Lecture 6 of the Quantitative Genetics series. From fitness at one locus to the breeder's equation, correlated response, drift and hitchhiking.
Selection is differential reproduction and survival of individuals due to genetic differences between the individuals.
The question that selection was formulated to answer is why organisms are suited to their circumstances. Three historical answers frame the topic.
The decisive difference is that Darwin's mechanism requires heritable variation that already exists and does not require variation to arise because it is needed. This makes it a quantitative theory rather than a narrative account, and allows the remainder of this lecture to be developed algebraically.
All three are necessary. Remove the surplus and there is nothing to select among; remove the variation and there is nothing to choose; remove the genetic basis and the choice has no consequences for the next generation.
Artificial selection is the intentional preferential breeding of some individuals in a population in order to improve the population's genetic characteristics.
Natural selection is differential reproduction and survival due to genetic differences, where the differences are not caused by intentional human interference.
Selection is differential survival and reproduction, and it can act at any stage of a life cycle:
The stages occupy different positions in the life cycle, and the life cycle itself differs between taxa:
| Mammals | Flowering plants |
|---|---|
| Zygote | Zygote |
| Adult males and females | Sporophyte |
| Gametes (meiosis) | Spores (meiosis), then gametophyte |
| Mating | Pollination |
| Fertilisation, zygote formation | Fertilisation, zygote formation |
In plants there is a free-living haploid phase, so selection can act on the gametophyte directly, a stage mammals do not have.
Viability selection is due to differences in survival until reproduction. Focus on a single locus with two alleles \(A\) and \(a\) at frequencies \(p\) and \(q\).
Genotypes survive to reproduction with probabilities \(W_{AA}\), \(W_{Aa}\), \(W_{aa}\). Fitness is then just a quantitative trait, and it has a mean like any other:
Exactly as in the one-locus model, define the fitness of an allele as the average effect of that allele, plus the mean:
These are consistent with the population mean, as they must be:
Higher survival for an allele means higher representation in the next generation. Selection therefore induces a change in allele frequencies. The derivation follows.
\(\Delta p\) is zero whenever \(W_A=W_a\), that is, whenever the two alleles have the same average fitness. It is also zero when \(p=0\) or \(q=0\). Selection needs both a fitness difference and standing variation to produce change, which restates Darwin's postulates algebraically.
In breeding, "survival" means the same thing as "being selected for breeding". In general we assume no differences in reproduction among the selected animals, they are all bred equally. So the whole viability-selection apparatus transfers directly to artificial selection, with \(W\) reinterpreted as the probability of being chosen.
"The rate of increase in the average fitness of a population is equal to the genetic variance of fitness of that population."
Fisher, R. A., Annals of Eugenics (London), 11, 53 (1941)
The theorem follows from the result we just derived in a few lines.
The middle step uses a result established earlier: \(2pq\alpha^2\) is the additive genetic variance derived in the one-locus lecture, here applied to fitness.
Since \(\bar W>0\) and \(V_A(W)\ge 0\), we have \(\Delta\bar W\ge 0\). Under viability selection, mean fitness never decreases. It also tells you when change stops: when the additive genetic variance in fitness is exhausted.
A particularly simple and practically important scheme:
The selection differential \(S\) is the average phenotypic superiority of the selected individuals, in phenotypic units.
The selection intensity \(i\) is the same superiority expressed in units of phenotypic standard deviation:
If the trait is normally distributed, \(i\) depends only on the proportion selected:
B <- 0.2
t <- qnorm(0.2, lower.tail = FALSE) # truncation point
phi <- dnorm(t) # density at t
i <- phi / B # selection intensity
Vp <- 2.0
S <- i * sqrt(Vp) # selection differential
R gives \(i=1.39981\). Falconer and Mackay (p. 379) tabulate \(i=1.400\), so the computation checks out against the standard reference.
