**2. Origin of data and genetic model**

Nine quantitative traits with 88 cases of combination cross-site, cross-isolate or crosstreatment of six generations (P1, P2, F1, F2, BC1 and BC2) for three crops (*Triticum Durum, Capsicum annum* and *Avena sp*) were collected from different works realised in our laboratory. Crops, traits and origin of data are reported in Table 1. For each trait parents of crosses were extreme. Transformations (such as Kleckowski transforms (Lynch and Walsh, 1998)) were applied to normalize the distribution of data or to make means independent of variances for several traits.


Table 1. Traits assessed in each crop and date of publication

#### **2.1 Best genetic model**

4 Agricultural Science

The aims of our study were to determine the importance of epistasis effects in heredity of quantitative traits and their consequences in the bias of four methods of estimation of

Nine quantitative traits with 88 cases of combination cross-site, cross-isolate or crosstreatment of six generations (P1, P2, F1, F2, BC1 and BC2) for three crops (*Triticum Durum, Capsicum annum* and *Avena sp*) were collected from different works realised in our laboratory. Crops, traits and origin of data are reported in Table 1. For each trait parents of crosses were extreme. Transformations (such as Kleckowski transforms (Lynch and Walsh, 1998)) were applied to normalize the distribution of data or to make means independent of

Number of head per plant , Spiklets per spike and Number of grains per spike (Bnejdi and

narrow-sense heritability.

variances for several traits.

**Two crosses/two sites** 

**Four crosses/ one site** 

**Four crosses/ Two sites** 

**Two crosses/ Five salt treatments** 

**Two crosses/ fifteen isolates** 

**(***Capsicum annuum* **L.)**

**Two crosses/ Two isolates** 

**Two crosses/ Six isolates** 

**Two crosses/ Two isolates**

**Pepper** 

**Oates (Avena sp.)**

El Gazzeh 2010b) **Four crosses/ one site** 

**Durum Wheat (***Triticum durum***)** 

**2. Origin of data and genetic model** 

Resistance to yellowberry (Bnejdi and El Gazzah, 2008)

Grain protein content (Bnejdi and El Gazzeh, 2010a)

Resistance to *Septoria tritici* (Bnejdi et al., 2011b)

Resistance to salt at germination stage (Bnejdi et al., 2011a)

Resistance to *Phytophthora nicotianae* (Bnejdi et al., 2009)

Resistance to *Phytophthora nicotianae* (Bnejdi et al., 2010b)

Table 1. Traits assessed in each crop and date of publication

Resistance to *P. coronate* Cda. f. sp. *avenae* Eriks (Bnejdi et al., 2010c)

Resistance to yellowberry (Bnejdi et al., 2010a)

Weighted least squares regression analyses were used to solve for mid-parent [M] pooled additive [A], pooled dominance [D] and pooled digenic epistatic ([AA], [DD] and [AD]) genetic effects, following the models and assumptions described in Mather and Jinks (1982). A simple additive-dominance genetic model containing only M, A and D effects was first tested using the joint scaling test described in Rowe and Alexander (1980). Adequacy of the genetic model was assessed using a chi-square goodness-of-fit statistic derived from deviations from this model. If statistically significant at *P* < 0.05, genetic models containing digenic epistatic effects were then tested until the chi-square statistic was non-significant.

### **3. Phenotypic resemblance between relatives**

We now will use the covariance (and the related measures of correlations and regression slopes) to quantify the phenotypic resemblance between relatives. Quantitative genetics as a field traces back to Fisher's 1918 paper showing how to use the phenotypic covariance to estimate genetic variances, whereby the phenotypic covariance between relatives is expressed in terms of genetic variances, as we detail below.

#### **3.1 Parent-offspring regressions**

There are three types of parent-offspring regressions: two **single parent - offspring regressions** (plotting offspring mean versus either the trait value in their male parent *Pf* or their female parent *Pm*), and the **mid-parent-offspring regression** (the offspring mean regressed on the mean of their parents, the mid-parent *MP* = (*Pf* +*Pm*)*/*2).

