# Lecture 45: Population Genetics III: Natural Selection

(version 8 November 2005)

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## Fitness

The fitness of an individual is the number of offspring it leaves. There are several components to the total fitness

• Viability -- whether or not an individual survives up to a given point
• Mating --- how many mates. This is especially an issue in males, which show a much higher variance of mating success than females
• Most female mate at least once
• Many males never mate, while some mate multiple times
• Fertility --- given an individual survives, how many offspring does it leave.

In population genetics, we look for differences in the fitness of different genotypes at a particular locus (or set of loci).

• For example, do AA individuals have (on average) more offspring than (say) aa individuals?
• We denote the expected fitness of a particular genotype (say Aa) by WAa
• If WAa = 12, Aa individuals leave (on average) 12 offspring.

In quantitative genetics, we look for differences in the fitness of different characters ( phenotypes).

• For example, do taller individuals have more offspring than shorter individuals.
• We denote the expected fitness of a particular character value z by W(z)
• If z = height, then W(60) = 2.5 means that individuals who are 60 inches have, on average, 2.5 offspring.
The mean population fitness , denoted Wbar (W with a bar over it), is the expected fitness of a randomly-drawn individual.

In predicting the response to selection, relative fitnesses are all that we need. Here, we can set any particular fitness to one and consider the fitness of other genotypes/phenotypes relative to our standard.

## Selection

Selection occurs when different classes of individuals have (on average) different fitnesses.

There is a response to selection if there are transmittable genetic differences between these groups.

Types of selection

• Viability selection: differences in the mean survivorship of different groups

• Fertility selection: differences in the mean fertility of different groups

• Fertility and viability selection together are often simply called natural selection to contrast these with:

• Sexual selection: differences in the mean ability to obtain mates for different groups

## Selection on a single locus with two alleles

Consider a locus with two alleles, A and a, and let p = freq(A) . What is the change in p after a single generation of selection?

We can also express this more compactly using marginal fitnesses . Let

• WA = p WAA + (1-p) WAa
• Wa = p WAa + (1-p) Waa
The mean fitness is simply the average of the marginal fitnesses,
• Wbar = p WA + (1-p) Wa
Note that
• The allele frequency following selection is: p' = p WA / Wbar
• The net change in the allele frequency is: Delta p = p (WA - Wbar) / Wbar

## Example: Directional selection

Consider a locus with two alleles, with genotype fitnesses

• Waa = 1, WAa = 2, WAA = 2.5

If p = freq(A) = 1/4, what is the change in the frequency of allele A following one generation of selection?

• WA = freq(A) WAA + freq(a) WAa = (1/4)*2.5 + (3/4)*2 = 2.215
• Wa = freq(A) WAa + freq(a) Waa = (1/4)*2 + (3/4)*1 = 1.25
• Wbar = freq(A) WA + freq(a) W a= (1/4)*2.215 + (3/4)*1.25 =1.491 Hence, the change in the frequency of allele A is
• Delta p = Freq(A) *( WA - Wbar)/ Wbar
• Delta p = 0.25*(2.215-1.491)/1.491 = 0.121
So that after one generation of selection, the frequency of A increases to 0.25 + 0.121 = 0.37

This is an example of directional selection where one homozygote genotype has the highest fitness. Here, the population becomes fixed for that allele --- its frequency goes to one.

## Example: Selection against a recessive allele

The fitnesses here are WAA = 1, WAa = 1, Waa = 1-s

Here allele A is ultimately fixed (if s > 0) or lost (if s < 0), but selection is very slow, especially as the allele become rarer. For example, suppose we have a recessive lethal, so that s = 1, so that Waa = 0, and we start at freq(A) = 0.1

 Change in A number of generations required change in freq(aa) 10% to 50% 3 generations 81% to 25% 50% to 90% 10 generations 25% to 1% 90% to 99% 93 generations 1% to 0.01% 99% to 99.9% 902 generations 0.01% to 0.0001%

Why does selection slow down so much as freq(a) becomes small?

Key: Only the recessive homozygote is selected against, with freq(aa) = freq(a)*freq(a) << 1

## Example: Overdominant Selection

Here, the heterozygote has the highest fitness:

• WAA = 1-t, WAa = 1, Waa = 1-s
Here, if A is rare, it increases, and if allele a is rare, it also increases.

