Genetic Merit of Parents and Heterosis in Cassava (Manihot esculenta Crantz)

Abstract

Cassava breeders are led to discard plants before evaluating families, failing to obtain important information, such as the genetic merit of the parents. In this work, we evaluated a Clonal Evaluation Trial (CET) with 1886 clones from 57 crosses in an augmented block design with the 15 parents used as check treatments. We applied a mixed-model framework incorporating pedigree information. Three traits (fresh root yield (FRY), dry matter content (DMC) and dry matter yield (DMY)) were evaluated, and genetic gains considering several selection intensities were investigated. Disregarding the family structure, the gains for FRY (19.96 to 30.80%) and DMY (16.63 to 27.56%) were much higher than the estimated for DMC (7.79 to 11.46%). The means of clones were very near to the mean of parents for all traits, suggesting the absence of heterosis. However, considering the data by family, heterosis varied from −4.38 to 7.53% for FRY, from −2.74 to 1.89% for DMC and from −4.36 to 6.89% for DMY. Heterosis for FRY is not negligible, although it is small. The analysis by family enables us to infer the genetic control of the traits under study. This is not possible when the family structure is disregarded.

1. Introduction

Cassava (Manihot esculenta Crantz), also called yuca or manioc, is native to the area which currently corresponds to the central region of Brazil [1]. The cassava plant, a shrub cultivated for about 10,000 years and whose roots, rich in starch (20–40%), are a source of calories for about 800 million people [2], is the sixth most planted crop in the world and the second most important source of starch, after maize [3]. The native populations of the Americas domesticated and disseminated the cassava plant throughout the American continent and, together with small farmers, are responsible for the conservation of the genetic variability of this species [4]. After the discovery of the Americas, the plant was introduced to Africa and later to Asia by Portuguese traders [2].

Due to its rusticity, it is very suitable to be produced under marginal conditions. At the global level, cassava occupied an area larger than 32 million hectares, yielding more than 330 million tons of fresh roots in 2022. Brazil had the fourth largest area harvested and the sixth total production in the world in this year [5]. Besides the importance of cassava as food, its starch, the main product of the crop, has a broad range of industrial applications [3].

Despite the relevance of this crop, its genetic improvement is recent [6,7]. Although it produces seeds, cassava is vegetatively propagated by farmers. A particularity of vegetative propagation is that it allows the immediate cloning of a superior genotype. For this reason, in species of vegetative propagation, each descendant of a given cross can be considered a potential cultivar [8]. This correct statement gives a false impression that the improvement of vegetative propagation species is extremely simple. However, in the improvement of any crop, obtaining genetic progress is an unavoidable objective, and in this aspect, cassava has particularities that make it difficult to obtain high genetic gains. Cassava has male and female flowers arranged in inflorescences (racemes or panicles). In the same inflorescence, female flowers open about 10 to 14 days before male flowers [9], which favors cross-pollination. Cross-pollination produces—and vegetative propagation maintains—the high heterozygosity of cassava [4,10].

Thus, due to the high heterozygosity, when two cassava clones are crossed, there is relatively greater segregation within than between progenies. This low segregation between progenies makes it difficult to select superior parents. In turn, large segregation within families requires the evaluation of a large number of genotypes per family, and the selected individuals need to be evaluated in a large number of environments, which makes the cassava breeding process slow and expensive [11].

There are also characteristics inherent to the biology of the cassava plant that slow down its improvement. Cassava plants grown from seeds (“seedlings”) tend to produce main roots. Thus, root yield is not evaluated in the stage of seedlings, which are further cloned. In turn, the low multiplication rate of cassava (1:5 to 1:10) delays the obtaining of planting material in sufficient quantities for setting up experiments with replicates and in several locations [3]. This initial slowness forces breeders to make a selection based on traits of higher heritability [3], such as plant size and the color of root peel and pulp; thus, when the amount of planting material necessary for setting up experiments with repetitions is reached, the families are already incomplete. For this reason, and induced by vegetative propagation, cassava breeders start to focus on individual genotypes (clones) instead of families, which compromises the obtaining of essential genetic parameters for the establishment of heterotic groups, for example.

Differences in flowering capacity are another inherent characteristic of the cassava plant that makes its improvement difficult. Some cassava clones flower 3–5 times in a cultivation cycle, others flower little or late, and others do not flower at all [9], which makes it difficult to carry out certain crosses and implement crosses in a diallel scheme, which allows the estimation of general (GCA) and specific (SCA) combining abilities, which in turn are important for the identification of heterotic groups.

Another common aspect of the improvement of vegetative propagation species is that, once a cross is carried out, the individuals are evaluated and selected without further recombination for a long period (about 8 years, in the case of cassava). Given that the probability of occurrence of a superior genotype in a cross is low and there is no recombination until the selective cycle is completed, it can be seen that this combination of factors constitutes an additional difficulty in obtaining genetic gains in cassava.

Genetic divergence, i.e., the difference between allelic frequencies in the parents involved in a cross, is one of the conditions for the occurrence of heterosis [12]. It turns out that a low divergence (and consequently low heterosis) is expected in the cross between highly heterozygous parents, which is one of the probable causes of the low genetic progress obtained in cassava improvement.

Based on the scenario described, the selection of parents in cassava is a bottleneck, and the breeding strategies currently employed do not provide tools for the breeder to make/predict the best crosses. Among the alternatives to know the breeding value of potential parents for using in crosses, the diallel design is widely known and used, but in cassava, it has not been largely employed for evaluating a great number of parents due to the difficulty of some genotypes in flowering [3], which compromises the obtaining of all the desired combinations in a field of crossing involving several parents. Diallel cross is a system that involves all possible crosses between a group of parents, which can be lineages, open-pollinated varieties, or an F₂ generation. One variation is the partial diallel cross, which consists of the cross between two groups of parents [13]. In general, the information that can be obtained in diallel crosses is the following [14]: general (GCA) and specific (SCA) combining ability, reciprocal (or maternal) effect, and heterosis.

Best linear unbiased prediction (BLUP) provides estimates of breeding values without mating designs, as pedigree-based variance-covariance relationship matrices describe the resemblance between relatives in a linear mixed model [15].