The slope of the (single) parent-offspring regression is estimated by

$$b\_{\
u/p} = \frac{\operatorname{Cov}(O, P)}{\operatorname{Var}(P)}, \text{ where } \operatorname{Cov}(O, P) = \frac{1}{n - 1} \{\sum\_{i = 1}^{n} O iP\_i - n\overline{O}, \overline{P}\}$$

Where *Oi* is the mean trait value in the offspring of parent *i (Pi)* and we examine *n* pairs of parent-offspring. One could compute separate regressions using males (*Pm*) and females (*Pf*), although the later potentially includes maternal effect contributions and hence singleparent regressions usually restricted to fathers.

$$\begin{aligned} b\_{o/p} &= \frac{\text{Cov}(O, P)}{\text{Var}(P)}\\ \text{Cov}(O, P) &= \frac{\sigma^2 A}{2} + \left(\frac{\sigma^2 AA}{4} + \frac{\sigma^2 AAA}{8} + \frac{\sigma^2 AAA}{16} + \dots \dots \right) \\ b\_{o/p} &= \frac{\text{Cov}(O, P)}{\text{Var}(P)} = \frac{\sigma^2 A}{2\sigma\_P^2} + \frac{1}{\sigma\_P^2} (\frac{\sigma^2 AA}{4} + \frac{\sigma^2 AAA}{8} + \frac{\sigma^2 AAA}{16} + \dots \dots) \\ b\_{o/p} &= \frac{\text{Cov}(O, P)}{\text{Var}(P)} = \frac{h^2}{2} + \frac{1}{\sigma\_P^2} (\frac{\sigma^2 AA}{4} + \frac{\sigma^2 AAA}{8} + \frac{\sigma^2 AAA}{16} + \dots \dots) \end{aligned}$$

Assuming an absence of epistasis we have

Impact of Epistasis in Inheritance of Quantitative Traits in Crops 7

2 

2 2 2 22

*F BC BC A A*

2 2

2 2

1 1 11 2 ( ) 2( ) ( ) ........ 2 2 22 *F BC BC*

*a ad d dd aa ad*

Separate generation means analysis revealed that the additive-dominance model was found adequate only for 18 cases. Therefore, the digenic epistatic model was found appropriate for 70 cases (Table 2). Additive and dominance effect were significant for all cases of combination. With regard to epistatic effects, the additive x additive effect was significant for 77 cases and the additive x dominance for 42 cases and dominance x dominance effects for 56 cases. Recent studies suggest that epistatic effects are present for inheritance of quantitative traits in many species. Examples are wheat (resistance to leaf rust, Ezzahiri and Roelfs 1989), wheat (resistance to yellowberry, Bnejdi and El Gazzah 2008), common bean (resistance to anthracnose, Marcial and Pastor 1994), barley (resistance to Fusarium head blight, Flavio et al. 2003), chickpea (resistance to *Botrytis cinerea*, Rewal and Grewal 1989), and pepper (resistance to *Phytophthora capsici*, Bartual et

To conclude for this part, the additive dominance model was rarely fitted and digenic epistatic model was frequently appropriate. Therefore epistasis is common in inheritance of quantitative traits and any model or methods assumed that epistasis was negligible were

The comparison of the four methods is reported in Table 3. In absence of dominance and epistatic effect, the methods were not biased. Therefore, in presence of epistasis narrowsense heritability based on the four methods was underestimated. Based in Full-Sib Analysis and Warner (1952) methods, bias was caused by dominance, interaction between homozygote loci, interaction between heterozygote loci and interaction between homozygote and heterozygote loci. Therefore based in Half-Sib Analysis and Parentoffspring regressions, bias was caused only with the presence of interaction between

The result of generations means analysis indicate that digenic epistasis model were frequently appropriate. So the additive model in which many methods of genetic quantitative were based was rarely adequate. Based on the result, the methods of Half-Sib Analysis and Parent-offspring regressions were underestimated with additive x additive

*F F*

2

*h*

*a*

 

*BC* represented respectively the variance of

2 12 <sup>222</sup> 2( ) 

In absence of epistasis

generation F2, BC1 and BC2

*F BC BC* Where <sup>2</sup>

Therefore in presence of epistasis

**4. Results and discussion** 

homozygote loci or fixable effect.

al. 1994).

biased.

