1

Population Genetics 7:

Genetic Drift

( ) ( ) knk

kn

P −

⎟⎟⎠

⎞⎜⎜⎝

⎛= 2/12/1

( )!!!knk

nkn

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛ Combinations Formula: n is the number of flips k is the number of successes

Assume a fair coin with p = ½: •  If you sample many times the most likely single outcome = ½ heads. •  The overall most likely outcome ≠ ½ heads

k heads from n flips Probability k =5 from n = 10 0.246 k =6 from n = 10 0.205

Sampling error

2

Sampling error

The long term average value for pH is 0.5; let’s call that E(pH).

How do we improve our changes of getting something close E(pH)?

If we flipped the coin 1000 times: we get very close to E(pH) in a single try, but not exactly.

N flips pH <0.35 pH = 0.35-0.45 pH = 0.45-0.55 pH = 0.55-0.65 pH <0.65 variance

10 0.16 0.21 0.25 0.21 0.16 0.025 20 0.06 0.19 0.50 0.19 0.06 0.0125 50 0.002 0.16 0.68 0.16 0.002 0.005

A note about HWE.

Probability of 50:50 heads : tails = 0.2256

Probability of 50:50 heads : tails = 0.0796

N = 12 flips

N = 100 flips

3

Genetic drift

Consider a diploid population:

•  Ideal population: no sampling errors because infinite population size

•  Natural population: finite size and finite sample of gametes [sampling errors]

Example:

Let’s assume: A = p = 0.75; a = q = 0.25; N = 500

This generation: 200 individuals reproduce [400 gametes]

This is a binomial sampling problem:

The probability of getting p = 0.75 and q = 0.25 in next generation is:

P = 0.046

( ) ( )100300 25.075.0300400

⎟⎟⎠

⎞⎜⎜⎝

⎛=P

Draw4:6

Draw7:3

Draw8:2

Restock Restock Restock

Generation 0 Generation 1 Generation 2 Generation 3

white = 0.5 white = 0.4 white = 0.7 white = 0.8

Draw4:6

Draw7:3

Draw8:2

Restock Restock Restock

Generation 0 Generation 1 Generation 2 Generation 3

white = 0.5 white = 0.4 white = 0.7 white = 0.8

Genetic drift

Genetic drift is the accumulation of random sampling fluctuations in allele frequencies over generations.

4

N1

e

1N

Genetic drift

The magnitude of change in allele frequencies is inversely proportional to the sample size:

Ideal population with finite size and finite gamete sample per generation. See last slide for example

Remember that natural populations are less than ideal in many more ways!

In most natural populations the effective size (Ne) will be less than the census size.

The magnitude of drift in natural populations is:

Drift and inbreeding effects are not independent!

Ne = 100

Ne = 1000

Ne = 10000

Ne = 50000

Genetic drift

5

Genetic drift

•  rate to fixation [under drift] slows with increasing in Ne

•  ultimate fate is fixation or loss ( if f(A1) = 0.5, P(fixed) = 0.5 )

If we run this simulation long enough it will go to fixation or loss; it just takes much longer

Genetic drift

What is the fate (on average) of a new mutant?

e

1N

The probability of fixation of a new mutant is its frequency (p or q) in the population:

This is al low as it gets. The fate of most new mutations is LOSS due to drift.

WAA = 0.5; WAa = 0.5; Waa = 1:

•  ideal population: probability of fixation = 1

•  population with Ne = 50: probability of fixation ~ 0.25

Probability of fixation actually declines as Ne decreases!

6

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 250

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

0

0.2

0.4

0.6

0.8

1

1 3 5 7 9 11 13 15 17 19 21 23 25

Generation

Allele frequency

*

*

* = fixation

Genetic drift

Changes in allele frequency due to drift are unpredictable!

Note if we ran more generations, more popns would go to fixation

10 Independent populations; each started with p = q = 0.5

Ne = 50; generations = 50

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12 13

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 130 allele frequency 1 0 allele frequency 1

num

ber o

f pop

ulat

ions

initial distribution; t = 0 generations distribution after t = 50 generations

Genetic drift

7

Genetic drift

The effects of drift are cumulative over time.

The effects of drift are predictable as averaged over time and populations:

1.  loss of variation within populations

2.  gain in variation between populations

Let x be the amount of change in p and q in a population due to drift.

As we have seen the long term average, E(x), due to drift will be zero because changes in p and q are equally likely to be positive or negative.

Given E(x) = 0, what happens to heterozygosity? Does heterozygosity change at all?

Does genetic drift affects heterozygosity?

Let’s start with HW at generation t:

Ht = 2pq

The allele frequencies, p and q, will change from generation to generation by the amount x:

Ht+1 = 2(p + x)(q – x)

Ht+1 = 2pq + 2x(q – p) – 2x2

Although E(x) = 0, the expected value of x-squared, E(x2), is always positive.

E(2pq + 2x(q – p) – 2x2)

2pq – 2x2

Heterozygosity is expected reduced by genetic drift. Nice, eh?

8

Genetic drift and inbreeding are not independent

1.  Unequal numbers in successive generations

(approx.) 1...11111

321 ⎥⎥⎦

⎤

⎢⎢⎣

⎡++++=

ge NNNNgN

2. Different numbers of males and females

(approx.) 4

14

11

fme NNN+=

3. Variance in reproductive success (other than male verse female)

( ) 224

+

−=

k

ve V

NN

Bottlenecks and founder effects

Bottleneck: is a single, extraordinarily large, reduction in population size

1.  Change in allele frequencies, as compared with pre-bottleneck population

2.  Reduction in diversity

9

Bottlenecks and founder effects

Effective population size is dominated by historical lows and can be very much lower than current census size.

0

20,000

40,000

60,000

80,000

100,000

120,000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time

popu

latio

n ce

nsus

siz

e

Ave N

Ne

Population crash Population recovered to historical high

(approx.) 1...11111

321 ⎥⎥⎦

⎤

⎢⎢⎣

⎡++++=

ge NNNNgN

Two species that have suffered extreme bottlenecks due to commercial harvesting

Northern right whale

Poor population recovery

Northern elephant seal

Excellent population recovery

10

Parental population Dispersal event to a neighbouring island

Island 1

New population

Island 2

Polydactyly caused by the homozygous recessive disease Ellis-van Creveld syndrome

Other symptoms of this disease include dwarfisms, abnormalities of the nails and teeth, and a hole between the two upper chambers of the heart.

11

Picture wing Drosophila

Direction of colonization

Direction of archipelago growth

Keynotes: •  Genetic drift influences both allele frequency and genotype frequency. •  Drift decreases diversity within populations and increases diversity between populations. •  Under genetic drift, the rate to fixation is determined by Ne and the probability of fixation by p. •  In specific cases the outcome of genetic drift is unpredictable. •  The effects of drift are predictable as an average over populations. •  Because drift reduces genetic variation in populations, a population’s ability to evolve in response to

new selective pressures might be reduced (remember Trudy MacKay’s experiments). Alternatively, some believe that drift could actually increase the rate of speciation (e.g., Hawaiian Drosophila).

•  Because the effect of drift is inversely proportional to the effective population size, its affects are

particularly important in rare and endangered species. •  Founder effects may play an important role in some speciation events