course outline hwe: what happens when hardy- weinberg assumptions are met inheritance: multiple...

Course outline

HWE:What happens when Hardy-Weinberg assumptions are met

Inheritance:Multiple alleles in a population;Transmission of alleles in a family

Unit 1: HWE and Inheritance

Evolution:When violations in H-W assumptions cause changes in the genetic composition of a population

Population Structure:When violations in H-W assumptions cause changes in the distribution of alleles within/across populations

Unit 2: Evolution and Pop. Structure

(a.k.a. violations in H-W assumptions)

Course outline

Evolution:When violations in H-W assumptions cause changes in the genetic composition of a population

Population Structure:When violations in H-W assumptions cause changes in the distribution of alleles within/across populations

Unit 2: Evolution and Pop. Structure

(a.k.a. violations in H-W assumptions)

Wed. 2/4: genetic driftMon. 2/9: natural selectionWed. 2/11: mutation

Mon. 2/16: migrationWed. 2/18: assortative matingMon. 2/23: inbreeding

Genetic DriftFeb. 4, 2015

HUGEN 2022: Population Genetics

J. ShafferDept. Human GeneticsUniversity of Pittsburgh

Objectives

After this lecture you will need to be able to:

1. explain the qualitative effects of genetic drift on a population• founder effects• bottleneck effects• rare disease alleles

2. use Binomial distribution to calculate probabilities of having i alleles in the next generation

3. calculate:• effective population size• probability of allele going to fixation at some point in the future• approximate number of generations until allele fixation

The big picture: Evolution• Definition:

– change in the genetic composition (allele frequencies) of a population across successive generations

• Evolution vs. Hardy-Weinberg– the H-W Law tells us that if the assumptions are met, genotype and

allele frequencies do NOT change from one generation to the next– for evolution to occur, H-W assumptions must be violated– Which processes drive evolution?

• mutation• natural selection• random changes in allele frequency (due to population size)

–genetic drift

Hardy-Weinberg assumptions

• diploid organism• sexual reproduction• nonoverlapping generations• random mating• large population size• equal allele frequencies in the sexes• no migration• no mutation• no selection

Definition of Drift

Random changes in allele frequency by chance in finite populations.

Key points:

Particularly important for small populations.

Due to the random sampling of gametes.

Why does drift happen?

Cause: random sampling of alleles

“law of large numbers” predicts random sampling of alleles will have a small effect in large populations

however…in small populations, random sampling of alleles can

greatly affect allele frequencies in the next generation

Why does drift happen?Simple Scenario:

Population of N = 4 individuals (8 alleles)

4 A alleles and 4 a allelesP(A) = 0.5 P(a) = 0.5

HWE says that in the next generation we will have:

P(AA) = p2 P(Aa) = 2pq P(aa) = q2

P(AA) = 0.25 P(Aa) = 0.5 P(aa) = 0.25





P(AA) = p2 P(Aa) = 2pq P(aa) = q2

P(AA) = 0.25 P(Aa) = 0.5 P(aa) = 0.25P(A) = 0.5P(a) = 0.5





P(AA) = p2 P(Aa) = 2pq P(aa) = q2

P(AA) = 0.25 P(Aa) = 0.5 P(aa) = 0.25P(A) = 0.5P(a) = 0.5

Will that really happen?

What about allele frequencies in the third generation?


generation 0 A A A A a a a a1 2 3 4 5 6 7 8 each allele uniquely labeled

P(A) = 0.5


generation 0 A A A A a a a a1 2 3 4 5 6 7 8 each allele uniquely labeled

random sampling (with replacement) used my TI-83 calculator to randomly pick alleles 1-8

P(A) = 0.5


generation 0 A A A A a a a a

generation 1 A A A a A A a a

1 2 3 4 5 6 7 8 each allele uniquely labeled

random sampling (with replacement)

2 3 1 8 1 1 5 7

used my TI-83 calculator to randomly pick alleles 1-8

P(A) = 0.5

P(A) = 0.625




generation 2 a A A A A A A a



2 3 1 8 1 1 5 7


7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75





generation 3 A a a A A A a A



2 3 1 8 1 1 5 7


7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75

1 8 7 2 2 1 8 1

P(A) = 0.625

Wright-Fisher model

assumes two alleles: P(A)=p P(a)=qassumes non-overlapping generations

Binomial Distribution:

probability of exactly i A alleles in the next generation

where N = population size x! = x (x-1) (x-2) ….. 1 0! =1

(2N)!(2N – i)! i! = piq2N-i

Wright-Fisher model

assumes two alleles: P(A)=p P(a)=qassumes non-overlapping generations

Binomial Distribution:

probability of exactly i A alleles in the next generation

where N = population size x! = x (x-1) (x-2) ….. 1 0! =1

(2N)!(2N – i)! i! = piq2N-i

NOTE: formula is forthe A allele, with P(A)=p

Example: Two-Allele Model for Drift

Start with 2N = 8, 4A and 4a alleles

~27% chance that the allele frequency stays the same

8! 4! 4!

P (i A alleles in next generation )

What is the probability that the next generation has exactly i = 4 A alleles?

P (4 A alleles in next generation ) =

p = 0.5 q = 0.5

0.273

(2N)!(2N – i)! i!

= piq2N-i

~73% chance that the allele frequency changes!(in one generation)

Exam

ple

0.54 0.54 =

What happens in long term?