The mixed model methodology is a powerful tool for prediction in which breeders can accommodate unbalanced data covariance among levels of random effects and can borrow information across evaluated individuals with varying accuracies [16].

The mixed model’s methodology was already successfully used in animal evaluation to predict the genetic merit of sires [17] and has been used in plant breeding, especially for predicting hybrid performance in allogamous crops, such as maize [18]. However, scarcer in cassava, the successful use of mixed models to predict clones for selection and use as parents in order to obtain the next cycle of genomic selection has been reported [19].

The objective of this work was to verify the feasibility of using breeding values as a criterion for selecting parents and to compare selection with and without the family structure in order to verify the occurrence of heterosis in cassava.

2. Materials and Methods

2.1. Genotypes

In our evaluation trials, we evaluated 1886 cassava clones belonging to 57 full-sib families (Supplementary Table S1), stemming from manual crosses of 15 cassava parents from the cassava germplasm bank and the cassava breeding program of Embrapa Cassava & Fruits, located in Cruz das Almas, Bahia, Brazil. The genotypes BGM 0131, BGM 0212, BGM 0254, BGM 1116, BGM 1632, BGM 1660, BGM 1709, BGM 1722, BGM 1807, BGM 2024 and BGM 2038 are accessions from the germplasm bank. BRS Dourada, BRS Gema de Ovo and BRS Jari are cultivars already released, while 2003 14-11 is a hybrid generated by the breeding program of that institution.

2.2. Crosses

The crosses were made in that referred institution in 2020. On the day before the opening of female flowers, the inflorescences were covered with voile bags (20 cm × 15 cm) in order to avoid contamination with undesirable pollen, while the male flowers from other parents were harvested, identified and stored in plastic pots. The next morning, the female flowers were uncovered, and the male flowers were rubbed against them. After the pollination, non-fertilized (immature) female flowers and the male flowers were eliminated (emasculation), and after identification of crossing (indication of both parents on a label), the fertilized female flowers were covered again in order to avoid the loss of seeds, as the cassava has dehiscent fruits.

The seeds, collected around 4 months after crosses, were stored in a cold chamber and later sown in tubes in a greenhouse, where the seedlings grew for 60 days before being transplanted to the field. The plants were cut 12 months after planting, and the ones capable of generating seven stem cuttings of about 20 cm were selected.

2.3. Experimental Design and Evaluations

The trial was carried out in 2022 at the experimental area of Embrapa Cassava & Fruits in Cruz das Almas, BA, Brazil (12°40′42.4″ S, 39°05′27.8″ W, 224 m asl). The soil was plowed and harrowed, and then furrows of about 10 cm depth were opened. Fertilization at planting consisted of Phosphorus (60 kg ha⁻¹ P₂O₅) and Potassium (40 kg ha⁻¹ K₂O) applied in the furrows, whereas Nitrogen (30 kg ha⁻¹ N) was applied as topdressing at 45 days after planting.

The experimental design was an augmented block design [20] in which the 15 parents were used as check treatments. The 15 parents were randomly distributed (repeated) into 4 blocks, and after the 1886 clones were also randomly distributed (non-repeated) in the blocks. The plots had one row of seven plants spaced 0.9 m apart and 0.7 m between the plants.

Cultural management comprised weeding within the first three months after planting and ant control with granulated baits within the first month of growing.

At 12 months after planting, the final stand of each plot was counted, and the harvest was performed by hand. The fresh root yield (FRY) of the whole plot was weighed with a digital scale (Brecknell ElectroSamson 45 kg × 0.01 kg, Fairmont, MN, USA), and the values were converted to yield per hectare (t ha⁻¹). After weighing, a sample of roots of about 5 kg was taken for determination of dry matter content (DMC) using the same digital scale mentioned earlier, according to [21]:

$$\mathrm{DMC} (\%)=158.3\times \left[{\displaystyle \frac{\mathrm{weight} \mathrm{in} \mathrm{air}}{\mathrm{weight} \mathrm{in} \mathrm{air}-\mathrm{weight} \mathrm{in} \mathrm{water}}}\right]-42$$

The estimate of dry matter yield (DMY, in t ha⁻¹) was obtained by multiplying FRY by DMC (divided by 100). All the 1886 clones were assessed for FRY, but just 1442 were evaluated regarding DMC (and consequently for DMY) because some plots did not produce enough roots (5 kg) to estimate the dry matter content, and in other cases, there was loss of labels during root transport. Therefore, for data analysis, a file with the 1886 observations of FRY, another with the 1442 observations of DMC and DMY, and a third with the 1442 observations of FRY, DMC and DMY were arranged to calculate the correlations.

The steps of the work are presented in Figure 1.

2.4. Data Analysis

The phenotypic data were subjected to mixed model analysis by Restricted Maximum Likelihood/Best Linear Unbiased Prediction (REML/BLUP) methodology [22]. In a linear formula, the model is described by:

$$\mathit{y}=\mathit{Xb}+\mathit{Zg}+\mathit{Wr}+\mathit{e}$$

where: $\mathit{y}$ is the vector of phenotypic records at the plot level; $\mathit{b}$ is the vector of fixed effect of intercept; $\mathit{g}$ is the vector of random effects of genotypes with $\mathit{g}~\mathrm{N}(0, \mathit{A}{\sigma }_{\mathrm{g}}^2)$, where N is a normal distribution; $\mathit{r}$ is the vector of random effects of blocks with $\mathit{r}~\mathrm{N}(0, {\mathit{I}}_{\mathit{r}}{\sigma }_{\mathrm{r}}^2)$; and $\mathit{e}$ is the vector of random effects of residuals with $\mathit{e}~\mathrm{N}(0, {\mathit{I}}_{\mathit{e}}{\sigma }_{\mathrm{e}}^2)$. $\mathit{X}$, $\mathit{Z}$ and $\mathit{W}$ are the incidence matrices for the respective effects; $\mathit{I}$ matrices are identity matrices of the corresponding order. The variance of $\mathit{g}$ is ${\mathit{A}\sigma }_{\mathrm{g}}^2$, where $\mathit{A}$ is the pedigree-based covariance matrix and ${\sigma }_{\mathrm{g}}^2$ is the additive genetic variance. The covariance of the random effect of genotypes, namely, the genetic covariance, was based on pedigree records and considers the probability of alleles identical by descent (IBD) from the coefficient of coancestry given by [23]. The pedigree R package version 1.4.2. [24] was used to calculate that relationship.