Two treatments have been developed separately: allele-frequency change under fitness differences, and selection on a phenotype. This section shows that they are equivalent.
The effect of selecting on a quantitative trait can be modelled at a single locus of small effect. Assume selection is on phenotype alone: we use no information from other traits or from relatives.
Two alleles, with genotypes differing by a small additive amount \(\delta\):
so on this scale the allele substitution effect is \(\beta_y=\delta\).
Under truncation selection with proportion \(B\) selected, an individual's probability of being selected rises with its phenotype. To first order, a genotype whose mean is higher by \(\delta\) has its selection probability raised by \(\delta\phi\), where \(\phi\) is the standard normal density at the truncation point:
| Genotype | \(aa\) | \(Aa\) | \(AA\) |
|---|---|---|---|
| Fitness | \(B\) | \(B+\delta\phi\) | \(B+2\delta\phi\) |
| \(\bar W\) | \(B+2p\delta\phi\) | ||
| \(W_a\) | \(B+p\delta\phi\) | ||
| \(W_A\) | \(B+(1+p)\delta\phi\) | ||
| \(\beta_W\) | \(\delta\phi\) | ||
Starting from nothing but allele-frequency change at a single locus of small effect, we have arrived at the breeder's equation. The two stories are one story. This also explains why the breeder's equation contains \(h^2\) and not \(H^2\): the derivation runs through \(2pq\beta_y^2=V_A\), the additive variance, because only average allele effects are transmitted.
The derivation above proceeded from the locus. The complementary derivation, from regression, generalises to breeding programmes. We assume non-overlapping generations and that all selected individuals breed equally.
Expressing the superiority in standard deviations, \(S=i\sigma_P\):
With pure phenotypic selection, this is the predicted gain per generation.
This simplest case generalises in two directions:
Genetic correlations describe the relationship between the breeding values for different traits. Consequently, genetic change in one trait induces change in another, whether we want it or not.
The correlated response in trait 2 from selection on trait 1 is
which requires the joint variance structure of the phenotype of trait 1 and the breeding values of both traits:
Note the symmetry with \(\Delta G=ih^2\sigma_P=i\,h\cdot h\,\sigma_P\). The direct response uses \(h_1\) twice; the correlated response replaces one \(h_1\) with \(h_2\) and multiplies by \(r_g\). If \(r_g\) is negative, selecting on trait 1 degrades trait 2, which is the classic dairy story of yield against fertility.
Genetic drift affects all loci, including those with effects on quantitative traits. Recall its consequences: alleles in a population become more related, genetic variation shrinks, and population means diverge.
With additive gene action and \(n\) individuals with additive genetic values \(g_{a,i}\):
The first term does not shrink with \(n\). However many animals you select, the shared drift component \(F_{ST}V_A\) remains, so the outcome of a selection scheme in a small population is less predictable, not merely slower.
Natural selection acts in all populations, including domestic animals, so the observed outcome is always a balance between artificial selection, natural selection and drift. Selection experiments are how that balance is studied.
Each requirement answers a specific alternative explanation: drift, chance, and environmental change over the years of the experiment.
Genes on a chromosome are physically linked and inherited together. When selection targets one gene, the surrounding region, including genes that were never selected, may also rise in frequency.
Hitchhiking is the process by which an allele may increase in frequency because it is physically linked to another gene that is the target of selection.
A selective sweep is the sweeping away of variation around a selected site following the fixation of a favourable allele.
The genomic signature is striking. Comparing homozygotes for the persistence allele with non-persistent homozygotes, the homozygous tracts surrounding the locus are much longer in the persistent group, on average by a factor of about 1000. That is a very long selective sweep: anything linked to the lactase persistence mutation is now frequent, including any recessive deleterious allele that happened to be on the same haplotype, and such recessives are now expressed more often simply because they are more common.
