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, suc. Ebook Selection methods in plant breeding part 2
Chapter 11 Applications of Quantitative Genetic Theory in Plant Breeding In the preceding chapters dealing with traits with quantitative variation, a number of important concepts were introduced, such as phenotypic value and genotypic 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 identically 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 parameters defined in the framework of the F∞ -metric (Section 8.3.2), or in terms of additive genotypic value and dominance deviation (Section 8.3.3), here the rather straightforward partitioning in terms of general combining ability and specific combining ability is elaborated 11.1 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 11.1, are defined as follows: S : = Eps,t − Ept (11.1) R : = Ept+1 − 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 candidates belonging to generation t of the population subjected to selection Ept+1 designates the expected phenotypic value calculated across the offspring of the selected candidates I Bos and P Caligari, Selection Methods in Plant Breeding – 2nd Edition, 225–287 c 2008 Springer 225 226 11 Applications of Quantitative Genetic Theory in Plant Breeding Fig 11.1 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.e the quantities S and R, can be estimated from the phenotypic values of a random sample of the (selected) ˆ will candidates and their offspring, i.e from pt , ps,t and pt+1 , As the symbol R 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 11.1 Prediction of the Response to Selection 227 The response to selection is now considered for three situations: The hypothetical case of absence of environmental deviations, as well as absence of dominance and epistasis Absence of environmental deviations, presence of dominance and/or epistasis 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 8.3.2 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 + The offspring of these candidates will have the same phei=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 phenotypic value, may have a homozygous or a heterozygous genotype Then the offspring of the selected candidates are expected to comprise plants with genotype bb for one or more loci, giving rise to an inferior phenotypic value compared to that of the selected candidates In the case of complete dominance, for instance, candidates with the highest possible phenotypic value for a trait controlled 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) Equation (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 = 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 (11.4) CSY := EpY − EpY,t 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 calculated 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.e βpY ,pX = cov(pY , 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 Example 11.1 presents an illustration Example 11.1 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.e after pollen 11.1 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 = g This implies SY = 12 (201 + 0) = 100.5 g Population B consisted of 1163 plants Immediately prior to pollen distribution 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 selection 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.e 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 designated by p = N (µ, σ ) where • • µ = Ep, and σ = var(p) Standardization, i.e the transformation of p into z according to p−µ =z σ implies that z has a standard normal distribution characterized by µz = and σz = Thus z = N (0, 1) Selection of candidates with a phenotypic value exceeding the phenotypic value minimally required for selection (pmin ) is called truncation selection Selection 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 (11.5) zmin = σ Thus ∞ v = P p > pmin = P (z > zmin ) = f (z).dz zmin where f (z) = √ e− z 2π is the density function of the standard normal random variate z In Fig 11.1 the shaded area corresponds with v Most statistical handbooks (e.g 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 (11.6) Example 11.2 gives an illustration of this Example 11.2 It was desired to select the 168 best yielding plants from the 5016 winter rye plants occurring at the central plant positions of the population which is mentioned in Example 11.7 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 = 1.83 The mean and the standard deviation of the phenotypic values for grain yield were calculated to be 50 dg and 28.9 dg, respectively When assuming a normal distribution for grain yield, substitution of these values in Equation (11.5) yielded: pmin = 50 + (28.9 × 1.83) = 102.9 dg To measure the selection differential in a scale-independent yardstick, a parameter, called selection intensity and designated by the symbol i, has been defined: S (11.7) i= σ There is a simple relationship between the proportion of selected candidates (v) and i if the phenotypic values of the considered trait follow a normal distribution, namely f (zmin ) (11.8) i= v where f (zmin ) represents the value at z = zmin of the density function of the standard normal random variate z Equation (11.8) is derived in Note 11.1 Note 11.1 Equation (11.6) implies that, in the case of a normal distribution of the phenotypic values, the expected phenotypic value of candidates with a phenotypic value larger than pmin amounts to Eps,t = E(p|p > pmin ) = + Ez s,t where 232 ã ã 11 Applications of Quantitative Genetic Theory in Plant Breeding pmin may be obtained from Equation (11.5) Ez s,t = E(z|z > zmin ), where zmin follows from Equation (11.5) The quantity Ez s,t is now