2 12

 

2 *<sup>F</sup>* , <sup>2</sup>

1 

*BC* and 2

2 12

<sup>1</sup> 2( ) <sup>2</sup>

2 12

22 2 2 22

2( )

22 2 2

 

*F BC BC A*

 

$$Cov(O, P) = \bigvee\_{\mathbf{2}} \sigma^2 A \text{ , giving } \begin{aligned} b\_{o/p} &= \frac{1/\sigma^2 A}{2} = \frac{h^2}{2} \\ \sigma\_P^2 &= \frac{h^2}{2} \\ h^2 &= 2b\_{o/p} \end{aligned}$$

#### **3.2 Full-sib analysis**

The covariance full-sib analysis is equal to:

$$\text{Cov}(\text{FS}) = \frac{1}{2}\sigma\_A^2 + \frac{1}{4}\sigma\_D^2 + \frac{1}{4}\sigma\_{AA}^2 + \frac{1}{8}\sigma\_{AD}^2 + \frac{1}{16}\sigma\_{DD}^2 + \frac{1}{8}\sigma\_{AAA}^2 \dots \dots$$

$$\frac{\text{Cov}(\text{FS})}{\sigma\_P^2} = \frac{h^2}{2} + \frac{1}{\sigma\_P^2}(\frac{1}{4}\sigma\_D^2 + \frac{1}{4}\sigma\_{AA}^2 + \frac{1}{8}\sigma\_{AD}^2 + \frac{1}{16}\sigma\_{DD}^2 + \frac{1}{8}\sigma\_{AAA}^2 \dots \dots)$$

So, when epistasis was considered negligible

$$\begin{aligned} Cov(FS) &= \frac{1}{2} \sigma\_A^2 \\ h^2 &= \frac{2Cov(FS)}{\sigma\_P^2} \end{aligned}$$

#### **3.3 Half-sib analysis**

Based on half-sib analysis, narrow-sense heritability was calculated as:

$$\begin{aligned} Cov(HS) &= \frac{1}{4}\sigma\_A^2 + \frac{1}{16}\sigma\_{AA}^2 + \frac{1}{64}\sigma\_{AAA}^2 + ... \\ \frac{Cov(HS)}{\sigma\_P^2} &= \frac{h^2}{4} + \frac{1}{\sigma\_P^2} (\frac{1}{16}\sigma\_{AA}^2 + \frac{1}{64}\sigma\_{AAA}^2 + ... ) \end{aligned}$$

So, epistasis was considered negligible and the narrow-sense heritability was determined as:

$$\begin{aligned} Cov(HS) &= \frac{1}{4} \sigma\_A^2\\ h^2 &= \frac{4Cov(HS)}{\sigma\_P^2} \end{aligned}$$

#### **3.4 Method of Warner (1952)**

Based on additive dominance model Warner in 1952 revealed that narrow-sense heritability could be estimated as:

2 12 <sup>222</sup> 2( ) *F BC BC* Where <sup>2</sup> 2 *<sup>F</sup>* , <sup>2</sup> 1 *BC* and 2 2 *BC* represented respectively the variance of generation F2, BC1 and BC2

In absence of epistasis

6 Agricultural Science

/

*o p*

2

222 2 2 2

*A D AA AD DD AAA*

( ) 11 1 1 1 1 ( .....) 2 4 4 8 16 8

111 1 1 1 ( ) ......) 2 4 4 8 16 8

2

Based on half-sib analysis, narrow-sense heritability was calculated as:

*Cov HS*

*Cov HS h* 

2

 

2

*Cov HS*

2 2

*P P*

*Cov FS*

/

*o p*

2

*h b*

22 2 2 2

2

*A*

2 2

2

*A*

*AA AAA*

2

*P*

22 2

*A AA AAA*

( ) 11 1 ( ......) 4 16 64

2

*P*

4()

11 1 ( ) ...... 4 16 64

So, epistasis was considered negligible and the narrow-sense heritability was determined

