Variation of the parental genome contribution in segregating populations derived from biparental crosses and its relationship with heterosis of their Design III progenies

Abstract

The variation of the parental genome contribution (PGC) and its relationship with the genetic architecture of heterosis have received little attention. Our objectives were to (1) derive formulas for the variance of PGC in selfing, backcross (BC) or intermated generations produced from biparental crosses of homozygous parents, (2) investigate the correlation $r(Z_{2} ,\Uppsi_{M} )$ of the PGC $(\Uppsi_{M} )$ estimated by a set M of markers, with Z ₂ (half the trait difference between each pair of BC progenies) in the Design III, and (3) interpret experimental results on this correlation with regard to the genetic basis of heterosis. Under all mating systems, the variance of PGC is smaller in species with a larger number and more uniform length of chromosomes. It decreases with intermating and backcrossing but increases under selfing. The ratio of variances of PGC in F₁DH (double haploids), F₂ and BC₁ populations is 4:2:1, but it is smaller in advanced selfing generations than expected for quantitative traits. Thus, altering the PGC by marker-assisted selection for the genetic background is more promising (i) in species with a smaller number and/or shorter chromosomes and (ii) in F₂ than in progenies of later selfing generations. The correlation $r(Z_{2} ,\Uppsi_{M} )$ depends on the linkage relationships between M and the QTL influencing Z₂ as well as the augmented dominance effects $d_{i}^{*}$ of the QTL, which include dominance and additive × additive effects with the genetic background, and sum up to mid-parent heterosis. From estimates of $r(Z_{2} ,\Uppsi_{M} )$ as well as QTL studies, we conclude that heterosis for grain yield in maize is caused by the action of numerous QTL distributed across the entire genome with positive $d_{i}^{*}$ effects.

Appendix

Here, we derive ${\text{Var}}(\Uppsi_{{c_{b} }} )$ for generation F _t+1 (t ≥ 1). According to Table 2, we have for F _t+1 ${\text{Cov}}(\Uppsi_{i} ,\Uppsi_{j} ) = \sum\nolimits_{r = 1}^{t} {\left( {{\frac{1}{2}}} \right)^{r + 2} e^{{ - 2r\left| {x - y} \right|}} }$. From Eq. [A3] of the Appendix of Frisch and Melchinger (2007), we obtain

$$ \int\limits_{0}^{{l_{b} }} {\int\limits_{0}^{{l_{b} }} {e^{{ - 2r\left( {x - y} \right)}} {\text{d}}x\,{\text{d}}y} } = {\frac{1}{{2r^{2} }}}\left( {2rl_{b} - 1 + e^{{ - 2rl_{b} }} } \right) $$

and together with Eq. (4), we get

$$ {\text{Var}}\left( {\Uppsi_{{c_{b} }} } \right) = {\frac{1}{{l_{b}^{2} }}}\int\limits_{0}^{{l_{b} }} {\int\limits_{0}^{{l_{b} }} {{\text{Cov}}\left( {\Uppsi_{i} ,\Uppsi_{j} } \right)} } {\text{d}}x\,{\text{d}}y = {\frac{1}{{l_{b}^{2} }}}\int\limits_{0}^{{l_{b} }} {\int\limits_{0}^{{l_{b} }} {\sum\limits_{r = 1}^{t} {\left( {{\frac{1}{2}}} \right)^{r + 2} e^{{ - 2r\left| {x - y} \right|}} {\text{d}}x\,{\text{d}}y} } } = {\frac{1}{{l_{b}^{2} }}}\sum\limits_{r = 1}^{t} {\left( {{\frac{1}{2}}} \right)^{r + 3} {\frac{1}{{r^{2} }}}\left( {2rl_{b} - 1 + e^{{ - 2rl_{b} }} } \right)} $$