derived The density function of the conditional random variable (z|z > zmin ) is f (z|z > zmin ) = f (z) f (z) = P (z > zmin ) v Thus ∞ ∞ Ez s = E(z|z > zmin ) = zf (z|z > zmin )dz = z=z z z f (z) dz v min ∞ ∞ 2 1 = √ · z ze− z dz = √ · e− z d v 2π zmin v 2π zmin f (z ) −1 − z2 ∞ −1 e − e− zmin = = √ = √ v z=zmin v 2π v 2π This means that Eps,t f (zmin ) =µ+σ v Because µ = Ep, Equation (11.1) can be written as f (zmin ) S=σ v Thus when applying truncation selection with regard to a trait with a normal distribution and selecting the proportion v the selection intensity is: i= f (zmin ) = Ez s,t v One can easily calculate i for any value for v and next Eps,t = µ + σi, see Example 11.3 Falconer (1989, Appendix Table A) presents a table for the relation between i and v Example 11.3 In Example 11.2 it was derived that the standardized minimum phenotypic value zmin is 1.83 when selecting the proportion v = 0.0335 In the case of a normal distribution of the phenotypic values the selection intensity amounts then to f (1.83) = 0.0335 √1 e− (1.83) 2π 0.0335 = 0.3989 × 0.1874 = 2.232 0.0335 11.1 Prediction of the Response to Selection 233 Thus Eps = 50 + 28.9 × 2.232 = 114.5 dg Among the 168 plants with the highest grain yield, the grain yield of the plant with the lowest phenotypic value amounted to 102 dg The actual minimum phenotypic value was thus 102 dg Their mean grain yield amounted to 117.5 dg, implying S = 117.5 − 50 = 67.5 dg and i= 67.5 = 2.34 28.9 Also the measurement of the response to selection (R) deserves closer consideration It requires determination of Ep in the two successive generations t and t + To exclude an effect of different growing conditions these two generations should preferably be grown in the same growing season This is possible by Testing simultaneously plant material representing generation t + (say population P t+1 ), obtained by harvesting candidates selected in generation t, and – from remnant seed – plant material representing generation t (say population Pt ) Testing simultaneously plant material representing generation t + 1, obtained by harvesting candidates selected in generation t (population P t+1 ), and plant material, also representing generation t + 1, obtained by harvesting in generation t random candidates (population Pt+1 ) Simultaneous testing of populations P t+1 and Pt Measurement of R by simultaneous testing of populations P t+1 and Pt will be biased if these populations differ due to other causes than the selection Such differences may be due to • • • the fact that the remnant seed is older and has, consequently, lost viability; the remnant seed representing Pt was produced under conditions deviating from the conditions prevailing when producing the seed representing P t+1 or a difference in the genotypic compositions of P t+1 and Pt which is not due to the selection This is to be expected when dealing with self-fertilizing crops: P t+1 tends to contain a reduced frequency of heterozygous plants in comparison to Pt 234 11 Applications of Quantitative Genetic Theory in Plant Breeding When testing populations P t+1 and Pt simultaneously, no allowance is made for the possible quantitative genetic effect of the reduction of heterozygosity occurring in self-fertilizing crops Simultaneous testing of populations P t+1 and Pt+1 The causes for the bias mentioned above not apply to simultaneous testing of populations P t+1 and Pt Furthermore, this method allows – for crossfertilizing crops – estimation of the coefficient of regression of the phenotypic value of offspring on parental phenotypic value Such an estimate may be interpreted in terms of the narrow sense heritability (Section 11.2.2) One should realize that R as defined by Equation (11.2) does not represent K a lasting response to selection if di = For self-fertilizing crops populai=1 tions after generation t + 1, obtained in the absence of selection, will – due to the ongoing reduction of the frequency of heterozygous plants – tend to have an expected genotypic value deviating from Ept+1 = Ept + R The same applies to selection after pollen distribution in cross-fertilizing crops: population P t+1 results then from a bulk cross and will, consequently, contain an excess of heterozygous plants compared to population Pt+2 obtained – in the absence of selection – from population P t+1 In the case of selection before pollen distribution, population P t+1 is in Hardy–Weinberg equilibrium and P t+1 and Pt+2 will then, in the absence of epistasis, have the same expected genotypic value A procedure to predict R is, of course, of great interest to breeders, because such prediction may be used as a basis for a decision with regard to further breeding efforts dedicated to the plant material in question As the prediction is based on linear regression theory, a few important aspects of that theory are reminded In the case of linear regression of y on x the y-value for some x-value is predicted by yˆ = α + βx, where β= E(x · y) − (Ex) · (Ey) cov(x, y) = var(x) Ex2 − (Ex)2 (11.9) and, because of Ey = α + β · Ex the intercept α is equal to α = Ey − β.Ex (11.10) yˆ = (Ey − β · Ex) + βx = Ey + β(x − Ex) (11.11) yˆ − Ey = β(x − Ex) (11.12) Thus implying ... the selection intensity amounts then to f (1.83) = 0.0335 √1 e− (1.83) 2? ? 0.0335 = 0.3989 × 0.1874 = 2. 2 32 0.0335 11.1 Prediction of the Response to Selection 23 3 Thus Eps = 50 + 28 .9 × 2. 2 32 =... 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.. .22 6 11 Applications of Quantitative Genetic Theory in Plant Breeding Fig 11.1 The density function for the phenotypic value p in generation t and in generation t + 1, obtained by selecting in