PBG 650 Advanced Plant Breeding Module 4: Quantitative Genetics Components of phenotypes Genotypic values Average effect of a gene Breeding values

What is Quantitative Genetics? Definition: “Statistical branch of genetics based upon fundamental Mendelian principles extended to polygenic characters” Primary goal: To provide us with a mechanistic understanding of the evolutionary process Lynch and Walsh, Chapter 1

Questions of relevance to breeders How much of the observed phenotypic variation is due to genetic vs environmental factors? How much of the genetic variation is additive (can be passed on from parent to offspring)? What is the breeding value of the available germplasm? Are there genotype by environment interactions? What are the consequences of inbreeding and outcrossing? What are the underlying causes? Are there genetic correlations among traits?

Questions of relevance to breeders Answers to these questions will influence response to selection choice of breeding methods choice of parents optimal type of variety (pureline, hybrid, synthetic, etc.) strategies for developing varieties adapted to target environments

Phenotypic Value P = G + E Components of an individual’s Phenotypic Value P = G + E For individual k with genotype AiAj P(ij)k =  + gij + e(ij)k P = phenotypic value G = genotypic value E = environmental deviation gij = Gij-mu For now we assume that changes in the environment will only affect mu (no gxe) For the population as a whole: E(E) = 0  = E(P) = E(G) Cov(G, E) = 0 Bernardo, Chapt. 3; Falconer & Mackay, Chapt. 7; Lynch & Walsh, Chapt. 4

Single locus model z z+a+d z+2a -a 0 d a Genotypic Value Coded Genotypic Value -a 0 d a Pbar = z + a Bernardo doesn’t endorse calling ’a’ the additive effect of a locus, because it loses its meaning when d is not equal to zero Think of it as genetic value of homozygotes Partial dominance 0<d<a The origin ( ) is midway between the two homozygotes no dominance d = 0 partial dominance 0 < d < +a or 0 > d > –a complete dominance d = +a or –a overdominance d > +a or d < –a degree of dominance =

Single locus model -a 0 d a 0 (1+k)a 2a 0 au a 0 a 2a+d Different scales have been used in the literature A1A1 A2A2 A1A2 -a 0 d a Falconer 0 (1+k)a 2a Lynch & Walsh Comstock and Robinson (1948) 0 au a 0 a 2a+d Hill (1971) Conversions can be readily made

Population mean Frequency Genotypic value Frequency x value A1A1 p2 a 2pq d 2pqd A2A2 q2 –a –q2a M is a function of a, d, and gene frequencies M = p2a + 2pqd – q2a = a(p2 – q2) + 2pqd = a(p + q)(p - q) + 2pqd = a(p - q) + 2pqd Mean on coded scale (centered around zero) This is a weighted average contribution from homozygotes and heterozygotes   Mean on original scale

Population mean  = P + a(p - q) + 2pqd M = a(p - q) + 2pqd When there is no dominance  a(p - q) When A1 is fixed  a When A2 is fixed  -a Potential range  2a A1 fixed – p=1 and q=0 A2 fixed – q=1 and p=0 If the effects at different loci are additive (independent), then M = Σa(p - q) + 2Σpqd  = P + a(p - q) + 2pqd

Means of breeding populations  = P + a(p - q) + 2pqd In an F2 population, p = q = 0.5 F2 = P + (1/2)d In a BC1 crossed to the favorable parent, p = 0.75, so after random mating The mean of the two BC1 populations will only be the same as the F2 if there is no dominance; For a single locus, means in a nonrandom-mated BC1 population will be higher than in the random-mated BC1 if dominance is present Mean for the F2 is the same regardless of whether or not it is random mated (because genotype frequencies are the same) Effects across loci can cancel each other out. For ½ A1A1, ½ A1A2  = P + ½(a + d) BC1(A1A1) = P + (1/2)a + (3/8)d In a BC1 crossed to the unfavorable parent, p = 0.25, so after random mating For ½ A1A2, ½ A2A2  = P + ½(d - a) BC1(A2A2) = P - (1/2)a + (3/8)d

