The Theory of Natural Selection

Lecture 02

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Populations and population genetics

Populations are the fundamental unit of evolution, the smallest object that can evolve

Gene Frequency vs Genotype Frequency

Genotype Frequencies (two alleles)

Proportion of AA = P, proportion of Aa = Q, Proportion of aa = R

Gene Frequency

p = P + 1/2Q, q = 1/2Q + R

Deriving Hardy Weinberg

With three genotypes, there are 9 possible pairings when mating

The question is what are P', Q', and R', the genotype frequencies in the next generation

P' = P² + 1/2PQ + 1/2PQ + 1/4Q² = P² + PQ + 1/4Q²

P' = ( P + 1/2Q)(P + 1/2Q), - remember that p = P + 1/2Q

P' = (p)(p) = p²

Q' = 1/2PQ + PR + 1/2PQ + 1/2Q² + 1/2QR + PR + 1/2QR = 2PR + PQ + QR + 1/2Q²

Q' = 2(PR + 1/2QR + 1/2PQ + 1/4Q²) = 2(P + 1/2Q)(R + 1/2Q) = 2pq

R' is a mirror image of the reasoning behind P' = p²

 

Assumptions

diploid, sexually reproducing species, discrete generations

no mutation, no migration, no selection

random mating

large population size

Three alleles:

Males		Females
		A	A'	a
		p	q	r
A	p	p2 of (AA)	pq of (AA')	pr of (Aa)
A'	q	pq of (AA')	q2 of (A'A')	qr of (A'a)
a	r	pr of (Aa)	qr of (A'a)	r2 of (aa)

Modelling natural selection in a population

To start, make lots of assumptions:

One locus completely determines phenotype

Two alleles at that locus

Sexual species in which Hardy-Weinberg assumptions hold except for the assumption of no selection

Selection against the homozygous recessive phenotype

Discrete Generations

Selection operates via differences in mortality between birth and maturity with no differences in fertility

Fitnesses:

If p' is the gene frequency after selection, then

And the change in gene frequencies is (remember, the increase in p is matched by the decrease in q)

    and   

Rearranging the equation for delta p, we can calculate the selection coefficient if we know the change in p:

Fitness Estimation

Industrial melanism is the example in the book, and it presents two models for estimation of fitness:

Relative survival (or fecundity) of genotypes within a generation or over generations

must be able to track individuals (mark recapture for mobile organisms, mapping quadrats for sessile)

Change in gene frequencies over one or more generations

Mutation/Selection Equilibrium

When they operate in opposite directions, an equilibrium gene frequency is reached that depends on the mutation rate and selection coefficient

Assume that the fitness of three genotypes are such that the recessive homozygote is fittest (no difference between the homozygous dominant and heterozygote (keeping it simple):

Genotypes	AA	Aa	aa
Relative Fitnesses	1 - s	1 - s	1

Then selection removes A alleles and mutation produces A alleles and a dynamic equilibrium is reached when the removal and addition rates are equal (from here on, a * means an equilibrium rate or frequency) -

 

This can be simplified even more if one assumes that the selection coefficient is much larger than the mutation rate, so that m contributes very little to the denominator:

Heterozygous Advantage

Selection for heterozygote can only change the gene frequencies so much as mating re-generates the homozygotes

Highest proportion of heterozygotes when gene frequencies are equal, if both homozygotes are equally "unfit" (if there is no difference in the selection coefficients.  In this case, the equilibrium frequency of p = q = 0.5

What if the situation is not symmetric?  Here we have fitnesses:

Genotypes	AA	Aa	aa
Relative Fitnesses	1 - s	1	1 - t

Here, the rate of loss of each allele is its selection coefficient times its frequency.  An equilibrium is reached when the rate of loss of each allele is equal (assuming no mutation, etc.)

p*s = q*t

To solve for p, substitute for q :

p*s = (1 - p*)t

   (for q rearrange and get) 

This means that selection does not automatically favor one allele over another and that elimination is not an inevitability

Frequency-Dependent Selection

Predators that hunt according to a type III functional response can result in frequency dependence

Proportion of common morphs eaten greater than proportion of common  morphs in the population

Rare morphs ignored and selectively favored

Rare morphs increase until they become the common morphs and are then searched for and eaten

Other Violations of Hardy-Weinberg

Population Subdivision and the Wahlund Effect

Under H-W, subdivision produces an increase in the number of homozygotes without the action of any selection

Take a population, give it a p and q and calculate the H-W proportions

Divide the population into two or three equally large subdivisions

If p and q for subpopulations differs from p and q of total population , you  will see the Wahlund Effect

(Has to do with the average of squared numbers)

Migration will unify gene frequencies between subpopulations

Migration can also offset loss of alleles through natural selection.  If the fitnesses of geneotypes are:

Genotypes	AA	Aa	aa
Relative Fitnesses	1 - s	1 - s	1

It is possible that, the rate of arrival of new A alleles will balance the rate of loss of A alleles.  If so, the equilibrium gene frequency is

The frequency of the A allele in the migrants is pm and m is not the mutation rate, as above, but now is the proportion of migrants in the population each generation.

 

Last modified on January 24, 2008