The significance of random effects was evaluated by deviance analysis [25]. For that, nested models were constructed, that is, models with (full models) and without (reduced models) the random effect under testing. The random effect in question was tested by the likelihood ratio test at the level of 0.05.

The theoretical selection accuracy of genotypes was estimated by the equation:

$${\mathrm{r}}_{\hat{\mathrm{g}}\mathrm{g}}=\sqrt{1-{\displaystyle \raisebox{1ex}{$\overline{\mathrm{PEV}}$}\!\left/ \!\raisebox{-1ex}{${\sigma }_{\mathrm{g}}^2$}\right.}}$$

where $\overline{\mathrm{PEV}}$ is the average prediction error variance of the genotypic BLUPs. The predictive ability of the model was assessed by cross-validation. The total phenotypic observations were randomly assigned to 80% for the training set and the remaining 20% as validation set, according to [26] with cassava data. The training set had its phenotypic observations provided to the model, which gave predictions (BLUP) of the remaining individuals with phenotypic observations suppressed (validation set). For FRY (1886 total), the validation set had 377 clones, and for DMC and DMY (1442 total), it had 288 clones. The BLUP predictions and the corrected phenotypic observations were correlated by the Pearson coefficient. This procedure was repeated 50 times, and the prediction ability was the average value. The corrected phenotypic observations were obtained by analyzing the raw data as an augmented block design, according to [20], considering the effects of genotypes and blocks as fixed. The block effect is calculated regarding only the observations of check treatments, which are replicated in each block.

Another important criterion to evaluate the quality of prediction is the bias of each genotypic value predicted. The corrected phenotypic values (P) were linearly regressed on the respective predicted genotypic values (PGV), and the regression coefficient ${\hat{\mathrm{b}}}_{\mathrm{P}, \mathrm{PGV}}$ was used to measure the degree of bias. It is usual to represent the bias as one minus the regression coefficient (bias = 1 − ${\hat{\mathrm{b}}}_{\mathrm{P}, \mathrm{PGV}}$). So, a positive bias indicates overestimated predicted values, whereas a negative bias indicates underestimated values. The bias was calculated for all 50 iterations, as explained for prediction ability. All the phenotypic analyses were performed using the sommer R package version 4.4.2 [27].

The phenotypic variance was considered as the sum of genetic and residual variance from the REML estimates. So, heritability was calculated by the equation:

$${\mathrm{H}}_{\mathrm{g}}^2={\displaystyle \frac{{\sigma }_{\mathrm{g}}^2}{{\sigma }_{\mathrm{g}}^2{+\sigma }_{\mathrm{e}}^2}}$$

2.5. Ranking, Selection Gains and Heterosis

All the parents and their hybrids had genotypic values predicted, and they were studied separately, investigating different strategies to select the best clones regarding each trait and the effects on gains. The parents were ranked based on their breeding values, their own performance and the contribution of their offspring.

The clones, candidates for selection in this work, were ranked, and six intensities of selection (50, 100, 150, 200, 250 or 300 best clones) were investigated. The gains were calculated regarding the overall mean of the parents or the mean of the parents of the selected clones in each intensity of selection. Besides that, we investigated the segregation of these clones inside the families, that is, how the clones are distributed among the families, how many parents are concentrated inside each group and how many clones are better than the best parent.

All the selection gains presented in this work took the mean of parents as a basis since in cassava, as in any crop, the major challenge is to obtain a new cultivar better than any of the established and widely used cultivars. The gains were expressed on a relative scale and were computed using only BLUP-predicted, and not phenotypic values, by the expression:

$$\mathrm{Genetic} \mathrm{gain} (\%)=\left[{\displaystyle \frac{\overline{\mathrm{x}}-{\overline{\mathrm{x}}}_0}{{\overline{\mathrm{x}}}_0}}\right]·100$$

where $\overline{\mathrm{x}}$ is the mean of the selected clones in each intensity of selection and ${\overline{\mathrm{x}}}_0$ is the mean of parents.

The midparent heterosis is the difference between the performance of F₁ (each family in this case) and the parents’ mean performance $(\mathrm{MPH}={\overline{\mathrm{F}}}_1-\mathrm{MP})$ [12,15]. We divided ${\overline{\mathrm{F}}}_1-\mathrm{MP}$ by MP and multiplied the result by 100 so as to express MPH as a percentage relative to the parents’ mean [28,29], according to the expression [29,30]:

$$\mathrm{MPH} \left(\%\right)=\left({\displaystyle \frac{{\overline{\mathrm{F}}}_1-\mathrm{MP}}{\mathrm{MP}}}\right)·100$$

where ${\overline{\mathrm{F}}}_1$ is the mean of each family, and MP is the mean of parents of this family, namely, $\mathrm{MP}=\left({\displaystyle \frac{{\mathrm{P}}_1+{\mathrm{P}}_2}{2}}\right)$, being ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ the breeding values of the mother and father, respectively.

Heterosis can also be expressed in relation to the best parent (best parent heterosis or heterobeltiosis), defined by the expression $\mathrm{BPH}={\overline{\mathrm{F}}}_1-\overline{\mathrm{BP}}$. In the families with at least one clone superior to its best parent, we replaced ${\overline{\mathrm{F}}}_1$ with the breeding value of the best clone of the family $\left(\overline{\mathrm{BC}}\right)$, obtaining the best parent relative performance (BPRP) [15], and dividing $\overline{\mathrm{BC}}-\overline{\mathrm{BP}}$ by $\overline{\mathrm{BP}}$ and multiplying by 100, we expressed the best parent relative performance as a percentage:

$$\mathrm{BPRP} (\%)=\left({\displaystyle \frac{\overline{\mathrm{BC}}-\overline{\mathrm{BP}}}{\overline{\mathrm{BP}}}}\right)·100$$

where $\overline{\mathrm{BC}}$ is the mean of the best clone (in the families in which clones superior to the best parent occurred), and $\overline{\mathrm{BP}}$ is the mean of its best parent.