<sup>1</sup> ( ) <sup>4</sup>

*Cov HS <sup>h</sup>* 

Based on additive dominance model Warner in 1952 revealed that narrow-sense heritability

2 ()

<sup>1</sup> ( ) <sup>2</sup>

*Cov FS <sup>h</sup>* 

*D AA AD DD AAA*

1 2

*<sup>A</sup> <sup>h</sup> <sup>b</sup>*

*P*

2 2

2 2

, giving

<sup>1</sup> <sup>2</sup> (,) <sup>2</sup> *Cov O P A*

**3.2 Full-sib analysis** 

**3.3 Half-sib analysis** 

**3.4 Method of Warner (1952)** 

could be estimated as:

as:

The covariance full-sib analysis is equal to:

*Cov FS h* 

So, when epistasis was considered negligible

*Cov FS*

2

 

2 2

*P P*

$$2\sigma\_{F\_2}^2 - \left(\sigma\_{BC\_1}^2 + \sigma\_{BC\_2}^2\right) = \frac{1}{2}a\_A^2 = \sigma\_{A\_1}^2$$

$$\frac{2\sigma\_{F\_2}^2 - \left(\sigma\_{BC\_1}^2 + \sigma\_{BC\_2}^2\right)}{\sigma\_{F\_2}^2} = \frac{\sigma\_A^2}{\sigma\_{F\_2}^2} = h^2$$

Therefore in presence of epistasis

$$2\sigma\_{\mathbb{F}\_2}^2 - \left(\sigma\_{\mathrm{BC}\_1}^2 + \sigma\_{\mathrm{BC}\_2}^2\right) = 2\left(a + \frac{1}{2}ad\right)^2 + \left(d + \frac{1}{2}dd\right)^2 + \frac{1}{2}ad^2 + \frac{1}{2}ad + \dots = 0$$

#### **4. Results and discussion**

Separate generation means analysis revealed that the additive-dominance model was found adequate only for 18 cases. Therefore, the digenic epistatic model was found appropriate for 70 cases (Table 2). Additive and dominance effect were significant for all cases of combination. With regard to epistatic effects, the additive x additive effect was significant for 77 cases and the additive x dominance for 42 cases and dominance x dominance effects for 56 cases. Recent studies suggest that epistatic effects are present for inheritance of quantitative traits in many species. Examples are wheat (resistance to leaf rust, Ezzahiri and Roelfs 1989), wheat (resistance to yellowberry, Bnejdi and El Gazzah 2008), common bean (resistance to anthracnose, Marcial and Pastor 1994), barley (resistance to Fusarium head blight, Flavio et al. 2003), chickpea (resistance to *Botrytis cinerea*, Rewal and Grewal 1989), and pepper (resistance to *Phytophthora capsici*, Bartual et al. 1994).

To conclude for this part, the additive dominance model was rarely fitted and digenic epistatic model was frequently appropriate. Therefore epistasis is common in inheritance of quantitative traits and any model or methods assumed that epistasis was negligible were biased.

The comparison of the four methods is reported in Table 3. In absence of dominance and epistatic effect, the methods were not biased. Therefore, in presence of epistasis narrowsense heritability based on the four methods was underestimated. Based in Full-Sib Analysis and Warner (1952) methods, bias was caused by dominance, interaction between homozygote loci, interaction between heterozygote loci and interaction between homozygote and heterozygote loci. Therefore based in Half-Sib Analysis and Parentoffspring regressions, bias was caused only with the presence of interaction between homozygote loci or fixable effect.

The result of generations means analysis indicate that digenic epistasis model were frequently appropriate. So the additive model in which many methods of genetic quantitative were based was rarely adequate. Based on the result, the methods of Half-Sib Analysis and Parent-offspring regressions were underestimated with additive x additive

Impact of Epistasis in Inheritance of Quantitative Traits in Crops 9

O, offspring; P, parent; A, additive; D, dominance; AA, additive × additive;

**5. References** 

*Euphytica,* 163: 225–230.

*China*, 9(10): 101-105.

