Quantitative Genetics Quantitative Genetics Floral trait variation in What we’ve covered so far: Mimulus. Used a population genetics approach Variation in these traits to study the dynamics of how selection affects the degree to which plants are attractive to operates in a one locus haploid model, a hummingbirds versus bees. one-locus diploid model and a two- The fact that a range of locus diploid model. phenotypes is seen suggests that these traits are controlled by multiple loci. From Bradshaw et al. 1998. Genetics R. A. Fisher 1890 – 1962 AD Quantitative genetics The genetic basis of many traits is only poorly known. Uniting Mendelian and quantitative We lack specific information about: genetics • The number of genes • In 1918, Fisher showed that a large number of Mendelian factors (genes) • The position of genes within the genome influencing a trait would cause a nearly • The fitness effects of particular alleles continuous distribution of trait values • The interactions among loci • Mendelian genetics can lead to an approximately normal • Even if we had more information about the distribution genetic basis of a trait, explicit models with multiple loci are astonishingly complex. Furthermore, the expression of many traits is affected to some degree by the environment. Quantitative genetics Quantitative genetics Evolutionary models fall into two camps: Building a quantitative genetic model • Population genetic models explicitly follow allele In quantitative genetics, the phenotypic value (P) of an frequency changes at specific loci individual (e.g. height) is attributed to the genotype of the • Behavior of these models is well understood only for individual and to its environment: one or two loci. P = G + E • Quantitative genetic models ignore genetic details The genotypic value (G) is a measure of the influence of (e.g. recombination rates, linkage disequilibria) and every gene carried by the individual on the phenotypic focus on traits, and on the average effect of many value. genes on trait values. The environmental deviation (E) is a measure of the • Such simplified multi-locus models are easier to analyse. influence of the environment of an individual, scaled such However, it remains unclear to what extent the genetic that the average value of E is zero within a population. details matter. Quantitative genetics Quantitative genetics Example: Average yield in wheat strains Example: Average yield in wheat strains (bushels /acre) Strain Strain Year Roughrider Seward Aggasiz Year Roughrider Seward Aggasiz Environmental values 1986 47.9 55.9 47.5 1986 47.9 55.9 47.5 63.8 - 45.63 = 18.17 1987 63.8 72.5 59.5 1987 63.8 72.5 59.5 1988 23.1 25.7 28.4 1988 23.1 25.7 28.4 1989 61.6 66.5 60.5 1989 61.6 66.5 60.5 1990 0.0 0.0 0.0 1990 0.0 0.0 0.0 1991 60.3 71.0 55.4 1991 60.3 71.0 55.4 1992 46.6 49.0 41.5 1992 46.6 49.0 41.5 1993 58.2 62.9 48.8 1993 58.2 62.9 48.8 49.0 - 52.18 = -3.18 1994 41.7 53.2 39.8 1994 41.7 53.2 39.8 1995 53.1 65.1 53.5 1995 53.1 65.1 53.5 Mean 45.63 52.18 43.49 Mean 45.63 52.18 43.49 Genetic values (G) Quantitative genetics Quantitative genetics Example: Average yield in wheat strains Breeding values Strain Year Roughrider Seward Aggasiz G + E = P 1986 2.27 3.72 4.01 Population mean Genetic value of a parent 45.63 + 18.17 = 63.8 1987 18.17 20.32 16.01 (70 bushels/acre) (80 bushels/acre) 1988 -22.53 -26.48 -15.09 1989 15.97 14.32 17.01 Expected genetic value 1990 -45.63 -52.18 -43.49 of offspring 1991 14.67 18.82 11.91 The genetic value of a genotype reflects the sum total 1992 0.97 -3.18 -1.99 effect of all alleles at the loci that affect the trait of 1993 12.57 10.72 5.31 interest. 1994 -3.93 1.02 -3.69 1995 7.47 12.92 10.01 Given that a parent in a sexual species passes half of its Mean 0 0 0 alleles to the offspring, what is the expected genetic G 45.63 52.18 43.49 value of the offspring? (assume a randomly chosen mate) Quantitative genetics Quantitative genetics Breeding values Breeding values Population mean Genetic value of a parent Population mean Breeding value (70 bushels/acre) (80 bushels/acre) (70 bushels/acre) of parent (A) Expected genetic value Actual genetic value of offspring of offspring With the alleles present in a new genetic context, the But this assumes that the alleles act in the same way in offspring mean may be higher or lower than expected. the offspring as in the parent... • The breeding value of the parent may be different from its genetic value Quantitative genetics Quantitative genetics Breeding values Breeding values a Example: To increase milk yield, dairy farmers estimate the breeding value of bulls from the average dairy Population mean Breeding value production of the bulls daughters. of parent (A) Say that the