3. Results

3.1. Phenotypic Analysis and Predictions

All the random effects considered in our initial model, that is, genotypes and blocks, were significant for the three traits, and so the full model was used in all analyses (Supplementary Table S2). The significance of the genotype effect means that our population has a genetic variance different from zero for the three traits, which enables us to explore it for breeding purposes.

The cassava population of clones, obtained from the 15 parents used in the crosses, showed an overall mean very close to the mean of parents for the three traits (

Table 1. Population parameters and results from REML/BLUP analysis for fresh root yield (FRY) (t ha⁻¹), dry matter content (DMC) (%) and dry matter yield (DMY) (t ha⁻¹).

Parameters	FRY	DMC	DMY
The mean of clones a	25.23	34.35	8.69
The mean of parents a	25.71	34.86	8.96
Minimum prediction for clones a	18.89	27.60	6.84
Maximum prediction for clones a	36.20	40.13	12.88
Clones above the mean b	0.45	0.50	0.46
Heritability	0.16	0.47	0.15
Selection accuracy	0.67	0.75	0.64
Prediction ability	0.19	0.46	0.20
Bias of prediction	0.0036	0.0024	−0.0307

^a: From BLUP predictions; ^b: Proportion of clones above the overall mean of clones.

). In DMC, 50% of the clones surpassed the overall mean of clones, while in FRY and DMY, these proportions were slightly lower (0.45 and 0.46, respectively).

The values of heritability for FRY (16%) and DMY (15%) were very similar and around three times lower than the heritability of DMC (47%).

A similar situation can be observed in prediction ability, where DMC had a prediction (46%) 138% greater than the average values of FRY and DMY, which were very close to each other (19 and 20%, respectively).

Concerning the bias of prediction, the three traits behaved alike, with values close to zero.

In terms of the theoretical selection accuracy, FRY and DMY also showed values similar to each other and lower than that of DMC, although this difference was much smaller than that observed in prediction ability.

3.2. Correlations and Rankings

Knowledge of the value and the signal of correlations between traits are important to support indirect selection and to infer simultaneous genetic gains by selection indices. The phenotypic and genotypic correlations among the three traits were positive and highly variable (Figure 2). FRY and DMY showed the highest correlations with each other at both phenotypic (0.98) and genotypic (0.85) levels. DMC was less correlated with the others, showing very low phenotypic (0.07) and genotypic (0.05) correlations with FRY and DMY (0.26 and 0.43, respectively).

The ranking of parents, based on their breeding values (BLUP predictions), varied greatly among the three traits (

Table 2. Ranking of parents for each trait and corresponding breeding values (between parentheses).

Rank.	FRY	DMC	DMY
1st	BGM 1709 (34.56)	BGM 1116 (39.50)	BGM 0254 (11.57)
2nd	2003 14-11 (30.79)	BGM 0254 (38.81)	BGM 1709 (11.47)
3rd	BGM 0254 (29.95)	BGM 1632 (37.34)	BGM 2024 (10.57)
4th	BGM 2024 (29.94)	BGM 2038 (36.75)	2003 14-11 (9.51)
5th	BGM 1660 (27.22)	BRS Gema de Ovo (36.73)	BGM 1660 (9.47)
6th	BGM 0212 (26.36)	BGM 1660 (36.26)	BGM 1632 (9.44)
7th	BGM 1632 (25.78)	BGM 2024 (35.61)	BGM 1116 (9.12)
8th	BRS Jari (25.69)	BGM 1709 (34.90)	BGM 0212 (9.09)
9th	BGM 0131 (23.66)	BGM 0212 (33.92)	BRS Gema de Ovo (8.39)
10th	BGM 2038 (23.27)	BGM 1807 (33.82)	BGM 2038 (8.25)
11th	BRS Gema de Ovo (23.26)	BRS Dourada (33.50)	BRS Dourada (7.73)
12th	BRS Dourada (22.05)	BGM 0131 (32.80)	BGM 1807 (7.65)
13th	BGM 1807 (21.95)	BRS Jari (31.28)	BRS Jari (7.59)
14th	BGM 1116 (21.37)	BGM 1722 (31.12)	BGM 0131 (7.54)
15th	BGM 1722 (19.79)	2003 14-11 (30.54)	BGM 1722 (7.02)

). Reflecting the correlations among the three traits, there was some concordance between FRY and DMY, but regarding DMC, the rankings were completely different. The Spearman correlations among the ranks of parents (0.79 between FRY and DMY, −0.05 between DMC and FRY, and 0.41 between DMC and DMY) reflected this trend (Supplementary Table S3).

The knowledge of parents’ breeding values is of utmost importance for future hybridizations. It is worth pointing out that BGM 1709 was classified as first in FRY and second in DMY. However, for DMC, BGM 1709 was classified as 8th, a lower position than that observed for FRY and DMY, which reflects the low correlations between DMC and the other traits (Figure 2). Also deserving attention is the parent BGM 0254, third in FRY, second in DMC and first in DMY. BGM 2024, BGM 1660 and 2003 14-11 ranked among the first five in FRY and DMY. It is worth pointing out that 2003, 14-11, the second-best parent in FRY and fourth in DMY, was the worst in DMC, and inversely, BGM 1116, 1st in DMC, was 14th in FRY. BGM 0212 performed as moderate in the three traits (6th, 9th and 8th, respectively). BGM 1722 was the parent with the worst performance in FRY and DMY, besides having the second-worst classification in DMC. BRS Jari is another parent with a bad classification (third worst) in DMC and DMY. Yet with a poor classification, there are the parents BGM 0131, BGM 1807 and BRS Dourada, in the worst third part of the table. The other parents varied a lot and did not have a pattern of behavior.

3.3. Selection Gains

Considering the mean of all 15 parents (left side in Figure 3), the estimated gains were high and varied a lot across the traits. FRY and DMY had the highest estimated gains (19.96 to 30.80% and 16.63 to 27.56%, respectively), while in DMC, the trait with the highest heritability, the gains (7.79 to 11.46%) was much lower than those of the other two traits.

When the mean of each group of clones (50, 100,…, 300) was compared with the mean of the corresponding parents (right side in Figure 3), the gains were smaller. However, the same trend of higher gains for FRY and DMY than for DMC remains. The gains varied from 14.18 to 21.48% in FRY and from 16.63 to 19.93% in DMY, while in DMC, the range was from 4.40 to 8.79%.