AD, additive × dominance; DD, dominance × dominance; AAA, additive × additive × additive;

Table 3. Bias of four methods of estimation of narrow-sense heritability in presence of epistasis

Bnejdi F, Saadoun M, Allagui MB, El Gazzah M (2009). Epistasis and heritability of resistance to *Phytophthora nicotianae* in pepper (*Capsicum annuum* L). *Euphytica,* 167: 39-42. Bnejdi F, El Gazzah M (2008). Inheritance of resistance to yellowberry in durum wheat.

Bnejdi F, El Gazzah M (2010a). Epistasis and genotype-by-environment interaction of grain protein concentration in durum wheat. *Genet Mol Biol*, 33(1): 125-130. Bnejdi F, El Gazzah M (2010b). Epistasis and genotype-by-environment interaction of grain yield related traits in durum wheat. *J Plant Breed Crop Sci,* 2 (2): 24-29. Bnejdi F, Rassa N, Saadoun M, Naouari M, El Gazzah M (2011a). Genetic adaptability to salinity level at germination stage of durum wheat. *Afri J Biot,* 10 (21): 4400-4004 Bnejdi F, Hammami I, Allagui MB, El Gazzah M (2010a). Epistasis and maternal effect in

resistance to *Puccinia coronata* Cda. f. sp. *avenae* Eriks in oats (*Avena* sp*.*) *Agri Sci* 

effect (Table 3). Because additive x additive effect can be fixed by selection, estimation of narrow-sense heritability with theses methods was recommended and efficiency in crops breeding. Linkage disequilibrium and absence of epistasis are compulsorily assumed in almost all the methodologies developed to analyze quantitative traits. The consequence, clearly, is biased estimates of genetic parameters and predicted gains, as linkage and genetic interaction are the rule and not the exception Viana (2004). The prediction of gains from selection allows the choice of selection strategies. Therefore the gain from selection was estimated from narrow-sense heritability. Breeding strategies applied for plant breeding aimed to increase the favourable gene frequency. The efficiency of any methodology of selection was associated with the best estimated of the additive genetic effect value.


M, mean; A, additive; D, dominance; AA, additive × additive; AD, additive × dominance; DD, dominance × dominance; C, cytoplasm effect.

Table 2. Best-fit models of nine traits with 88 cases of combinations Cross-site, crosstreatment and or cross-isolate for three crops.

In presence of epistasis effect, Parent-offspring regressions and Half-Sib Analysis were the best methods. In fact, these methods were biased only with interaction between homozygote loci represented by "additive x additive" effect. However, both the methods of Warner (1952) and Full-Sib Analysis were biased with dominance, additive x dominance, dominance x dominance and additive x additive effects. The interaction between the homozygote loci can be fixed by selection. But the fixation of interaction between heterozygote loci prerequisite maintain of heterozygote. Depending upon the methods, the bias in the estimation of narrow-sense heritability in presence of epistasis was more pronounced.

The presence of epistasis complicated the procedure of amelioration of quantitative traits and revealed the limitation of most quantitative studies based on the assumption of negligible epistasis. However, the exploitation of epistasis in the breeding program such as the superiority of heterozygous genotypes over their corresponding parental genotypes was of great importance.


O, offspring; P, parent; A, additive; D, dominance; AA, additive × additive; AD, additive × dominance; DD, dominance × dominance; AAA, additive × additive × additive;

Table 3. Bias of four methods of estimation of narrow-sense heritability in presence of epistasis

### **5. References**

8 Agricultural Science

effect (Table 3). Because additive x additive effect can be fixed by selection, estimation of narrow-sense heritability with theses methods was recommended and efficiency in crops breeding. Linkage disequilibrium and absence of epistasis are compulsorily assumed in almost all the methodologies developed to analyze quantitative traits. The consequence, clearly, is biased estimates of genetic parameters and predicted gains, as linkage and genetic interaction are the rule and not the exception Viana (2004). The prediction of gains from selection allows the choice of selection strategies. Therefore the gain from selection was estimated from narrow-sense heritability. Breeding strategies applied for plant breeding aimed to increase the favourable gene frequency. The efficiency of any methodology of selection was associated with the best estimated of the additive genetic