1 2 3 4 5 6 7 8


2 3 1 8 1 1 5 7

7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75

1 8 7 2 2 1 8 1

P(A) = 0.625






1 2 3 4 5 6 7 8


2 3 1 8 1 1 5 7

7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75

1 8 7 2 2 1 8 1

P(A) = 0.625

A a4 6

some alleles are lost






1 2 3 4 5 6 7 8


2 3 1 8 1 1 5 7

7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75

1 8 7 2 2 1 8 1

P(A) = 0.625

A a4 6


A a3 5






1 2 3 4 5 6 7 8


2 3 1 8 1 1 5 7

7 1 1 2 2 1 2 8

P(A) = 0.5

P(A) = 0.625

P(A) = 0.75

1 8 7 2 2 1 8 1

P(A) = 0.625

A a4 6


A a3 5

(none lost this gen.)


generation 3 A a a A A A a A1 8 7 2 2 1 8 1

P(A) = 0.625





P(A) = 0.625

generation 4 a a A A A A A A7 8 1 1 1 1 1 1

P(A) = 0.75



A 2



P(A) = 0.625


P(A) = 0.75



A 2

generation 5 A A A a A A A A1 1 1 7 1 1 1 1

P(A) = 0.875a

8



P(A) = 0.625


P(A) = 0.75



A 2

generation 5 A A A a A A A A1 1 1 7 1 1 1 1

P(A) = 0.875a

8

generation 6 A A A A A A A A1 1 1 1 1 1 1 1

P(A) = 1.0a

7



P(A) = 1.0



P(A) = 1.0


P(A) = 1.0

allele fixation



P(A) = 1.0


P(A) = 1.0

generation t A A A A A A A A1 1 1 1 1 1 1 1

P(A) = 1.0

allele fixation

note: allele fixation is because they are all A alleles;

Genetic Drift Simulations

• http://popgensimulator.com/

• If p0 = 0.5 …• What happens when N = 4? (i.e., 2N = 8)• N=25?• N=100?• N=1000?

• if p0 is …• p0 = 0.25?• p0 = 0.10?• p0 = 0.01?

http://popgensimulator.com/

http://popgensimulator.com/

Long-term results

Eventually (at time < infinity) one allele is fixed.

• It can be either allele.• With 2N = 8, it happens pretty fast, usually.• At any given point in time, the probability that A is the allele that

will become fixed in the future is the current allele frequency, pt

important slide

Long-term results

Eventually (at time < infinity) one allele is fixed.

• It can be either allele.• With 2N = 8, it happens pretty fast, usually.• At any given point in time, the probability that A is the allele that

will become fixed in the future is the current allele frequency, pt

Additional comments:

• if a bunch of separate populations all have the same starting allele frequency, p0… given drift, each population goes to fixation. We expect p0 populations to become fixed for A and

q0 populations to become fixed for a

• The expected (approximate) time, t, to fixation of A due to drift is:

important slide

tfixation = -4Ne(1-p0)ln(1-p0) p0

where Ne is the effective population size

A lot of variation around this estimate

Effective Population Size

How do you think this would affect our assumptions/calculations?

Example: Population of 1000 people, but only 1 male

Solution: Effective population size

Nf females and Nm males

Ne =

More complicated formulae exist for populations that are changing in size over time.

(not covered in this course)

Nm + Nf

4 Nm Nf

How does drift operate in real human populations?

Migration, environmental disasters/epidemics, social factors (religion)

Why important for humans?

Until recently (last 5000 yrs), most human populations were small - ergo, drift could occur

• Drift mostly comes into play when the population is genetically isolated.

New small isolated populations form recently due to:

How does drift operate in real human populations?

Bottleneck

Founder Effect

Genetic effects on a population started by a small group of individuals

Large population is reduced, then re-expands

As a result, alleles in the founder group become the alleles in the population

Ex. If 100 alleles emigrate to the desert, THAT IS the new population

Founder effect example

In a large population, q = 0.001 for a recessive disease. Call the disease allele “a.”

50 individuals join a religious cult and go off and form an isolated commune.

If one of those individuals carries the “a” allele, what’s the allele frequency in the new population?

How might this affect the new population going forward?

Why Founder Effects are Important

Because the founder effect occurs at every locus, there will be some loci with very different allele frequencies than those in the population from which the founders came.

Thought experiment:

- Genome consists of 1000 loci with disease alleles.- Disease allele at each locus has frequency q = 0.001.- Choose a new population of 100 alleles at each locus.

Results of one random example of choosing this new population:

of the 1000 loci of interest:- 900 loci: 0 copies of the disease allele in the new population (q = 0)- 95 loci: 1 copy of the disease allele in the new population (q = .01)- 4 loci: 2 copies of the disease allele in the new population (q = .02) - 1 locus: 3 copies of the disease allele in the new population (q = .03)

Take home message: Founder effect = new population has decreased risk for many genetic diseases but greatly increased risk for few genetic diseases

What happens after the founder effect?

(1) Genetic drift:

What happens in the first few generations?

(2) Other violations to H-W assumptions:• Inbreeding• Mutation• Natural selection

After we found a small population, what happens next?

Drift eliminates alleles (randomly)Remove a few founder alleles,

but increase others

(more on these in upcoming lectures)

Overall Result• Lots of small populations have genetic variation caused by founder effects and drift.• Different populations will have different “common” genetic diseases

(especially recessive diseases)

Summary

• Genetic drift– drift simulations

• effect of sample size• effect of starting allele frequency

– allele fixation– founder effect, bottleneck effect, etc.

• Calculations:

– binomial formula– effective population size• probability of allele going to fixation at some point in the

future• approximate number of generations until allele fixation

course outline hwe: what happens when hardy- weinberg assumptions are met inheritance: multiple...

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