Chapter 11 Applications of Quantitative Genetic Theory in Plant Breeding In the preceding chapters dealing with traits with quantitative variation, a num- ber of important concepts were introduced, such as phenotypic value and geno- typic value (Chapter 8), expected genotypic value (Chapter 9) and genotypic variance (Chapter 10). The present chapter focusses on applications of these concepts that are important in the context of this book. Thus the response to selection, both its predicted and its actual value, is considered. The prediction of the response is based on estimates of the heritability.
Procedures for the estimation of this quantity are elaborated for plant material that can identi- cally be reproduced (clones of crops with vegetative reproduction, pure lines of self-fertilizing crops and single-cross hybrids). It is shown how the heritability value depends on the number of replications. In addition to the partitioning of the genotypic value in terms of parame- ters defined in the framework of the F∞ -metric (Section 8.2), or in terms of additive genotypic value and dominance deviation (Section 8.3), here the rather straightforward partitioning in terms of general combining ability and specific combining ability is elaborated.1 Prediction of the Response to Selection When dealing with selection with regard to quantitative variation the concepts of selection differential, designated by S, and response to selection, designated by R, play a central role. These concepts, see also Fig.1, are defined as follows: S : = Eps,t − Ept (11.2) where • Eps,t designates the expected phenotypic value of the candidates (plants, clones, families or lines) in generation t of the considered population with a phenotypic value greater than the phenotypic value minimally required for selection (pmin ).
Eps,t designates thus the expected phenotypic value of the selected candidates. • Ept designates the expected phenotypic value calculated across all candi- dates belonging to generation t of the population subjected to selection. • Ept+1 designates the expected phenotypic value calculated across the off- spring of the selected candidates. Caligari, Selection Methods in Plant Breeding – 2nd Edition, 225–287.
226 11 Applications of Quantitative Genetic Theory in Plant Breeding Fig.1 The density function for the phenotypic value p in generation t and in generation t + 1, obtained by selecting in generation t all candidates with a phenotypic value greater than pmin. The selection differential (S) in generation t and the response to the selection (R) are indicated. The shaded area represents the probability that a candidate has a phenotypic value larger than the minimally required phenotypic value (pmin ) In Section 8.2 it was derived that Ep = EG This implies that one may write EG t instead of Ept and EG t+1 instead of Ept+1. The quantities Eps,t , Ept and Ept+1 , i.
the quantities S and R, can be estimated from the phenotypic values of a random sample of the (selected) candidates and their offspring, i. from pt , ps,t and pt+1 , As the symbol R̂ will be used to indicate the predicted response to selection, the values estimated for S and R will be written in terms of pt , ps,t and pt+1 .1 Prediction of the Response to Selection 227 The response to selection is now considered for three situations: 1. The hypothetical case of absence of environmental deviations, as well as absence of dominance and epistasis 2. Absence of environmental deviations, presence of dominance and/or epistasis 3.
Presence of environmental deviations, dominance and/or epistasis Absence of environmental deviations, dominance and epistasis In the absence of environmental deviations, dominance and epistasis, both the genotypic value and the phenotypic value of a candidate can be described by a linear combination of the parameters a1 ,. , aK defined in Section 8. Selection of candidates with the highest possible phenotypic value implies selection of candidates with genotype B1 B1. BK BK and with genotypic K value m + ai.
The offspring of these candidates will have the same phe- i=1 notypic and genotypic value as their parents. This applies to self-fertilizing crops as well as cross-fertilizing crops, when the selection occurs before pollen distribution. Under the described conditions R will be equal to S. Absence of environmental deviations, presence of dominance and/or epistasis In the case of absence of environmental deviations but presence of dominance and/or epistasis, selected candidates, with the same highest possible pheno- typic value, may have a homozygous or a heterozygous genotype.
Then the offspring of the selected candidates are expected to comprise plants with geno- type bb for one or more loci, giving rise to an inferior phenotypic value com- pared to that of the selected candidates. In the case of complete dominance, for instance, candidates with the highest possible phenotypic value for a trait con- trolled by loci B1 −b1 and B2 −b2 will have genotype B1 ·B2 ·. Selection of such candidates will yield offspring including plants with genotype b1 b1 b2 b2 , b1 b1 B2 · or B1 · b2 b2 , having an inferior genotypic and phenotypic value. Under these conditions R will be less than S.
Presence of environmental deviations, dominance and/or epistasis In actual situations environmental deviations, dominance and epistasis should be expected to be present. Among the selected candidates their phenotypic values will tend to be (much) higher than their genotypic values. Furthermore, except in the case of identical reproduction, the genotypic composition of the selected candidates will deviate from that of their offspring. Under these conditions R will be (much) smaller than S.