**Best fit- model Number of cases** 

M, mean; A, additive; D, dominance; AA, additive × additive; AD, additive × dominance; DD,

Table 2. Best-fit models of nine traits with 88 cases of combinations Cross-site, cross-

In presence of epistasis effect, Parent-offspring regressions and Half-Sib Analysis were the best methods. In fact, these methods were biased only with interaction between homozygote loci represented by "additive x additive" effect. However, both the methods of Warner (1952) and Full-Sib Analysis were biased with dominance, additive x dominance, dominance x dominance and additive x additive effects. The interaction between the homozygote loci can be fixed by selection. But the fixation of interaction between heterozygote loci prerequisite maintain of heterozygote. Depending upon the methods, the bias in the estimation of narrow-sense heritability in presence of epistasis

The presence of epistasis complicated the procedure of amelioration of quantitative traits and revealed the limitation of most quantitative studies based on the assumption of negligible epistasis. However, the exploitation of epistasis in the breeding program such as the superiority of heterozygous genotypes over their corresponding parental genotypes was

M + A + D 18

M + A + D + AA 2

M + A + D+ AA + DD 26

M + A + D + AA + AD 13

M + A + D + DD + AD 3

M + A + D + AA + DD + AD 18

M + A + D + AA + DD + AD + C 8

dominance × dominance; C, cytoplasm effect.

was more pronounced.

of great importance.

treatment and or cross-isolate for three crops.

effect value.


**2** 

*Brazil* 

**Genetic Diversity Analysis of** 

**and Chloroplast DNA Regions** 

Helio Almeida Burity4 and Vivian Loges1

*Federal Rural University of Pernambuco,* 

*Heliconia psittacorum* **Cultivars and** 

**Interspecific Hybrids Using Nuclear** 

Luiza Suely Semen Martins2, Luciane Vilela Resende3,

*2Department of Biology, Biochemical Genetics Laboratory/Genome,* 

*3Department of Agronomy, Federal University of Lavras, Minas Gerais, 4Agronomic Research Institute of Pernambuco, Recife, Pernambuco,* 

Walma Nogueira Ramos Guimarães1,\*, Gabriela de Morais Guerra Ferraz1,

*1Department of Agronomy, Federal Rural University of Pernambuco, Recife, Pernambuco,* 

Heliconia cultivation has intensified in Brazil as a cut flower, especially in the Northeast region. This ornamental rhizomatous herbaceous plant from the *Heliconia* genus, belongs to the Musaceae family, now constitutes the Heliconiaceae family in the Zingiberales order. The various species of *Heliconia* are subdivided into five subgenera: *Heliconia, Taeniostrobus, Stenochlamys, Heliconiopsis* and *Griggisia*; and 28 sections (Kress *et al*., 1993). In *Heliconia* genus, the number of species ranges from 120 to 257 and there are also a great number of cultivars and 23 natural hybrids (Berry and Kress, 1991; Castro *et al*., 2007), these plants can be found either in shaded places, such as forests or at full Sun areas, such as forest edges and roadsides (Castro and Graziano, 1997). They are native from Tropical America (Berry and Kress, 1991), found at different altitudes, from sea level up to 2.000 meters in Central and South America, and up to 500 meters in the South Pacific Islands (Criley and Broschat, 1992). Heliconia hybrids comprise many of the major cultivars as cut flowers, like *H. psittacorum* x *H. spathocircinata* cv. Golden Torch, cv. Golden Torch Adrian, cv. Alan Carle and *H. caribaea* x *H. bihai* cv. Carib Flame, cv. Jacquinii, cv. Richmond Red (Berry and Kress, 1991). Many heliconia species are identified through their morphological differences, such as the size and color of its flowers and bracts. These characteristics can be influenced either by geographic isolation or by environmental factors, such as light and nutrients (Kumar *et al*., 1998). *H. psittacorum* clones, even when closely grown, can vary in blooming, size and color of bracts,

as well as post harvest durability (Donselman and Broschat, 1986).

**1. Introduction** 

 \*

Corresponding author