daughters of a particular bull mated to several Actual genetic value cows produce 100 liters of milk per day, on average, in a of offspring (mean + a/2) herd with an average production of 75 liters. The breeding value of a genotype (A) is obtained by In terms of dairy production, adding twice the deviation of the mean of the offspring from the population mean ...what is the breeding value (A) of the bull? 125 liters • The quantity a represents the portion of a genotype’s ...what is the phenotypic value of the bull? (ouch) genetic value that gets passed on to offspring (sometimes ...the deviation from mean (a) is 50 litres called the genic deviation) Quantitative genetics Quantitative genetics Breeding values Breeding values Phenotypic value Example: Now say that a particular cow produces 100 liters of milk per day compared to an average of 75 liters. When mated with different bulls, her daughters produce Population mean Breeding value 80 liters of milk per day. Genetic value In terms of dairy production, ...what is the breeding value (A) of the cow? 85 liters What might cause differences between genetic value & breeding value? 100 liters ...what is the phenotypic value of the cow? • If alleles at some loci affect traits differently depending on ...what contributes to this difference? the rest of the genotype (Interactions) • Dominance (D) (interactions at the same locus) Environmental effects, interactions among genes, • Epistasis (I) (interactions at different loci) interactions between genes and the environment... Quantitative genetics Quantitative genetics Expanding the model Expanding the model Similarly, good interactions between alleles at different genes (= epistasis) are not faithfully transmitted: For example, if a parent is homozygous at a locus, it 5 6 7 8 9 6 4 A A A cannot transmit this status to its children, because only " " " " " ! " # $ " one allele is passed to the offspring. Mom Dad If B is recessive to b, a high fitness BB parent mated to a low fitness bb parent produces only Bb (low fitness) offspring. 6 4 7 A 9 ! " " $ " Dominance effects (D) Offspring Interaction/epistasis effects (I) Quantitative genetics Quantitative genetics Expanding the model Expanding the model The average effect of an allele accounts for the chance that P = G + E the allele is paired with any other genes currently found within the population (e.g., accounting for the chance that G = A + D + I it is found in a heterozygote or homozygote). The genetic value equals the breeding value (A) plus the interaction values (D and I) The breeding value of an individual (A) represents the But it is primarily the breeding value (A) that determines average effects of all of his/her alleles. the phenotype of an individual’s offspring. Quantitative genetics Quantitative genetics From individuals to populations From individuals to populations Quantitative genetics is particularly concerned with describing the variation within a population and with estimating the genetic component of this variation. The phenotypic variance (VP) measures the extent to which individuals within a population differ in a trait. Quantitative genetics Quantitative genetics From individuals to populations From individuals to populations The phenotypic variance within a population may be The genotypic variance can be due to genetic and/or further subdivided into environmental differences additive, dominance, and among individuals: interaction components: VP = VG + VE VG = VA + VD+ VI (Ignoring interactions between genes & environment) Quantitative genetics Quantitative genetics From individuals to populations Calculating variances !n (X X)2 V = i=1 i − X n 1 − The additive genetic variance Example: Milk yield in cows (pounds/day) Cow Yield (VA) equals the variance in 1 75 breeding values within a 2 88 (75 75)2 + (88 75)2... 3 52 VP = − − population and measures the 4 83 9 5 82 = 425.6 degree to which offspring 6 43 resemble their parents. 7 100 8 48 9 79 10 100 Mean 75 Quantitative genetics Quantitative genetics Calculating variances Breeding values a n 2 !i=1(Xi X) VX = − n 1 Population mean Breeding value Example: Milk yield in− cows (pounds/day) (75) of parent (A) Cow Yield Offspring yield A = (offspring - mean) x 2 + mean Actual genetic value 1 75 74.5 2 88 81.5 A = a + mean of offspring (mean + a/2) 3 52 65.5 4 83 79 83 = (79 - 75) x 2 + 75 5 82 78.5 6 43 66 7 100 84 93 = (84 - 75) x 2 + 75 8 48 64.5 9 79 79.5 10 100 78 Mean 75 Quantitative genetics Quantitative genetics Calculating variances Inheritance !n (X X)2 V = i=1 i − X n 1 Example: Milk yield in− cows (pounds/day) These variance components can be used to determine Cow Yield Offspring yield A 1 75 74.5 74 how much relatives should resemble one another.