Genetic variation and fitness

pp pq

qp qqq

p

qp

BA

B

A

Hardy Weinberg law

2 2 2( ) 2 1p q p pq q

AA AB BB SumAfter crossing p2 2pq q2 1Frequency of B 2pq / 2 q2 pq+q2

Assume a gene with two alleles A and B that occur with frequency p and q = 1-p.

2

2 2 2

( )

( 2 ) ( )

pq q q p qq

p pq q p q

What is the frequency after crossing?

According to the Hardy Weinberg law gene frequencies are constant.

How can evolution occur?

Assumptions of the Hardy Weinberg law

1. No mutations to generate new alleles

(no genetic variability)

2. Mating is random

3. The population is closed

4. The population is infinitively large

5. Individuals are equivalent

None of these assumptions is fully met in nature.

Thus, gene frequencies permanently change

Therefore, evolution must occur!

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Frequency p of allele A

Fre

qu

en

cy

z

2pq

ppqq

The frequency of heterozygotes is

highest at p = q = 1/2

Inbreeding

GM1 GF1 GF2A,B C,D G,H

Grandparents

Parents

Childrens

GM2E,F

M F

Ch

The probability that Ch gets allele C is 0.125.

What is the probability for a children to get a certain allele from their grandparents?

GM1 GF1 GF1A,B C,D C,D

GM2E,F

M F

Ch

The probability that Ch gets allele C is 0.25.

P(C)=0.25

P(C)=0.125

P(C)=0.25 P(C)=0.25P(C)=0

P(C)=0.25

GM1 GF1 GF1A,B C,C C,C

GM2E,F

M F

Ch

The probability that Ch gets allele C is 0.5.

P(C)=0.5 P(C)=0.5

P(C)=0.5

GF1 is already inbred The mean probability to get an allele X from one of the members of a lineage is called

the coefficient of inbreeding FX.

Sewall Wright defined this coefficient as

𝐹 𝑋=∑𝑖=1

𝑛

[( 12 )𝑘+1

(𝐹 𝐴+1)]n is the number of connecting links between the two parents of X through common ancestors and FA is

the coefficient of inbreeding of the common ancestor A.

Mutation rates

dpp

dt

dqq

dt

The change in gene frequency is assumed to be proportional to actual gene frequency

multiplied with the mutation rate.

0

0

t

t

p p e

q q e

M D M kD

Assume the number of mutation events M in a genome is proportional to the total amount of

the mutation inducing agent D, the dose

M kD

N N

Mutation rate m

The change of gene frequency follows an exponential function

Equilibrium conditions

(1 ) dp

p q p pdt

The change in p is the sum of forward and backward mutations

At equilibrium dp/dt = 0

(1 )p q p p

Under constant forward and backward

mutation rates p and q will achieve equilibrium frequencies.

Otherwise they will permanently change.

Constant immigration of individuals causes a permaent linear change in allele frequency

Nonrandom mating

If mating is totally random a population is said to be panmictic.

A special type of nonrandom mating is inbreeding.

Inbreeding results in the accumulations of homozygotes.

Inbreeding depression due to homozygosity in Italian marriages

1903-1907.

0 10 20 30 40

Notrelated

Secondcousins

3/2cousins

Firstcousins

De

gre

e o

f re

late

dn

ess

z

Percent offspring mortality (< 21 years))

Assortative mating describes a situation where breeding occurs among individual

with similar genetic structure.

The opposite is called disassortative mating.

Domingue et al. 2014, PNAS

Qua

ntile

of g

enet

ic

sim

ilarit

y of

pai

rsQuantile of cross sex genetic

similarity

American pairs have a slight (about 4.5% effect) affinity to partners of similar

genetic predisposition

Positive assortative mating increases the degree of inbreeding

Individuals are not equivalent

If individuals are not equivalent they have different numbers of progenies.

Selection sets in

Zygotes

AdultsParents

GametesChildren

Ontogenetic selection

Viability selection

Mating success

Gametic selection

Compatability selection

Five levels of natural selection

What is the unit of selection?