Average effects We have defined the mean in terms of genotypic values Genes (alleles), not genotypes, are passed from parent to offspring Average effect of a gene (i) mean deviation from the population mean of individuals who received that gene from their parents (the other gene taken at random from the population) coded values a, d and –a are expressed in terms of genotypes Genotypes shown in table are consequences of random combinations of A1 (or A2) with gametes in the population subtract M = a(p - q) + 2pqd Gamete A1A1 a A1A2 d A2A2 -a Freq x value Average effect of a gene A1 p q pa + qd 1=q[a+d(q-p)] A2 pd - qa 2=-p[a+d(q-p)]

Average effect of a gene substitution Average effect of changing from A2 to A1  = 1 - 2 q[a+d(q-p)] – (-p)[a+d(q-p)] = a+d(q-p) Average effect of changing from A1 to A2 = - Relating this to the average effects of alleles: 1 = q 2 = -p a and d are intrinsic properties of genotypes 1, 2, and  are joint properties of alleles and the populations in which they occur (they vary with gene frequencies)

Breeding value of individual Aij = i + j Genotype Average effect of a gene Average effect of a gene substitution A1A1 21 2q A1A2 1 + 2 (q - p) A2A2 22 -2p Breeding value of an individual is judged by the mean value of its progeny. Breeding value can be measured. Breeding value is 2 x GCA. Sum across all loci affecting the trait. For a population in H-W equilibrium, the mean breeding value = 0 The expected breeding value of an individual is the average of the breeding value of its two parents For an individual mated at random to a number of individuals in a population, its breeding value is 2 x the mean deviation of its progeny from the population mean.

Regression of breeding value on genotype Breeding values can be measured provide information about genetic values lead to predictions about genotypic and phenotypic values of progeny Additive genetic variance variance in breeding values variance due to regression of genotypic values on genotype (number of alleles) ● genotypic value ○ breeding value

Genotypic values a d a(q-p)+d(1-2pq) (q-p)+2pqd -a Genotypic values have been expressed as deviations from a midparent To calculate genetic variances and covariances, they must be expressed as a deviation from the population mean, which depends on gene frequencies subtract M = a(p - q) + 2pqd Genotypic values Genotype Scaled Adjusted for mean A1A1 a 2q(a-pd) 2q(-qd) A1A2 d a(q-p)+d(1-2pq) (q-p)+2pqd A2A2 -a -2p(a+qd) -2p(+pd) Remember  = a + d(q - p)  Substitute a =  - d(q - p)

P = G + E G = A + D Dominance deviation Gij =  + i + j + ij Components of an individual’s Phenotypic Value P = G + E G = A + D In terms of statistics, D represents within-locus interactions deviations from additive effects of genes Arises from dominance between alleles at a locus dependent on gene frequencies not solely a function of degree of dominance (a locus with completely dominant gene action contributes substantially to additive genetic variance) Gij =  + i + j + ij If there is no intralocus variance due to D, then you can say that there is only additive gene action within a locus (may not hold true across loci)

Partitioning Genotypic Value Genotype Genotypic Value (adj. for mean) Breeding Value (additive effects) Dominance Deviation A1A1 2q(-qd) 2q -2q2d A1A2 (q-p)+2pqd (q - p) +2pqd A2A2 -2p(+pd) -2p -2p2d When p = q = 0.5 (as in a biparental cross between inbred lines) Genotype Genotypic Value Breeding Value Dominance A1A1 -(1/2)d  -(1/2)d A1A2 (1/2)d A2A2 --(1/2)d -

Dominance deviations from regression Genotypic Value A1A1 2q - 2q2d A1A2 (q-p)+2pqd A2A2 -2p - 2p2d -2p2d 2pqd -2q2d

Interaction deviation Components of an individual’s Phenotypic Value P = G + E P = A + D + E When more than one locus is considered, there may also be interactions between loci (epistasis) G = A + D + I P = A + D + I + E ‘I’ is expressed as a deviation from the population mean and depends on gene frequencies For a population in H-W equilibrium, the mean ‘I’ = 0