The net result (instead of fixation) is a selective equilibrium, where the allele frequencies converge to an equilibrium frequency of

• Freq(A) = s/(s+t)
Here, selection maintains both alleles, rather than removing all but one.

At the equilibrium point, the marginal fitnesses of both alleles are equal (homework problem).

Note that here, at equilibrium, the mean fitness is at its maximum value.

This is a stable equilibrium point. If we deviate slightly from it, the allele frequencies return to the equilibrium frequency.

## Example: Underdominant Selection

Suppose the heterozygote has the lowest fitness.

• One example is chromosome inversions, where the homozygotes have equal fitness, but the heterozygote shows reduced fitness
What happens?

The result is an unstable equilibrium , wherein if the allele frequency starts below the equilibrium point, it is lost, while the allele is fixed if its frequency starts above the equilibrium point.

Here at equilibrium the mean fitness is at its minimum value. unstable equilibrium point. If we deviate slightly from it, the allele frequencies go to either zero or one..

## Summary of one-locus fitness models

 Fitnesses Outcome WAa > WAA, Waa Stable polymorphic equilibrium with A and a WAa < WAA = Waa Unstable equilibrium, which allele fixed depends on the initial frequencies WAA > WAa , Waa Allele A fixed WAA = WAa > Waa Allele A fixed

## Key points for selection in an infinite population

• Unmeasurably small selection coefficients can cause substitutions (directional selection) or can maintain a polymorphism (overdominant selection)

• Different types of selection are required to account for allelic substitutions and the maintenance of polymorphisms.

• Thus, effects that cannot be measured in the lab are nonetheless evolutionarily critical

## Multiple alleles at one locus

By using marginal fitnesses, the above results easily extend to multiple alleles at one locus. Let pi be the frequency of allele i.

As above, the mean population fitness Wbar is simply the average of the marginal fitnesses, so that with k alleles,

### Wbar = freq(A1)WA1 + freq(A2)WA2 + ... + freq(Ak)WAk

Hence, those alleles whose marginal fitness exceed the mean fitness will increase, while those alleles whose marginal fitness is less than the mean fitness will decrease

Note that the marginal fitness changes as the frequencies of allele change (i.e. they are not constant), as they depend on the distribution of genotypes in the population.

At an equilibrium point, the mean fitness of all segregating alleles is equal.

## Selection on two or more loci

When selection acts on only a single locus, then generally mean population fitness increases each generation.

This is often not the case when selection acts on multiple loci. Why? Linkage disequilibrium

• If the selection-recombination equilibrium value for linkage disequilibrium is non-zero, then the mean fitness at the equilibrium value is not at its maximum value.

The analysis of even two-locus selection models is extremely complicated, and we will not cover this further.

## Example: Decrease in mean fitness

Suppose the genotype AaBb has fitness 1.1, while all other genotypes have fitness 1.

If we cross AABB and aabb parents, the F1 are all AaBb, and hence the mean population fitness is 1.1. However, each generation the mean population fitness declines each generation to an equilibrium value of 1.025 (if the two loci are unlinked).

## Interactions of Drift and Selection

In a finite population,

• an allele favored by selection can still become lost
• an allele selected against can still become fixed.
How often do these respective events occur (i.e., when does drift overpower the effects of selection and vise-versa?)

## Kimura's expression

Motto Kimura (1957, 1964) showed that for an allele with the simple fitnesses of 1: 1+ s : 1+2s for the genotypes aa: Aa: AA, that

the probability of fixation, U(p), that allele A is fixed given it starts at allele frequency p, is given by

Of greatest interest is the probability that an allele introduced as a single copy, so that p = 1/(2N). Here, Kimura's expression simplifies

Key points:

• Drift dominates selection when 4 N | s | << 1, as here U(p) = p.

• Even a selectively-favored allele has a low chance (2s) of becoming fixed
Example Consider an allele with s = 0.01 in three different populations:
• For N = 10
• p = 1/20 = 0.05, while U(p) = 0.06
• For N = 100
• p = 1/200 = 0.005, while U(p) = 0.02
• For N = 1000
• p = 1/20 = 0.0005, while U(p) = 0.02