The best clones for each trait were concentrated in a few families (

Table 3. The number of families and the number of parents crossed in different intensities of selection.

Trait	Number of Clones	Number of Families	Number of Parents Crossed
FRY	50	6	6
	100	10	8
	150	16	13
	200	19	13
	250	23	14
	300	26	14
DMC	50	8	7
	100	10	7
	150	10	7
	200	16	10
	250	19	11
	300	26	13
DMY	50	7	7
	100	12	10
	150	14	12
	200	21	14
	250	23	14
	300	27	15

FRY, DMC, and DMY mean fresh root yield, dry matter content and dry matter yield, respectively.

). The best 50 clones came from only 6 families for FRY, 8 for DMC and 7 for DMY, over a total of 57 families. As the number of selected clones increases, the number of families involved also increases, but even when selecting the best 300 clones, less than half of the families (26, 26 and 27) are involved. For the three traits, the ranges in the number of families involved, according to the different numbers of clones selected, are alike, indicating a similar segregation within each trait.

Contrary to what was observed regarding the number of families, the number of parents crossed increases sharply with the number of clones selected. When selecting the best 300 clones, the number of parents crossed was 14 for FRY, 13 for DMC and 15 for DMY.

3.4. Heterosis

Supplementary Tables S4–S6 present the data of midparent heterosis [MPH (%)] and best parent relative performance [BPRP (%)] of each family for the traits FRY, DMC and DMY, respectively.

3.4.1. Fresh Root Yield (FRY)

We can observe in Supplementary Table S4 that the midparent heterosis (MPH) values varied from −4.38 (family 2020-66) to 7.53% (family 2020-57) with a mean of 0.49%. Of the 57 families, 26 had negative values of MPH. All 15 parents are present in the genealogy of these 26 families with negative MPH, as well as in the pedigrees of the 31 families with positive MPH.

In a crop of vegetative propagation, such as cassava, an outstanding individual can be cloned. For this reason, in the 42 families with at least one clone better than its best parent, we calculated the best parent relative performance (BPRP), whose mean value was 8.04%, and the range was from 0.03 (family 2020-30) to 23.53% (family 2020-01). All parents are involved in these 42 crosses, and considering the 15 families with no descendant superior to the best parent, 11 of the 15 parents are involved. The four exceptions were the parents BGM 0131, BGM 1632, BGM 2024 and BRS Dourada. It is interesting that nine (2020-12, 2020-17, 2020-25, 2020-50, 2020-53, 2020-58, 2020-63, 2020-65 and 2020-66) out of these 15 families with no offspring superior to the best parent belong to the 26 with negative MPH and six (2020-07, 2020-15, 2020-34, 2020-49, 2020-64 and 2020-67) are part of the 31 with positive MPH.

It can be seen that the 15 parents participated in both the 26 crosses with negative MPH and the 31 crosses with positive MPH. Regarding the comparison with the best parent (BPRP), the 15 parents were part of the 42 crosses with at least one clone superior to the best parent, while 11 of these 15 were also parents in the 15 crosses without any offspring superior to the best parent. Another interesting result is that four of the 16 families with BPRP higher than 10% (2020-03, 2020-22, 2020-24 and 2020-29, respectively, 7th, 4th, 6th and 15th regarding the values of BPRP) are part of the 26 families with negative MPH. These results show a complete lack of predictability in cassava crosses.

3.4.2. Dry Matter Content (DMC)

The information about DMC is presented in Supplementary Table S5. The values of MPH varied from −2.74 to 1.89%, with a mean of −0.29%. It can be observed that 33 of the 57 values of MPH are negative. All 15 parents are present in these 33 crosses, as well as in the other 24, whose MPH values are positive.

The BPRP values of the 30 families with at least one clone superior to the best parent ranged from 0.03 to 9.73%. The 15 parents are part of the genealogy of these 30 families. Regarding the 27 families that did not produce any clone better than the best parent, 14 parents (except BGM 0212) are involved in their genealogy.

Fourteen of the 33 negative MPH values correspond to families with no descendants superior to the best parent, which means that 19 values correspond to crosses with at least one descendant superior to the best parent. This demonstrates that the same unpredictability already mentioned in relation to FRY occurs in DMC.

It is interesting that the range of variation of MPH and BPRP for DMC was smaller than for FRY. While for FRY, the values of MPH ranged from −4.38 to 7.53%, for DMC, the range was from −2.74 to 1.89%. For BPRP, the variation for FRY was from 0.03 to 23.53% and for DMC from 0.03 to 9.73%. This explains the lower gains obtained in DMC (Figure 3) despite its heritability (0.47) being the highest among the three traits evaluated (Table 1).

When observing the data of FRY and DMC, we perceive a big discrepancy in the family’s classification. Considering the 10 best families for MPH in FRY (2020-01, 2020-15, 2020-27, 2020-31, 2020-36, 2020-39, 2020-43, 2020-57, 2020-64 and 2020-74) and in DMC (2020-03, 2020-05, 2020-07, 2020-28, 2020-34, 2020-45, 2020-48, 2020-55, 2020-60 and 2020-81), we can see that there was no coincidence, which reflects the low genetic correlation between these traits (0.05%) obtained in this work.

3.4.3. Dry Matter Yield (DMY)

DMY is an “artificial” trait created from FRY and DMC, which summarizes these two traits into one in order to select clones with reasonable performance in both since, due to the negative correlation between FRY and DMC [31], the occurrence of genotypes with the best performance in both traits is not easy.

DMY data are presented in Supplementary Table S6. It is possible to observe that 28 families had negative MPH of DMY. It is very similar to what happened in FRY (Supplementary Table S4), in which 26 families had negative MPH. When comparing the Supplementary Tables S4 and S6, we can see that 24 genotypes are common, i.e., they have negative MPH both in FRY and in DMY. The mean value of MPH in DMY is 1.16%, and the variation from −4.36 to 6.89%. In FRY, the mean MPH was 0.49%, varying from −4.38 to 7.53%. In the 38 families with clones superior to the best parent, the values of BPRP in DMY varied from 0.02 to 18.33%, with a mean of 7.16%. In FRY, the mean BPRP was 8.04%, varying from 0.03 to 23.53%. These similarities between the distributions of DMY and FRY data are a consequence of the high genetic correlation between these two traits (85%, Figure 2), and it can be attributed to the fact that FRY is expressed in a unit of larger magnitude (t ha⁻¹) than DMC (%) so that when the values are multiplied (to obtain DMY), FRY predominates over DMC. Thus, DMY is largely a duplication of FRY.

