Quantitative Genetics

Quantitative Traits• Mendel worked with traits that were all discrete, either/or traits: yellow or

green, round or wrinkled, etc. Different alleles gave clearly distinguishable phenotypes.

• However, many traits don’t fall into discrete categories: height, for example, or yield of corn per acre. These are “quantitative traits”.

• The manipulation of quantitative traits has allowed major increases in crop yield during the past 90 years. This is an important part of why today famine is rare, a product of political instability rather than a real shortage of food. Until very recently, crop improvement through quantitative genetics was the most profitable aspect of genetics.

• Early in the history of genetics is was argued that quantitative traits worked through a genetic system quite different from Mendelian genetics. This idea has been disproved, and the theory of quantitative genetics is based on Mendelian principles.

Types of Quantitative Trait• In general, the distribution of

quantitative traits values in a population follows the normal distribution (also known as Gaussian distribution or bell curve). These curves are characterized by the mean (mid-point) and by the variance (width). Often standard deviation, the square root of variance, is used as a measure of the curve’s width.

• 1. continuous trait: can take on any value: height, for example.

• 2. countable (meristic) can take on integer values only: number of bristles, for example.

• 3. threshold trait: has an underlying quantitative distribution, but the trait only appears only if a threshold is crossed.

Punchline and Basic Questions

• The basic tenet of quantitative genetics: the variation seen in quantitative traits is due to a combination of many genes each contributing a small amount, plus environmental factors.

• Or: phenotype = genetics plus environment.

• Basic questions (plus answers):– 1. What is the genetic basis of quantitative traits? (they are

caused by normal genes following Mendel’s rules).– 2. How can we separate the effects of genetics from the effects

of the environment? (by inbreeding to eliminate genetic variation).

– 3. How can we predict and control the outcome of a cross? (by artificial selection).

Quantitative Traits are Caused by Mendelian Genes

• In 1909 Herman Nilsson-Ehle from Sweden did a series of experiments with kernel color in wheat.

• Wheat is a hexaploid, the result of 3 different species producing a stable hybrid, an allopolyploid. There are thus 3 similar but slightly different genomes contained in the wheat genome, called A, B, and D.

• Each genome has a single gene that affects kernel color, and each of these loci has a red allele and a white allele. We will call the red alleles A, B, and D, and the white alleles a, b, and d.

• Inheritance of these alleles is partially dominant, or “additive”. The amount of red pigment in the kernel is proportional to the number of red alleles present, from 0 to 6.

Wheat Kernel Color• The cross: AA BB DD x aa bb dd. Red x white.• F1: Aa Bb Dd phenotype: pink, intermediate between the parents. Now self

these.• F2: alleles follow a binomial distribution:

– 1/64 have all 6 red alleles = red– 6/64 have 5 red + 1 white = light red– 10/64 have 4 red + 2 white = dark pink– 15/64 have 3 red + 3 white = pink– 10/64 have 2 red + 4 white = light pink– 6/64 have 1 red + 5 white = very pale pink– 1/64 have all 6 white = white

• Add a bit of environmental variation and human inability to distinguish similar shades: you get a quantitative distribution.

• This demonstrates that a simple Mendelian system: 3 genes, 2 alleles each, partial dominance--can lead to a quantitative trait.

More Wheat Kernel Color

Separating Genetics from Environment

• Three experiments by Wilhelm Johannsen, from Denmark, using the common bean (Phaseolus vulgaris). Johannsen coined the words “genotype” and “phenotype”.

• First Johannsen experiment: he weighed a group of beans, then grew them up and weighed their progeny (after selfing them).

Johannsen’s First Experiment

The relation between the seed weight of the parental generation and the filial generation in a variety of brown beans.

The figures in the table represent the filial numbers of beans of the different weight categories

weight of the filial beans

weight categories weight of the parental beans

10 20 30 40 50 60 70 80 90 average

20 - 1 15 90 63 11 - - - 43,8

30 - 15 85 322 310 91 2 - - 44,5

40 5 17 175 776 956 283 24 3 - 44,2

50 - 4 57 305 521 196 51 4 - 48,9

60 - 1 23 130 230 168 46 15 2 51,9

70 - - 5 53 175 180 64 15 2 56,0

total 5 38 170 1676 2255 928 187 33 2 47,92

according to W. JOHANNSEN; 1903, 1926

Interpretation

• In general, heavy parents gave heavy offspring and light parents gave light offspring. That is, there is a significant correlation between parent and offspring weights.