In regions subject to hitchhiking the effect can be much greater than simple models predict, and molecular studies confirm that the real effect often exceeds prediction. Note also that in the infinitesimal model \(V_A\) does not change due to selection, apart from the Bulmer effect, so hitchhiking is precisely one of the phenomena the infinitesimal model cannot capture.
Change in allele frequency. Selection acts at a locus with two alleles \(A\) and \(a\) at frequencies \(p=0.99\) and \(q=0.01\). The allele \(a\) is a recessive lethal, so the fitnesses are \(W_{AA}=1\), \(W_{Aa}=1\), \(W_{aa}=0\).
Recall from earlier lectures that \(V_A=2pq\alpha^2\) and that the allele substitution effect is \(\alpha=W_A-W_a\).
1. Mean fitness and \(\alpha\).
Allele fitnesses: \(W_A=pW_{AA}+qW_{Aa}=0.99(1)+0.01(1)=1\), and \(W_a=pW_{Aa}+qW_{aa}=0.99(1)+0.01(0)=0.99\).
2. Additive genetic variance in fitness.
3. Change in allele frequency.
Check via Fisher's theorem: \(\Delta\bar W=V_A/\bar W=1.98\times10^{-6}/0.9999\approx 1.98\times10^{-6}\), which is positive, as it must be.
Interpretation. Even against a lethal, progress is very slow once the allele is rare: \(\Delta p\approx 10^{-4}\) per generation. This corresponds to the result in the Foundations course: selection against a recessive is slow at low frequency because most copies are carried by heterozygotes. It is also why marker-assisted removal of recessive defects is more effective than phenotypic culling.
Intensity of selection.
1. \(t=\)qnorm(0.2, lower.tail=FALSE)\(=0.8416\);
\(\phi(t)=\)dnorm(0.8416)\(=0.2800\).
So the selected group is on average 1.98 phenotypic units above the population mean.
2. Python for the plot:
import numpy as np, matplotlib.pyplot as plt
from scipy.stats import norm
B = np.arange(0.01, 1.00, 0.01)
t = norm.ppf(1 - B)
i = norm.pdf(t) / B
plt.plot(B, i); plt.xlabel("Proportion selected (B)"); plt.ylabel("Selection intensity (i)")
plt.show()
Shape. \(i\) falls monotonically from very high values as \(B\to 0\) to zero at \(B=1\), and the curve is steep at small \(B\). Selecting 1% instead of 2% buys a large increase in \(i\); selecting 50% instead of 60% buys very little. Combined with \(\Delta F\propto i^2\) from the drift lecture, this is why extremely intense selection is a poor bargain: gain rises ever more slowly while inbreeding rises ever faster.
Correlated response. In Danish Holstein cattle, the heritability of milk protein percentage is \(h_p^2=0.60\) and of milk yield \(h_m^2=0.20\). The genetic correlation between them is \(r_g=-0.60\). Set the phenotypic variance of both traits to 1.
Predict the change in milk protein percentage that results from selecting the \(B=0.20\) of cows with the highest milk yield.
Use \(\Delta G_2=i\,r_g\,h_1h_2\,\sigma_{p2}\), where trait 1 is the trait selected on (milk yield) and trait 2 is the correlated trait (protein percentage).
From Exercise 6.2, \(B=0.20\) gives \(i=1.400\). The heritabilities in standard-deviation form are \(h_1=\sqrt{0.20}=0.4472\) (yield) and \(h_2=\sqrt{0.60}=0.7746\) (protein %), and \(\sigma_{p2}=1\).
So one generation of selecting the top 20% on milk yield is expected to reduce protein percentage by about 0.29 phenotypic standard deviations.
Interpretation. This is the principal practical implication of the section. The unfavourable genetic correlation means that selecting hard on yield degrades composition, and no amount of care in measuring yield avoids it. The only remedies are to select on an index that weights both traits, or to accept the loss. Historically, dairy programmes that selected on yield alone did exactly this, which is why balanced indices replaced them.
