**1. Introduction**

Epistasis is the interaction between alleles of different genes, i.e. non-allelic interaction, as opposed to dominance, which is interaction between allele of the same gene, called interallelic or intra-genic interaction (Kearsey and Pooni, 1996). Statistical epistasis describes the deviation that occurs when the combined additive effect of two or more genes does not explain an observed phenotype (Falconer and Mackay, 1996).

The heritability of a trait, an essential concept in genetics quantitative, "certainly one of the central points in plant breeding research is the proportion of variation among individuals in a population that" is due to variation in the additive genetic (i.e., breeding) values of individuals:

h2 = VA/VP = Variance of breeding values/ phenotypic variance (Lynch and Walsh, 1998). This definition is now termed "heritability in the narrow-sense" (Nyquist, 1991). Estimation of this parameter was prerequisite for the amelioration of quantitative traits. As well as choosing the selective procedure, that will maximize genetic gain with one or more selection cycles. Various methods were developed in the past, Warner (1952), Sib-Analysis, Parent-offspring regressions etc. Theses methods considered that additive-dominant model is fitted, assuming epistasis to be negligible or non existent. Because of the complexity of theoretical genetics studies on epistasis, there is a lack of information about the contribution of the epistatic components of genotypic variance when predicting gains from selection. The estimation of epistatic components of genotypic variance is unusual in genetic studies because the limitation of the methodology, as in the case of the triple test cross, the high number of generations to be produced and assessed (Viana, 2000), and mainly because only one type of progeny, Half-Sib, Full-Sib or inbred families, is commonly included in the experiments (Viana, 2005). If there is no epistasis, generally it is satisfactory to assess the selection efficiency and to predict gain based on the broad-sense heritability. Therefore, the bias in the estimate of the additive variance when assuming the additive-dominant model is considerable. The preponderance of epistasis effect in the inheritance of quantitative trait in crops was recently reported by many geneticists (Pensuk et al., 2004; Bnejdi and El Gazzah, 2008; Bnejdi et al. 2009; Bnejdi and El-Gazzah, 2010a; Shashikumar et al. 2010). Epistasis can have an important influence on a number of evolutionary phenomena, including the genetic divergence between species.

Impact of Epistasis in Inheritance of Quantitative Traits in Crops 5

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.

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

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

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 single-

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**2.1 Best genetic model** 

**3. Phenotypic resemblance between relatives** 

expressed in terms of genetic variances, as we detail below.

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

(,)

*Var P* , where

**3.1 Parent-offspring regressions** 

/

parent regressions usually restricted to fathers.

*Cov O P <sup>b</sup> Var P*

(,) ( )

/

*o p*

*o p*

*o p*

*o p* ( ) *Cov O P <sup>b</sup>*

/ 2 2

 

*Var P*

*Var P*

Assuming an absence of epistasis we have

/ 2

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 narrow-sense heritability.