Selected maternal plants coincide with the selected paternal plants in the case of self-fertilizing crops, as well as in case of hermaphroditic cross-fertilizing 228 11 Applications of Quantitative Genetic Theory in Plant Breeding crops if the selection is applied before pollen distribution. In other situations, the set of selected maternal parents providing the eggs differs from the set of selected paternal parents providing the pollen. Then one should determine Sf for the candidates selected as maternal parents and Sm for the candidates selected as paternal parents. Because both sexes contribute equal numbers of gametes to generate the next generation we may write S = 12 (Sf + Sm ) (11.3) does not only apply at selection in dioecious crops, but also when selecting in hermaphroditic cross-fertilizing crops when the selection is done after pollen distribution.
In the latter case there is no selection with regard to paternal parents. This implies Sm = 0 and consequently S = 12 Sf. Actual situations tend to be more complicated. Consider selection before pollen distribution with regard to some trait X.
In the case of an association between the expression for trait X and the expression for trait Y, the selection differential for X implies a correlated selection differential with regard to Y, say CS. Thus CSY := EpY − EpY,t (11.4) s,t where • EpY ,t designates the expected phenotypic value with regard to trait Y of s the candidates selected in generation t because their phenotypic value with regard to trait X being greater than minimally phenotypic value (pXmin ) and • Ept designates the expected phenotypic value with regard to trait Y cal- culated across all candidates belonging to generation t of the population subjected to selection with regard to trait X. When considering a linear relationship between the phenotypic values for traits X and Y, the coefficient of regression of pY on pX , i. cov(pY , pX ) βpY ,pX = var(pX ) may be used to write CSY = βpY ,pX SX The indirect selection (see Section 12.3) for trait Y, via trait X, may be followed, after pollen distribution, by direct selection for Y.
The effective selection differential for Y comprises then a correlated selection differential.1 presents an illustration.1 Van Hintum and Van Adrichem (1986) applied selection in two populations of maize with the goal of improving biomass. Population A consisted of 1184 plants. Mass selection for biomass (say trait Y) was applied at the end of the growing season, i.1 Prediction of the Response to Selection 229 distribution. The mean biomass (in g/plant), calculated across all plants, was pY = 245 g.
For the 60 selected plants it amounted to pYs = 446 g. Thus Sf = 446 − 245 = 201 g and Sm = 0 g This implies SY = 12 (201 + 0) = 100. Population B consisted of 1163 plants. Immediately prior to pollen dis- tribution the following was done.
The volumes of the plants (say trait X) were roughly calculated from their stalk diameter and their height. The 181 plants with the highest phenotypic values for X were identified. These plants were selected as paternal parents. The 982 other plants were emasculated by removing the tassels.
At the end of the growing season among all 1163 plants, the 60 plants with the highest biomass were selected. For the 1163 plants of population B it was found that: pY = 246 g, and pX = 599 cm3. For the 181 plants selected as paternal parents (because of superiority for X) it was established that: pYs = 320 g, pXs = 983 cm3 , and CSYm = 320 − 246 = 74 g. For the 60 plants selected for Y the following was established: pYs = 418 g pXs = 931 cm3 and SYf = 418 − 246 = 172 g The selection differential in population B amounted thus to SY = 12 (74 + 172) = 123 g Due to the correlated selection differential because of selection among the paternal parents with regard to trait X, this is clearly higher than the selec- tion differential in population A.
230 11 Applications of Quantitative Genetic Theory in Plant Breeding If the considered trait has a normal distribution, Eps,t , i. the expected phenotypic value of those candidates with a phenotypic value larger than the value minimally required for selection, may be calculated prior to the actual selection. This will now be elaborated. A normal distribution of the phenotypic values for some trait is often desi- gnated by p = N (µ, σ 2 ) where • µ = Ep, and • σ 2 = var(p).
the transformation of p into z according to p−µ =z σ implies that z has a standard normal distribution characterized by µz = 0 and σz = 1. Selection of candidates with a phenotypic value exceeding the phenotypic value minimally required for selection (pmin ) is called truncation selection. Selec- tion of superior performing candidates up to a proportion v implies applying a value for pmin such, that v = P (p > pmin ) Standardization of pmin yields the standardized minimum phenotypic value zmin : pmin − µ zmin = (11.dz zmin where 1 f (z) = √ e− 2 z 1 2 2π is the density function of the standard normal random variate z.1 the shaded area corresponds with v. Most statistical handbooks (e.
Kuehl, 2000, Table I) contain for the standard normal random variate z 11.1 Prediction of the Response to Selection 231 a table presenting zmin such P(z > zmin ) is equal to some specified value v. Then one can calculate pmin according to pmin = µ + σzmin (11.2 gives an illustration of this.2 It was desired to select the 168 best yielding plants from the 5016 winter rye plants occurring at the central plant positions of the pop- ulation which is mentioned in Example 11. The proportion to be selected amounted thus to: 168 v= = 0.0335 5016 The standardized minimum phenotypic value zmin should thus obey: 0.0335 = P(z > zmin ) According to the appropriate statistical table, his implies zmin = 1. The mean and the standard deviation of the phenotypic values for grain yield were calculated to be 50 dg and 28.
When assuming a normal distribution for grain yield, substitution of these values in Equa- tion (11.