• However, there is also a considerable variation among the offspring weights. This is due to variations in both genetics and environment.

• Most offspring of extreme parents (very heavy or very light) are more average than their parents. This is a phenomenon called “regression to the mean”. Extreme members of a population benefit from very lucky environmental conditions, which can’t be inherited.

Johannsen’s Second Experiment

• Johannsen then worked on separating environmental effects from genetic effects. He did this by inbreeding the beans for 10 generations.

• Inbreeding means doing the closest possible cross, in this case, selfing them.

• Half if the remaining heterozygosity (percentage of heterozygotes) disappears for each generation of selfing. This is because when a Aa heterozygote is selfed, 1/4 of the offspring are AA, 1/4 are aa, and 1/2 are Aa. Selfing the AA and aa offspring gives only homozygous offspring forever. Thus is each generation, 1/2 the offspring become homozygotes and all their offspring stay homozygotes.

• After 10 generations of selfing, the percentage of heterozygotes is less than 1/1000 of the original level.

Results• Johannsen created 19 inbred lines.

The inbred lines had some variation, but less than the original random-bred population. The remaining variation was due to environmental variations.

• The mean weight of each line was different, but it was stable across generations. The reason is that the lines are genetically different from each other, but they are genetically (more or less) identical. The variance was also stable between generations.

Johannsen’s Third Experiment• Johannsen then started to work

out the basis of artificial selection: how to improve a species as efficiently as possible.

• Start with a random-bred population. Take the best ones to be parents of the next generation. The next generation has a mean that is shifted in the desired direction .

• This procedure doesn’t work for inbred populations, because there is no genetic variation to inherit. The next generation’s mean is the same as the previous generation’s mean despite having selected the best parents.

Selection• Edward M. East from the United States worked out

the formal basis for modern artificial selection, following the work of George Shull on maize. East worked on both maize and tobacco.

• East measured the length of the tobacco corolla (the straight part of the flower). He crossed 2 inbred lines with different lengths, then selfed the F1 to get and F2, then selfed the F2’s to get a series of F3 lines.

• The variation in the plants can be observed in the width of the distribution curves. Environmental variation is constant among all plants.

• Genetic variation is minimal in both the inbred parental lines and in the F1’s. The F1’s are heterozygous, but they are genetically uniform, because all of their parents were homozygous. The F2 displays the maximum variation: for every gene, all possible genotypes (AA, Aa, and aa) are present in the population. The F3’s show less variation than the F2’s, as inbreeding starts to eliminate heterozygosity.

Heterosis• Both the parental lines and the F1’s are

genetically uniform. However, the parental lines are relatively small and weak, a phenomenon called “inbreeding depression”: Too much homozygosity leads to small, sickly and weak organisms, at least among organisms that usually breed with others instead of self-pollinating.

• In contrast, the F1 hybrids are large, healthy and strong. This phenomenon is called “heterosis” or “hybrid vigor”.

• The corn planted in the US and other developed countries in nearly all F1 hybrid seed, because it produces high yielding, healthy plants (due to heterosis) and it is genetically uniform (and thus matures at the same time with ears in the same position on every plant).

Mathematical Basis of Quantitative Genetics

• Recall the basic premise of quantitative genetics: phenotype = genetics plus environment.

• In fact we are looking at variation in the traits, which is measured by the width of the Gaussian distribution curve. This width is the variance (or its square root, the standard deviation).

• Variance is a useful property, because variances from different sources can be added together to get total variance.

• However, the units of variance are the squares of the units used to measure the trait. Thus, if length in centimeters was measured, the variances of the length are in cm2. This is why standard deviation is usually reported: length ± s.d. --because standard deviation is in the same units as the original measurement. Standard deviations from different sources are not additive.

• Quantitative traits can thus be expressed as: VT = VG + VE where VT = total variance, VG - variance due to genetics, and VE =

variance due to environmental (non-inherited) causes.• This equation is often written with an additional covariance term: the degree

to which genetic and environmental variance depend on each other. We are just going to assume this term equals zero in our discussions.