The range of variation in FRY is larger than in DMC. In Supplementary Table S4, the smallest breeding value of FRY is 18.89 t ha⁻¹, and the largest is 36.20 t ha⁻¹. Therefore, the range between the extreme breeding values in FRY is 91.62%. In Supplementary Table S5, the extreme values of the breeding value of DMC are 27.60% and 40.13%, which means a range of 45.39%. Given that FRY and DMC are highly important in cassava, it is crucial to consider these questions in the breeding of this crop.

4. Discussion

4.1. Phenotypic Analysis and Predictions

When observing the maximum and minimum prediction for clones (Table 1), it is possible to notice a large range, mainly for FRY and DMY, varying from 18.89 to 36.20 and from 6.84 to 12.88 t ha⁻¹, respectively, and even in DMC, the range (27.60 to 40.13%) was expressive, indicating the existence of a significant amount of genetic variability in the population. However, the means of clones and the means of parents are very similar for the three traits (25.71 vs. 25.23 in FRY, 34.86 vs. 34.35 in DMC and 8.96 vs. 8.69 in DMY), so only a small proportion of the clones (45% in FRY, 50% in DMC and 46% in DMY) had genotypic mean higher than the parents’ mean (Table 1).

Our estimate of heritability for FRY (0.16) is in agreement with the value (0.14) obtained by [19]. However, the heritability of DMC from the cited work (0.18) is much lower than ours (0.47). It is important to stress that although those authors have obtained a clonal population derived from a larger number of parents (100) than we have (15), they did not account for genetic relationships among the clones to perform the analysis.

In turn, [26] obtained heritabilities varying a lot among six cassava populations for both FRY and DMC, with mean values of 0.19 and 0.39, respectively. On the other hand, [28] obtained a much higher estimate of heritability for FRY (0.37) and a slightly lower estimate for DMC (0.40) when evaluating a much smaller number of crosses (15) and parents (6) compared to our study.

The selection accuracy is the correlation between the predicted and the true genetic values of the individuals [32], with accuracy values being considered medium when around 0.60, moderate when ranging from 0.70 to 0.89 and high if above 0.90 [33]. By these criteria, the values obtained in our work (0.67 in FRY, 0.75 in DMC and 0.64 in DMY) are medium for FRY and DMY and moderate for DMC. Values of 0.71 for FRY and DMC and 0.72 for DMY were observed [34].

As expected and stated in the literature, there is a trend to get lower prediction ability for traits with lower heritability [26,34], which was corroborated by our results. FRY (H² = 0.16) and DMY (H² = 0.15) showed a prediction ability of 0.19 and 0.20, respectively, while DMC (H² = 0.47) had a prediction ability of 0.46. When evaluating 888 cassava accessions from a germplasm bank, [35] also obtained much higher prediction ability for DMC, using molecular markers to make the predictions. The authors stated that this trait is less influenced by environmental factors, which explained the better predictions for all the statistical models evaluated by them and the higher estimate of heritability of this trait. In general, the genomic heritability obtained by those authors (0.51 for FRY, 0.59 for DMC and 0.46 for DMY) was higher than those found here for all three traits.

Apart from the lower values of prediction ability achieved, the use of a mixed model relying only on pedigree information resulted in very precise prediction since the values of bias of prediction (0.0036 for FRY, 0.0024 for DMC and −0.0307 for DMY) were all close to zero.

4.2. Correlations and Rankings

The phenotypic (0.98) and genotypic (0.85) correlations between FRY and DMY were much higher than the correlations with DMC (Figure 2). The reason for that is the fact explained earlier about the predominance of FRY in DMY due to the unit of measure. Genetic correlations of 0.99 between FRY and DMY were reported by [34,36].

The genetic correlation between FRY and DMC was 0.05, whereas [36] obtained a correlation of −0.07. Those values are in accordance with the observations of [31], who evaluated cassava data of 14 years and reported a value of 0.21 in the stage of single row trial and negative values (−0.13, −0.14 and −0.42) in the stages of preliminary yield trial (PYT), advanced yield trial (AYT) and uniform yield trial (UYT), respectively.

Due to the low or even negative genetic correlation, it is not possible to select a genotype that is the best in each of these traits, but genotypes with acceptable performance in both [31].

4.3. Selection Gains

The higher gains in FRY and DMY than in DMC (Figure 3) have been addressed in the literature [34,37] and suggest the predominance of dominance effects in the genetic control of FRY, while in DMC, additive effects are the major cause [37,38].

Looking at the number of families involved in each selection intensity, we verified that the best clones were concentrated in few families, and even considering the selection of 300 individuals, less than half of the 57 families (26 for FRY and DMC and 27 for DMY) had clones selected (Table 3). We can deduce that 31 families for FRY and DMC and 30 for DMY did not produce any clone of the best 300. The number of parents involved in the crosses that generated the 300 best clones was 14 for FRY, 13 for DMC and 15 for DMY (Table 3). Observing the genealogy of the 31 families that did not generate any of the best 300 clones in FRY and DMC, it can be seen that 13 and 15 parents, respectively, were involved. This seems to show us that the best offspring depends more on specific combinations of parents than on their breeding values.

Since the best clones came from a few families, breeders should concentrate their efforts only on those most promising crosses. Also working with full-sib families of cassava [34] observed that most of the superior clones came from few crosses, and there was a parent that stood out in terms of performance, as we found here, and [39], investigating the combining ability of nine potential cassava parents, by diallel design, found that three of the five best-performing crosses for FRY involved the same parent.

4.4. Heterosis

The exploitation of heterosis was one of the main factors responsible for the increases in agricultural productivity observed throughout the 20th century. In cassava crops, its exploitation has only recently been considered. In this section, we discuss, based on the results obtained in this work and information from the literature, the possibility of exploring heterosis in cassava breeding.