Selection changes frequencies of genes.

The gene is therefore a natural unit of selection.

However, selection operates on different stages of individual development.

Intragenomic conflict occurs when genes are selected for at earlier

stages of development that later may be disadvantageous.

This can occur if they are transmitted by different rules

Examples of such genes

• Transposons

• Cytoplasmatic genes

Individuals are not equivalent

The ultimate outcome of selection are changes in gene frequencies due to differential mating success.

Phe

noty

pic

fre

que

ncy

Phe

noty

pic

fre

que

ncy

Phe

noty

pic

fre

que

ncy

Phenotypic character value Phenotypic character valuePhenotypic character value

Parent Offspring

Diversifying selection Stabilizing selectionDirectional selection

Selection changes the frequency distribution of character states

EvoDots.exe

Selection triggers the frequency of alleles

A B Sum

Initial allele frequencies p q 1

Crossing AA AB,BA BB

Frequencies

Before Selection pp 2pq qq 1

Relative fitness w11 w12 w22

After selection w11p2 2w12pq w22q2 w11p2+2w12pq+w22q2

The absolute fitness W of a genotype is defined as the per capita growth rate of this genotype.

Using the Pearl Verhulst model of population growth absolute fitness is given by the growth parameter r of the logistic growth function for each genotype i.

dN(i) K NrN

dt K

The relative fitness w of a genotype is defined as the value of r with respect to the highest value of r of any genotype. w = W / Wmax.

The highest value of w is arbitrarily set to 1. Hence 0 ≤ w ≤ 1

The value s = 1 - w is defined the selection coefficient that measures selective advantage.s = 1 means highest selection pressure. s = 0 means lowest selection pressure.

A general scheme for two alleles

Absolute fitness is therefore equivalent to the reproduction rate of a focal population

A B Sum

Initial allele frequencies p q 1

Crossing AA AB,BA BB

Frequencies

Before Selection pp 2pq qq 1

Relative fitness w11 w12 w22

After selection w11p2 2w12pq w22q2 w11p2+2w12pq+w22q2

How do allele frequencies change after selection?

11 122 2

11 12 22

12 222 2

11 12 22

p(w p w q)p '

w p 2w pq w q

q(w p w q)q '

w p 2w pq w q

11 122 2

11 12 22

11 122 2

11 12 22

p(w p w q)p p ' p p

w p 2w pq w q

p(w p w q)dpp

dt w p 2w pq w q

The change of frequency of p is then

The general framework for studying allele frequencies after selection.

22212

211

22121211

2)]()([

qwpqwpwwwqwwppq

dtdp

The basic equation of classical population genetics

The dominant allele has the highest fitness

w11 = w12 > w22

w11 = w12 = 1

w22 = 1 - s

2

2

dp sp(1 p)

dt 1 s(1 p)

Rat poisoning with Warfarin in Wales shows how fast

advantageous alleles become dominant

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Generation

f(p

)

w22=0w22=0.3w22=0.5w22=0.7

w22=0.90

20

40

60

80

100

1975 1976 1977 1978

Year

Fre

qu

en

cy o

f re

sist

an

t

z

ind

ivid

ua

ls Start of Warfarin poisoning

End of Warfarin poisoning

22212

211

22121211

2)]()([

qwpqwpwwwqwwppq

dtdp

Lactose tolerance in the Neolithic

Poison tolerance in rats

If lactose tolerant children had a 20% better survival

probability, lactose tolerance would have been

common after about 100 generation (1500 years)

Heterozygotes have the highest fitness (heterosis effect)

w11 < w12 > w22

w12 = 1

w11 = 1 - s , w22 = 1 - t

2 2

dp p[1 p][ sp t(1 p)]

dt 1 sp t(1 p)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Generation

f(p

) w11=w22=0 w11=w22=0.3w11=w22=0.5

w11=w22=0.7

w11=w22=0.9

The heterosis effect stabilizes even highly disadvantageous alleles in a population Sickle cell anaemia

In heterozygote advantage, an individual who is heterozygous at a particular gene locus has a greater fitness than a homozygous

individual.

