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? • Are there genetic correlations among traits? What are the consequences of inbreeding and outcrossing? What are the underlying causes? 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 Components of an individual’s Phenotypic Value P=G+E P = phenotypic value G = genotypic value For individual k with genotype AiAj P(ij)k = + gij + e(ij)k E = environmental deviation 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 Genotypic Value Coded Genotypic Value A2A2 A1A2 A1A1 z z+a+d z+2a P -a P P d P a -a 0 d a The origin ( P ) is midway between the two homozygotes no dominance partial dominance complete dominance overdominance d=0 0 < d < +a or 0 > d > –a d = +a or –a d > +a or d < –a d degree of dominance = a Single locus model Different scales have been used in the literature A2A2 -a 0 A1A2 0 A1A1 d a Falconer (1+k)a 2a Lynch & Walsh 0 au a 0 a 2a+d Conversions can be readily made Comstock and Robinson (1948) Hill (1971) Population mean A1A1 Frequency p2 A1A2 A2A2 2pq q2 Genotypic Frequency value x value a p2 a d –a M = p2a + 2pqd – q2a = a(p2 – q2) + 2pqd = a(p + q)(p - q) + 2pqd = a(p - q) + 2pqd 2pqd –q2a Mean on coded scale (centered around zero) This is a weighted average contribution from homozygotes and heterozygotes Mean on original scale Population mean M = a(p - q) + 2pqd When there is no dominance When A1 is fixed When A2 is fixed Potential range a(p - q) a -a 2a If the effects at different loci are additive (independent), then M = Σa(p - q) + 2Σpqd = P + a(p - q) + 2pqd 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 BC1(A1A1) = P + (1/2)a + (3/8)d For ½ A1A1, ½ A1A2 = P + ½(a + d) In a BC1 crossed to the unfavorable parent, p = 0.25, so after random mating For ½ A1A2, ½ A2A2 BC1(A2A2) = P - (1/2)a + (3/8)d = P + ½(d - a) Average effects • • We have defined the mean in terms of genotypic values • Average effect of a gene (i) Genes (alleles), not genotypes, are passed from parent to offspring – mean deviation from the population mean of individuals who received that gene from their parents (the other gene taken at random from the population) subtract M = a(p - q) + 2pqd Gamete A1 A2 A1A1 A1A2 A2A2 a p d q -a p q Freq x value Average effect of a gene pa + qd 1=q[a+d(q-p)] 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 Breeding value of individual Aij = i + j Genotype A1A1 A1A2 A2A2 Breeding Value Average effect of a Average effect of a gene gene substitution 21 1 + 2 22 2q (q - p) -2p • 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 • • 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 A1A1 A1A2 A2A2 Scaled a d -a Adjusted for mean 2q(a-pd) 2q(-qd) a(q-p)+d(1-2pq) -2p(a+qd) (q-p)+2pqd -2p(+pd) Remember = a + d(q - p) Substitute a = - d(q - p) Dominance deviation Components of an individual’s Phenotypic Value P=G+E G=A+D Gij = + i + j + ij • 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) Partitioning Genotypic Value Genotypic Value Breeding Value Genotype (adj. for mean) (additive effects) Dominance Deviation A1A1 2q(-qd) 2q -2q2d A1A2 (q-p)+2pqd -2p(+pd) (q - p) -2p +2pqd -2p2d A2A2 When p = q = 0.5 (as in a biparental cross between inbred lines) Genotype Genotypic Value Breeding Value A1A1 -(1/2)d A1A2 A2A2 (1/2)d --(1/2)d 0 - Dominance -(1/2)d (1/2)d -(1/2)d Dominance deviations from regression Genotypic Value A1A1 A1A2 2q (q-p)+2pqd -2q2d 2q2d 2pqd -2p2d A2A2 -2p - 2p2d 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