.org/10.1016/S0065-2113(08)60438-1
Johnson WC, Gepts P (2002) The role of epistasis in controlling seed yield and other agronomic traits in an Andean x Mesoamerican cross of common bean (Phaseolus vulgaris L.). Euphytica 125:69–79. https://doi.org/10.1023/A:1015775822132
Kashiwagi J, Krishnamurthy L, Gaur PM, et al (2008) Estimation of gene effects of the drought avoidance root characteristics in chickpea (C. arietinum L.). F Crop Res 105:64–69. https://doi.org/10.1016/j.fcr.2007.07.007
Kearsey MJ, Jinks JL (1968) A general method of detecting additive, dominance and epistatic variation for metrical traits. Heredity (Edinb) 23:403–409
Kearsey MJ, Pooni HS (1996) The genetical analysis of quantitative traits. Chapman & Hall
Lanna AC, Mitsuzono ST, Gledson T, et al (2016) Physiological characterization of common bean (Phaseolus vulgaris L.) genotypes, water-stress induced with contrasting response towards drought. AJCS 10:1–6
Littell RC, Milliken GA, Stroup WW, et al (2006) SAS for Mixed Models, 2nd edn. SAS Institute Inc., Cary, North Carolina
Mather K, Jinks JL (1971) Biometrical genetics. Cornell University, Ithaca, NY – USA
McLean RA, Sanders WL, Stroup WW (1991) A unified approach to mixed linear models. Am Stat 45:54–64. https://doi.org/10.1080/00031305.1991.10475767
Moreto AL, Ramalho MAP, Bruzi AT (2012) Epistasis in an Andean × Mesoamerican cross of common bean. Euphytica 186:755–760. https://doi.org/10.1007/s10681-011-0578-8
Phillips PC (1998) Anectdotal , Historical and Critical Commentaries on Genetics The Language of Gene Interaction. Genetics 149:1167–1171
Poehlman JM (1979) Breeding Field Crops, 2nd edn. Avi, Westport, Connecticut
Polania J, Poschenrieder C, Rao I, Beebe S (2017) Root traits and their potential links to plant ideotypes to improve drought resistance in common bean. Theor Exp Plant Physiol 29:143–154. https://doi.org/10.1007/s40626-017-0090-1
Rocha F, Coan MMD, Coimbra JLM, et al (2010) Root distribution in common bean populations used in breeding programs. Crop Breed Appl Biotechnol 10:40–47. https://doi.org/10.12702/1984-7033.v10n01a06
Schmutz J, McClean PE, Mamidi S, et al (2014) A reference genome for common bean and genome-wide analysis of dual domestications. Nat Genet 46:707–713. https://doi.org/10.1038/ng.3008
Toaldo D, de Morais PPP, Battilana J, et al (2013) Selection in early generations and the occurrence of heterosis for the character root distribution. Euphytica 190:335–344. https://doi.org/10.1007/s10681-012-0746-5
Toledo JFF de, Arias CAA, De Oliveira MF, et al (2000) Genetical and environmental analyses of yield in six biparental soybean crosses. Pesqui Agropecu Bras 35:1783–1796. https://doi.org/10.1590/s0100-204x2000000900011
Tracy SR, Nagel KA, Postma JA, et al (2020) Crop Improvement from Phenotyping Roots: Highlights Reveal Expanding Opportunities. Trends Plant Sci 25:105–118. https://doi.org/10.1016/j.tplants.2019.10.015
Van Ginkel M, Ortiz R (2018) Cross the best with the best, and select the best: HELP in breeding selfing crops. Crop Sci 58:17–30. https://doi.org/10.2135/cropsci2017.05.0270
Velho LPS, de Melo RC, Bernardy JPF, et al (2018) Root distribution and its association with bean growth habit. An Acad Bras Cienc 90:. https://doi.org/10.1590/0001-3765201820170341
Vencovsky R, Barriga P (1992) Genética Biométrica no Fitomelhoramento. Sociedade Brasileira de Genética, Ribeirão Preto, São Paulo
Vernetti FJ (1983) Soja: Genética e Melhoramento. Fundação Cargil, Campinas

No competing interests reported.

supplementary.docx

Implications of additive x additive epistasis for common bean breeding

Status:

Version 1

Abstract

Introduction

Results

Discussion

Why does additive x additive epistasis occur in certain traits but not in others?

Is there any relationship between additive x additive epistasis and the mean progeny performance?

Methods

Genotypes and assessment site

Evaluated traits

Statistical analysis

Declarations

References

Additional Declarations

Supplementary Files

Status:

Version 1