Reported values of selection coefficients

0

2

4

6

8

10

12

14

16

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Selection coefficient

Pe

rce

nta

ge

z N = 394

0

2

4

6

8

10

12

14

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Selection coefficient

Pe

rce

nta

ge

z

N = 172

Endler (1986) compiled selection coefficient

(s = 1 – w) for discrete polymorphic traits

Survival difference

Reproductive difference

Survival differences are:

• mostly small.

• Reproductive difference are larger.

• The proportion of significant differences in reproductive success is higher than for the survival difference.

• In many species only a small proportion of the population reproduces successfully.

All values

Only statistically significant values

Classical population genetics predicts a fast elimination of disadvantageous alleles.

Polymorphism should be low.

Natural populations have a high degree of polymorphism

Balancing selection within a population is able to maintain stable frequencies of two or more phenotypic forms (balanced polymorphism).

This is achieved by frequency dependent selection where the fitness of one allele depends on the frequency of other alleles.

Shell NocturnalPartly

nocturnalGeneral habitat

Exposed Very

exposedDark 9 5 0 0 0Medium 8 15 7 14 0Light 0 1 2 10 17White 0 0 0 1 3Polymorphic 0 0 8 10 14

Habitat

Shell colour and habitat preference of European Helicidae

Cepaea nemoralis

The fundamental theorem of natural selection

𝜎𝑤2 ∝𝑤 ∆𝑤

Sir Ronald Aylmer Fisher1890-1962

The Fisher equation is a tautology. It is a simple restatement of the definitions of mean and variance.

Nevertheless, it is the basic description of evolutionary changeBecause mean fitness and its variance cannot be negative,

the fundamental theorem states that fitness always increases through time

Evolution has a direction

Allele Gen. 1 Gen. 2 Gen. 3 Gen. 4 Gen. 5 Gen. 6 Gen. 7 Gen. 8

a 0.48 0.58 2.58 2.17 11.70 15.72 11.74 53.95b 0.44 1.88 2.90 6.01 1.26 6.43 43.04 53.62c 0.82 1.10 2.73 3.60 11.28 31.95 3.86 22.92d 0.28 0.24 3.15 3.00 7.38 30.23 21.26 50.59e 0.59 1.97 1.98 1.67 10.61 5.25 25.58 25.17f 0.88 0.84 2.81 4.59 3.11 1.93 47.73 117.87g 0.05 0.20 2.51 3.03 4.06 12.26 5.04 125.64h 0.59 1.81 3.41 4.98 14.13 6.43 15.26 92.09i 0.16 1.20 0.57 7.51 4.22 0.84 24.90 22.20j 0.86 1.80 1.13 2.24 4.23 18.55 17.25 94.49k 0.22 0.68 0.22 6.67 6.38 0.30 17.35 92.48Variance 0.10 0.51 1.45 4.93 20.31 125.92 212.91 1693.89Mean 0.46 1.07 2.12 4.20 8.22 21.32 37.16 203.74Difference in mean 0.61 1.05 2.08 4.02 13.10 15.84 166.58 0.65 2.23 8.74 33.08 279.18 588.74 33940.04

Fitness

0.10

1.00

10.00

100.00

1000.00

0.10 10.00 1000.00

s2

Selection effect Change in fitness

By definition variance and mean fitness have positive values.

Adaptive landscapes

Sewall G. Wright

(1889-1988)

Species occupy peaks in adaptive landscapes where altitude denotes fitness..

Species increase in fitness through time

Fitn

ess

Genetic composition / morphological structure

Tim

eGlobal peak

Local peak

Species A

Species A

Species A

Species A

Species B Species DSpecies C

Theodosius Dobzhansky (1900-1975)

High adaptive peaks are hard to climb but when reached they might allow for fast further evolution but also for long-term survival and

stasis.