4.4.1. Fresh Root Yield (FRY)

The similarity between the mean of clones (25.23 t ha⁻¹) and parents (25.71 t ha⁻¹) for FRY (Table 1) suggests null heterosis. However, in Supplementary Table S4, we can see that MPH varied from −4.38 to 7.53%. Negative values of MPH in cassava were also obtained by [28,29,30].

As cassava is a species of vegetative propagation, the selected genotypes (seedlings) are cloned, and after successive evaluations, the ones that are confirmed to be superior become new cultivars. Therefore, considering that in any crop, a new cultivar needs to be better than the most planted cultivars, that is, to have a better performance than these cultivars (higher or lower depending on the characteristic in question) in at least one of the most important traits for the crop, it makes sense to check the occurrence of genotypes that outperforms the best parent. For this reason, in this study, only for families with at least one clone superior to its best parent, the best parent relative performance [BPRP (%)], the difference, in percentage terms, between the performances of the best clone in the family and that of its best parent, was calculated. In Supplementary Table S4, the BPRP was 8.04% on average and varied from 0.03 to 23.53% in FRY.

These results demonstrate that the heterosis for FRY can be low, but it is not null, as can be wrongly deduced when comparing the overall mean of clones and parents, that is, disregarding the family structure. Another important reason for considering the family structure is that, in the case of a full-sib family, as employed in this work, the larger part of genetic variation (${\displaystyle \frac{1}{2}}{\sigma }_{\mathrm{A}}^2+{\displaystyle \frac{3}{4}}{\sigma }_{\mathrm{D}}^2$) is within the families [40]. This fact was experimentally proven by [9], who, by the mean across 3 environments, observed that the variance within full-sib families was 9.74 times higher than the variance between families. Although, on average, the heterosis (MPH) obtained in the present study (0.49%) can be considered low, it can reach greater values (7.53%), and when considering the best clone inside the families (BPRP), even very high values (23.53%) can be obtained.

Although the experimental results pointed out the possibility of genetic progress for FRY in cassava breeding, this progress tends to be low [9]. Heterosis depends on genetic divergence, dominance [12], epistasis [41] and/or overdominance [15]. More recently, gene networks, allele bias, epigenomic and transcriptomic factors have been pointed out as additional causes of heterosis [42]. Dominance [37,39,43,44,45] and epistasis [26,43,45] have been reported in cassava for FRY.

If dominance occurs in the control of a given trait when performing crosses between related individuals or self-fertilization, deleterious recessive alleles (genetic load) are put in homozygosis. Under dominance, even without deleterious recessive alleles, recessive homozygotes will have a lower mean if dominance acts predominantly to increase the mean. When the allelic relationship is one of overdominance, both homozygotes of a locus have a lower mean than the corresponding heterozygote. A third cause of the decrease in the mean when individuals are exposed to homozygosis is the breakdown of favorable epistatic interactions between heterozygous loci [15]. This decrease in the performance of individuals exposed to inbreeding is termed inbreeding depression, which reduces fitness, leading to the death of less adapted individuals, reduction in size and even extinction of populations [46]. Thus, inbreeding depression is, to some extent, the opposite of heterosis [40]. Inbreeding depression has been reported for FRY in cassava [47,48,49,50].

Given the fact that the main requirements for heterosis in cassava are fulfilled, the low realized heterosis in this crop, considering FRY, is intriguing. A possible reason is the low genetic divergence among the parents [(p − r)² in the equation presented by [12]: $\left({\displaystyle {\sum }_{\mathrm{i}=1}^{\mathrm{n}}}{({\mathrm{p}}_{\mathrm{i}}-{\mathrm{r}}_{\mathrm{i}})}^2{\mathrm{d}}_{\mathrm{i}}\right)$, where p and r are the allele frequencies in each parent and d is the dominance effect]. By this equation, we can see that considering only two alleles per locus, the divergence [(p − r)²], and hence the heterosis, will be zero in the loci in which both parents are heterozygous (p = r = 0.5) even if the dominance in these loci is different from zero. Thus, knowing that cassava is a highly heterozygous species [3,9,10], it makes sense to assume that the low divergence among parents can explain (at least partially) the low heterosis realized in FRY nowadays.

In homozygous individuals, the frequency of an allele can only be 1 (if the individual has a given allele) or 0 (if the individual has another allele). This boosts the genetic divergence that will reach the maximum [(p − r)² = 1] in the contrasting loci. By this example, we understand that this is exactly what breeders search for when they obtain inbred lines for subsequently identifying the best combinations. This highlights the importance of establishing heterotic groups and, hence, evaluating families. One consequence of the crosses among heterozygous parents is the larger segregation within families, which impairs the selection of parents and requires larger populations for evaluation [11]. At this point, we can notice that crosses among homozygous parents would trigger the obtaining of more uniform progenies (more segregation between them), favoring the establishment of heterotic groups and exploitation of heterosis. For these reasons, the use of inbred lines has been raised as a way to increase genetic gains in cassava [3,9,11], as well as in other crops propagated vegetatively, such as potato [51,52] and sweet potato [53]. Besides increases in genetic gains, the use of lines enables the following advantages in cassava breeding [9]: (a) reduction of genetic load, (b) discovery of useful recessive traits, (c) implementation of the back-cross scheme, (d) facilitated germplasm exchange and conservation, (e) development of superior hybrids by design, (f) facilitated maintenance of superior clones, (g) facilitated conventional and molecular genetic studies, (h) shortened length of breeding cycles, and (i) a more dynamic and efficient breeding method.

Heterosis is a complex subject, both in theoretical aspects and in implementation. Maize, the first crop in which heterosis was exploited for breeding, is an example of that. The first studies were published in 1908, but the technology of hybrid cultivars just began to be adopted around 1930. In cassava, this technology will certainly face difficulties. One of them is the fact that, in cassava, due to its vegetative propagation, there will not be a market for botanical seeds, which means that the development of cassava lines can just be made by public institutions. Another is that, due to the long cycle of cassava, the duration of the process for obtaining lines through self-fertilization must be 12 to 15 years [9]. For this reason, the technique of obtaining doubled haploids has been developed [54,55]. A third difficulty is that in cassava, flowering and branching are closely linked processes, while for farmers, plants that do not fork (and therefore do not flower) are more interesting [3,9], which has stimulated the development of flowering induction protocols [56].