Heritability

• One property of interest is “heritability”, the proportion of a trait’s variation that is due to genetics (with the rest of it due to “environmental” factors). This seems like a simple concept, but it is loaded with problems.

• The broad-sense heritability, symbolized as H (sometimes H2 to indicate that the units of variance are squared). H is a simple translation of the statement from above into mathematics:

H = VG / VT • This measure, the broad-sense heritability, is fairly easy

to measure, especially in human populations where identical twins are available. However, different studies show wide variations in H values for the same traits, and plant breeders have found that it doesn’t accurately reflect the results of selection experiments. Thus, H is generally only used in social science work.

Additive vs. Dominance Genetic Variance

• The biggest problem with broad sense heritability comes from lumping all genetic phenomena into a single Vg factor. Paradoxically, not all variation due to genetic differences can be directly inherited by an offspring from the parents.

• Genetic variance can be split into 2 main components, additive genetic variance (VA) and dominance genetic variance (VD).

VG = VA + VD

• Additive variance is the variance in a trait that is due to the effects of each individual allele being added together, without any interactions with other alleles or genes.

• Dominance variance is the variance that is due to interactions between alleles: synergy, effects due to two alleles interacting to make the trait greater (or lesser) than the sum of the two alleles acting alone. We are using dominance variance to include both interactions between alleles of the same gene and interactions between difference genes, which is sometimes a separate component called epistasis variance.

• The important point: dominance variance is not directly inherited from parent to offspring. It is due to the interaction of genes from both parents within the individual, and of course only one allele is passed from each parent to the offspring.

Narrow Sense Heritability• For a practical breeder,

dominance variance can’t be predicted, and it doesn’t affect the mean or variance of the offspring of a selection cross in a systematic fashion. Thus, only additive genetic variance is useful. Breeders and other scientists use “narrow sense heritability”, h, as a measure of heritability.

h = VA / VT

• Narrow sense heritability can also be calculated directly from breeding experiments. For this reason it is also called “realized heritability”.

Heritability in a Selection Experiment

• There are 3 easily measured parameters in a selection experiment: the mean of the original random-bred population, the mean of the individuals selected to be the parents, and the mean of the next generation. These factors are related by the narrow sense heritability:

• The denominator is sometimes called the “selection differential”, the difference between the total population and the individuals selected to be parents of the next generation. The numerator is sometimes called the “selection response”, the difference between the offspring and the original population, the amount the population shifted due to the selection.

meanoriginalmeanparent

meanoriginalmeangenerationnexth _____

Example

• In Drosophila, the mean number of bristles on the thorax (top surface only) is 6.4.

• From this population, a group was chosen which had an average bristle number of 7.2.

• The offspring of the chosen group had an average of 6.6 bristles.

• h = (next gen - original) / (selected - original)• h = (6.6 - 6.4) / (7.2 - 6.4)• h = 0.2 / 0.8• h = 0.25

Realized Heritability• The value of h as measured by selection

experiments is remarkably constant over many generations, and selection can be continued for a very long time without apparently running out of genetic variation.

• In one experiment using Tribolium (flour beetles), the original population had a mean weight of 2.4 mg, with a range of 1.8 to 3.0 mg. After 125 generations of selection, the mean weight was 5.1 mg, more than twice the original weight and far outside the original range.

• Similar experiments at the University of Illinois on maize for high protein and high oil content have shown consistent improvement for more than 100 generations (i.e. 100 years, since 1896).

Summary of Equations

• You will need to know these equations, and then demonstrate them in solving problems.

• VT = VG + VE

• for an INBRED population, VT = VE, because VG = 0

• H = VG / VT

• VG = VA + VD

• h = VA / VT

• h = (next gen - original) / (selected - original)• Note that h can be calculated from either of the 2

equations.

Example Problem

• In a quest to make bigger frogs, scientists started with a random bred population of frogs with an average weight of 500 g. They chose a group with average weight 600 g to be the parents of the next generation. A few other facts: VE = 1340, VA = 870, VD = 410.

• What is the genetic variance? VG = VA + VD = 1280• What is the total variance? VT = VG + VE = 2620• What is the broad sense heritability? H = VG / VT = 0.49• What is the narrow sense heritability? h = VA / VT = 0.33• What is the mean weight of the next generation?

h = (next gen - original) / (selected - original) 0.33 = (next_gen - 500) / (600 - 500) = 533 g