Population genetics and Hardy-Weinberg equilibrium

Mendelian-Darwinian Synthesis-Population Genetics

• Although Mendel’s and Darwin’s work were published within 5 years of each other, a synthesis of their ideas was not truly met until 1930’s

• Recognition that the relative abundance of traits in a population is tied to the relative abundance of alleles that influence them

• Under what circumstances will the relative abundance of alleles change within a population (i.e. the population evolves)?

Population genetics integrates Darwin’s evolution by natural selection with Mendelian genetics

Evolution is change in allele frequency across generations

Population genetics begins with a model of what happens to allele and genotype frequencies in an idealized population

A A A AA A

Suppose that in a population there are two alleles, A and A at a locus of interest. There are three possible genotypes.

Homozygous, A Heterozygous Homozygous, A

A A

A A

A A

A A A A

AAAA

AAAA A

A

A

A

A

A

A

A

A

A

A

AA A

A A

A A

If gametes join at random…

a will meet with a

half the time , and with a

the other half the time.

A

A

A

Similarly, a will join with the two egg types at a 50:50 ratio.

A

A

A

A

A

A

A

A

A

A

A

AA

A

AA

A

A

A

A

A

A

A

A

A

Half of all eggs have A and half have A

Half of all sperm have A and half have A

AA

AA

AA

AA

AA

AA

AA

AA

AA

AA

AAAA

Zygotes

AA

AA

AA

AA

AA AA

AA

AA

AA

AA

AA

AA

Homozygous, Afreq. = 0.25 = p2

Heterozygous freq = 0.5 = 2pq

p = freq. of the A allele = 0.5q = freq. of the A allele = 0.5

Homozygous, Afreq. = 0.25 = q2

p2 + 2pq + q2 = 1

Regardless of allele frequencies - no matter what the values of p and q - genotype frequencies will go to, and remain at…

Hardy-Weinberg equilibrium equation

How are different alleles inherited? - Law of Segregation

each diploid individual carries 2 non-blending copies of each gene

• each gamete (sperm/egg) receives only one of these genes

• each gene is segregated randomly – there is no way of knowing which copy of the gene a specific gamete will receive

• different forms (alleles) of the gene are thus also segregated randomly amongst the gametes

How are allele frequencies and genotypes related in a

population?

For a simple 2 allele (A1 and A2) locus, possible genotypes are A1 A1, A1 A2 and A2 A2

If we know the relative frequencies of A1 and A2, we can predict the relative frequencies of each genotype

In a hypothetical 2 allele closed population

frequency of A1 in gene pool = p

frequency of A2 in gene pool = q

Since there are only 2 alleles in the population:

p + q = 1

Sperm

A1 A2

A1

Eggs

A2

A1A1

A1A2

A1A2

A2A2

Sperm

A1 A2

A1

Eggs

A2

A1A1

A1A2

A1A2

A2A2

Sperm

A1 A2

A1

Eggs

A2

A1A1

A1A2

A1A2

A2A2

Genotypic Outcome Probabilities

A1A1 Homozygotes = p x p = p2

A1A2 Heterozygotes = (p x q) + (p x q) = 2pq

A2A2 Homozygotes = q x q = q2

p2 + 2pq + q2 = 1

Yule’s Numerical Example:The Simplest Case

•If the frequency of each of 2 alleles in the population is exactly equal, the frequency of each allele = 0.5

•i.e. A1 = 0.5, A2 = 0.5

•Since any 2 gametes in a randomly mixed “gene lottery” will have an equal chance of being picked, the probability of a sperm having the A1 allele = 0.5

Conclusion 1: allele frequencies in a population will not change, generation after generation

Conclusion 2: if allele frequencies are given by p & q, the genotype frequencies are p2, 2pq, q2

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

So what is the value of this null model with entirely unrealistic

assumptions?

1) We can quantify what will happen if there is selection on an allele

2) Likewise if there are mutations3) etc.

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

When individuals with some genotypes survive at higher rates than individuals with other genotypes, allele frequencies can change from one generation to the next.

ie. natural selection causes evolution

Violation of no-selection assumption violatesConclusion 1: allele frequencies in a population will not change, generation after generation

Persistent selection can cause substantial changes in allele frequencies over time

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

Mutation can cause appreciable changes in allele frequencies over very long periods of time

Mutation is a weak force of evolution

Nonetheless it provides the raw material upon which natural selection acts

Mutation - selection balance

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

Migration is a potent force in evolution

Migration is a potent force in evolution

Migration is most important in preventing populations from diverging

Violation of no-migration assumption violates Conclusion 2: if allele frequencies are given by p & q, the genotype frequencies are p2, 2pq, q2

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

In populations of finite size, chance events - in the form of sampling error in drawing gametes from the gene pool - can cause evolution

Selection is differential reproductive success that happens for a reason; genetic drift is differential reproductive success that just happens

Genetic drift is most important in small populations

GenerationGeneration

Ave

rag

e h

eter

ozy

go

sity

Fre

qu

ency

of

alle

le A

1Popln = 4

Popln = 400

Popln = 40

As alleles drift to fixation or loss, the frequency of heterozygotes in the population declines

Violation of no-drift assumption violates both Conclusion 1: allele frequencies in a population will not change, generation after generation

Conclusion 2: if allele frequencies are given by p & q, the genotype frequencies are p2, 2pq, q2

5 Assumptions for H-W Principle

1) no selection

2) no mutation

3) no migration to or from population

4) no random events that cause some individuals to pass on more of their genes than others

5) random mating

Coral releasing gametes into the water

Violation of random mating assumption violates Conclusion 2: if allele frequencies are given by p & q, the genotype frequencies are p2, 2pq, q2

Inbreeding:decreases the frequency of heterozygotes and increases the frequency homozygotes.

Inbreeding can lead to “Inbreeding Depression”

• Most mutations are deleterious when homozygous, but not when heterozygous

• Close relatives are likely to have inherited the same deleterious mutations from their ancestors, and carry them in the heterozygous state.

• When close relatives mate, they produce homozygotes for the mutation.

PP PpPp pp

PP Pp

PP PpPP Pp

Inbreeding increases the chance that deleterious homozygotes are produced

Assortative Mating by Height in Humans

155

160

165

170

175

180

185

190

195

200

135 140 145 150 155 160 165 170 175 180

Mother's Height

Fa

the

r's

He

igh

t = 0.49

How do you know if a population is in H-W?

• Look at observed genotype frequencies in the population…

• From these, calculate allele frequencies..• From allele frequencies, calculate the genotype

frequencies predicted by H-W…• Compare observed genotype frequencies to

predicted

Wild flower population

AA, n = 44

Aa, n = 46

aa, n = 10