Fitn

ess

A B

C

D EF

Species

To evolve into new species they first have to cross adaptive valleys

Genetic composition / morphological structure

Evolution without change in fitness

Neutral evolution and genetic drift

Motoo Kimura (1924-1994)

A1

A2

A3

A4

A5

Time

Assume a parasitic wasp that infects a leaf miner. Take 100 wasps of which 80 have a yellow abdomen and 20

have a red abdomen. A leaf eating elephant kills 5 mines containing red and 3 mines containing yellow wasps.

By chance the frequencies of red and yellow changed to 15 red and 77 yellow ones.

The new frequencies are red: 15/(15+77) = 0.16yellow: 1-0.16 = 0.84

During many generations changes in gene frequencies can be viewed as a random walk

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80Time

N

i0 = 20 i80 = 12

A random walk of allele occurrences

(1/ )

2ln(1/ ) ln(1/ )ln( )

2

Ep

p pT N

Var

The Foley equation of species extinction probabilities applied to allele frequencies

0

200

400

600

800

1000

1200

1400

1 10 100 1000 10000 100000

Initial number of allele A

Su

rviv

al t

ime

z

At low allele frequencies survival times are approximately logarithmic

functions of frequency

Survival times of alleles

Effective population size

If we have N idividuals in a population not all contribute genes to the next generation

(reproduce).

The effective population size is the mean number of individuals of a population that

reproduce.

Consider a diploid population of effective population size Ne.

Let ue be the neutral mutation rate at a given locus.

Neutral mutations are those that don’t significantly effect fitness.

The number of new neutral mutations is 2Neue.

The frequency of heterozygotes in a neutral population is

e e

e e

4N uH

4N u 1

At fairly high population sizes neutral theory predicts high levels of

polymorphism.

Neutral genetic drift explains the high degree of polymorphism in natural populations.

For a mutation rate of u0 = 10-6 we get

0.001

0.01

0.1

1

0 20000 40000 60000 80000Ne

H

u0 = 0.000001

Genome complexity and genetic drift

Assume a newly arisen neutral allele within a haploploid population of effective size Ne.

Given a mutation rate of u of this allele uNe mutations will occur within the population.

Mutations are removed

Mutations can be fixed by genetic drift

Selective effect of mutation

N e

-10-3 -10-4 -10-5 -10-6 -10-7 -10-8

104

105

106

107

108

NeutralNegative

VertebrataLand plants

Invertebrates

Unicellular eucaryotes

Procaryotes

The low effective population sizes of higher organisms increase the speed of evolution to a power because a

much higher proportion of mutations can be fixed through genetic drift.

In accordance with the Eigen equation only small effective

population sizes allow for larger genome sizes.

Lynch and Connery 2003

y = 0.0522x-0.548

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000

Gen

ome

size

(MB)

Nu

Eukaryotes

Prokaryotes

ProkaryotesUnicellular EukaryotesInvertebrates

Land plantsVertebrates

Populations must not become to small

Time

Pop

ulat

ion

size

Bottleneck of very low population size

Bottlenecks

increase the degree of inbreeding,

decrease the genetic variability,

increase the effect of genetic drift

Recovery

Extinction

The population after recovery might have a significantly altered genetic composition compared to the original population (founder effect).

Man has extraordinary low genetic variation suggesting a bottleneck in sub-Saharan populations before 60,000 years.

Samaritans

Strictly inbreeding ethnic group of about 700 people.

Neanderthals

Lived in very small groups of highly inbreed people. Total population size was at most several

thousand in whole Europe.

Today’s reading

All about selection: http://en.wikipedia.org/wiki/Natural_selectionPolymorphism: http://en.wikipedia.org/wiki/Polymorphism_(biology)Fundamental theorem of natural selection: http://stevefrank.org/reprints-pdf/92TREE-FTNS.pdfand http://users.ox.ac.uk/~grafen/cv/fisher.pdf