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Quantitative Quantitative Genetics

Floral trait variation in What we’ve covered so far: Mimulus. Used a 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. 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 • In 1918, Fisher showed that a large number of Mendelian factors (genes) • The position of genes within the influencing a trait would cause a nearly • The fitness effects of particular 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 In quantitative genetics, the phenotypic value (P) of an frequency changes at specific loci individual (e.g. height) is attributed to the 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 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) • (D) (interactions at the same locus) Environmental effects, interactions among genes, • (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 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. 2 88 81.5 88 3 52 65.5 56 They also determine how much evolutionary change will 4 83 79 83 VP = 425.6 be accomplished when certain parents reproduce while 5 82 78.5 82 6 43 66 57 VA = 205.5 others do not. 7 100 84 93 8 48 64.5 54 9 79 79.5 84 10 100 78 81 Mean 75 75.2

Quantitative genetics Quantitative genetics Inheritance Inheritance Example Broad-sense (H2): The extent to which The roughrider strain of wheat is genetically uniform. variation in phenotype is caused by variation in genotype Therefore, all variation in yield among plants is due to (= V /V ). G P environmental differences among plants. H2 = 0 Broad-sense heritability will be 1 if all of the phenotypic But are only applicable in a certain place at a variation within a population is due to genotypic certain time. differences among individuals. Observing H2 = 0 in roughrider does not mean that wheat Broad-sense heritability will be 0 if all of the phenotypic yield would have no genetic component in other variation is caused by environmental differences. populations of wheat (though they would have to be genetically variable). Quantitative genetics Quantitative genetics Inheritance Inheritance

Narrow-sense heritability (h2): The extent to which Example variation in phenotype is caused by genes transmitted What is the narrow-sense heritability in the cow from parents (= V /V ). A P example? 2 h will be 1 only if there is no variation due to genetic V = 425.6 interactions (dominance or epistasis) or to environmental P differences. When h2 = 1, P = G = A. VA = 205.5 Narrow-sense heritability can be zero even if broad sense 205.5 heritability is not (but not vice-versa) because all the h2 = = 0.48 within a population may be due to 425.6 dominance or epistasis.

Quantitative genetics Quantitative genetics Inheritance Inheritance Example: human birth weight

Cause of variation % of total Genetic 18 2 H2= V /V H = 0.18 G P Additive 15 2 Non-additive 1 h = 0.15 =(VA + VD+ VI)/ VP Sex 2 Environmental 82 2 Maternal genotype 20 The majority of h = VA/VP Maternal environment 24 phenotypic variation is Age of mother 1 environmental in origin Parity (birth order) 7 Intangible 30 Quantitative genetics Quantitative genetics Relationship between parents and offspring Relationship between parents and offspring (We assume that mating is random and that epistasis and gene- by-environment interaction can be ignored) The between the phenotype of a parent and

its offspring equals Cov(O,P) = 1/2 VA More importantly, the regression of the phenotypic value of offspring on that of their parents equals

Cov(O,P)/VP = 1/2 VA /VP

Regression(offspring,parent) = 1/2 h2 h2 = 2 Regession(offspring,parent)

Quantitative genetics Quantitative genetics Other relationships Relationship between parents and offspring If we measure both parents: V Regression[mid-parent,offspring] = A = h2 VP h2 = 0.8 If we measure full siblings: 1/2V + 1/4V Corr[sibling,sibling] = A D VP The expected correlation between many other types of relatives has also been calculated Quantitative genetics Quantitative genetics Relationship between parents and offspring Sample heritabilities (From Falconer, 1989) h2 Man

2 Stature 0.65 h = 1 Serum immunoglobulin level 0.45 Cattle Adult body weight 0.65 % butterfat 0.40 Milk-yield 0.35 Pigs h2 = 0 Back-fat thickness 0.70 Weight gain per day 0.40 Litter size 0.05

Quantitative genetics Quantitative genetics The importance of heritability The importance of heritability

Narrow-sense heritability predicts how a trait changes in 2 the next generation when a subset of parents breed. R = h S Selection differential (S): The phenotypic mean of parents chosen to breed minus the population mean. Characters with a high heritability (e.g. back-fat thickness Response to selection (R): The phenotypic mean of in pigs) will respond rapidly to selection, whereas offspring of these parents minus the population mean. characters with low heritability (e.g. litter size in pigs) will respond slowly. The fundamental formula of quantitative genetics: R = h2S Quantitative genetics Quantitative genetics The importance of heritability Example (Clayton, Morris, and Robertson 1957) R = h2S The estimate of narrow-sense heritability of abdominal 0.1 S = 38 - 30 = 8 bristle number in Drosophila was 0.52 h2 = 0.7: 0.08 R = 0.7 x 8 = 5.4 Next generation mean = 35.4 Parental mean: 35.3 bristles y

c 2 n 0.06 h = 0.05: Among those allowed to breed (those selected), the mean e

u R = 0.05 x 8 = 0.4 q number of bristles was 40.6 e Next generation mean = 30.4

r 0.04 F The selection differential was S = 40.6 - 35.3 = 5.3 0.02 Predicted response to selection: R = 0.52 x 5.3 = 2.8 Phenotypic 10 20 30 40 50 60 Value Actual mean in next generation: 37.9 bristles Selected Actual response to selection: R = 37.9 - 35.3 = 2.6 Mean mean (30) (38)

Quantitative genetics Quantitative genetics Long-term response to selection Long-term response to selection R = h2S Often, however, heritability remains roughly the same Heritability is not a constant attribute of a population. over a number of Over time, the heritability of a trait will change as: generations. • allele frequencies change This can be seen if we plot response against the • disequilibria change cumulative selection • variance is reduced differential; if heritability remains constant, this gives a straight line. Quantitative genetics Quantitative genetics Limits to selection Long-term response to selection Eventually, the response to selection may reach a plateau Factors affecting total response before a plateau is reached

(1) The total response will be less when few individuals are chosen to breed, since less genetic variation is preserved among these individuals.

Quantitative genetics Quantitative genetics Long-term response to selection Long-term response to selection

Factors affecting total response Factors affecting total response before a plateau is reached before a plateau is reached

(2) The total response will be less when selection (3) The total response will be less if few loci contribute occurs rapidly because of genetic hitchhiking (some to the trait, since those few loci will go to fixation and alleles that act in the opposite direction may get since the array of possible combinations of alleles is dragged along and fix, especially when S is high). much more limited. Quantitative genetics Conclusion

By focusing on the mean and variance, quantitative genetics provides a summary description of the genetic and environmental basis of traits. Although an individual’s phenotype is a complex product of how its genes and environment interact, it is the additive effect of genes (A) that is passed from parents to offspring. Consequently, it is the narrow-sense heritability (h2 = VA/VP) that describes the resemblance between parents and offspring and the response to selection (R = h2 S).