Another equation for heterosis was proposed by [41]: $\mathrm{h}=2\delta -\alpha$, in which δ represents the dominance effects and α the additive x additive epistasis. By this equation, it can be seen that the magnitude and the sign (positive or negative) of the value of heterosis will depend on the magnitude of epistasis in comparison to dominance.

The negative heterosis (MPH) values obtained here in FRY are corroborated by other authors [28,29,30]. Loss of epistatic combinations of the additive x additive type may be a cause of negative heterosis. When favorable combinations between loci existing in the parents are undone, they can result in a progeny (F₁) with a lower mean than that of the parents [15]. Another possible cause of negative heterosis is the occurrence of multiple alleles. When more than one allele occurs per loci, even if in each locus that controls a given trait the relationship between the alleles is of partial dominance, complete dominance or overdominance, not all allelic combinations will contribute to increasing the expression of the trait, so that negative values of heterosis are possible when the sum of the negative values exceeds the positive ones [57].

4.4.2. Dry Matter Content (DMC)

In Table 1, we can see that the mean of DMC was 34.35% for clones and 34.68% for parents. In FRY, the mean of clones (25.23 t ha⁻¹) and the mean of parents (25.71 t ha⁻¹) were also very close. However, when the families are examined individually, the values of MPH in FRY can reach 7.53% (Supplementary Table S4), which can be considered low but not null, while in DMC, the MPH values reached the maximum of 1.89%. In DMC, we observed that in both the overall mean (Table 1) and the family mean (Supplementary Table S5), there is a trend of similar values for parents and their offspring, which suggests the predominance of additive effects for this trait. These results corroborate the works that point to the importance of additive effects for DMC in cassava [9,35,37,39,43].

4.4.3. Dry Matter Yield (DMY)

Cassava roots are constituted by 65 to 70% of water. The remaining 30 to 35% is dry matter, whose main component (65 to 91%) is starch [58]. Cassava is the second source of starch in the world, being surpassed only by maize [9]. In sweet cassava, which normally needs to be cooked, the starch exerts influence on cooking [59]. So, in cassava, the ideal clone would be that with the greatest value of FRY and DMC. However, the negative correlation between these traits [31,34] makes it difficult to reach this target. Therefore, the trait DMY is introduced as a product of FRY and DMC, through which one seeks to identify clones with reasonable performance in both crop traits.

The high genetic correlation (0.85) between FRY and DMY obtained in this work and by [34,36] (0.99) arises from the predominance of FRY over DMC, so the variation in DMY largely reflects the variation in FRY. For this reason and by the larger range of variation in FRY than in DMC, we can deduce that increases in DMY depend mainly on increases in FRY, whose heritability is lower than that of DMC. This reinforces the statement of [9,60] of changing the selection method in cassava, shifting from phenotypic recurrent selection employed currently to reciprocal recurrent selection as a way to increase the genetic gains in FRY.

5. Perspectives

In addition to the complexity of heterosis, some characteristics of the cassava crop, such as vegetative propagation (which discourages private sector investments in breeding) and the difficulty of some clones in flowering, will constitute additional difficulties to the introduction of inbreeding in the improvement of this species. However, given that there are reports of heterosis occurring in FRY, further research on this topic is a great and exciting challenge for those involved in its improvement.

The lack of heterotic groups is a serious obstacle to cassava breeding. The implementation of diallel crosses, a classic means of obtaining combining abilities (GCA and SCA), is hampered by the great variation between clones regarding flowering time. Mixed models make it possible to obtain breeding values (related to GCA) without the need for mating designs [15]. For this and many other reasons, the use of mixed models should be intensified in cassava breeding. Also, in this context, given that the self-fertilization of a heterozygous genotype (S₀) reduces heterozygosis by 50% and that the cross between more homozygous parents increases the probability of occurrence of superior offspring, the cross between partially inbred materials (S₁ for example) from different parents can be a simple and direct means of checking the occurrence of heterosis and at the same time starting the process of defining heterotic groups in cassava.

Another difficulty in cassava breeding is the long duration of each cycle [9]. The main advantage of genomic selection in a cassava breeding program is to allow, with the help of molecular markers, accurate and early prediction of the agronomic potential of clones in order to reduce the duration of breeding cycles [34]. It is important to remember that when crossing at least partially inbred (less heterozygous) parents, the probability of occurrence of superior offspring is greater than when these parents are highly heterozygous (as is currently the case), from which it can be deduced that the introduction of inbreeding may help to increase the gains with genomic selection.

Due to the long cycle, it takes 12 to 15 years to obtain completely homozygous lines through successive self-fertilizations [9]. The doubled haploid technique allows the obtaining of completely homozygous lines in a period of 1 to 2 years and has already been applied to cassava crops [54]. Given the possibility of reducing the time to obtain lines, the development of this technique and its application to cassava breeding should be intensified.

Erect plants, preferred by cassava growers, are more difficult to flower, while reality (whether in the context of genomic selection or for obtaining lines) demands intensification of crosses and, therefore, flowering. It is, therefore, vital that the great progress made recently in inducing flowering in cassava [56] be continued and internalized by cassava breeding programs.

6. Conclusions

In this work, we explored the selection with and without the family structure. Without considering the family structure, the overall mean of the parents was very similar to the overall mean of the descendants for the three traits evaluated, suggesting the absence of heterosis. However, when comparing the means of the parents and their descendants in each family, it was observed that considerable values of heterosis can be obtained in FRY. The analysis of data per family enables us to make inferences about the genetic control of the traits, which is not possible when the family structure is disregarded.

Another important result of this work was the verification of the fact, already pointed out by other authors, that in cassava, the breeding values of the parents do not allow for predicting the performance of the offspring, considering the trait FRY. From this, it is deduced the need to define heterotic groups and, therefore, to evaluate families. Crossing less heterozygous parents would facilitate selection within families. For this reason, following previous studies, we defend here that the pertinence of introducing inbreeding as a way of establishing heterotic groups and exploiting heterosis in cassava breeding should be investigated more deeply.