10. Genetic variation and fitness

pp pqqp qqq

pqpBA

BA

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 q q

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

Freq

uenc

y

z2pq

ppqq

The frequency of heterozygotes is

highest at p = q = 1/2

Mutation rates

dp pdt

dq qdt

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 kDN N

Mutation rate

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.

Immigration of alleles Assume a population has an allele A with

frequency p.

Due to migration the next generation gets individuals from outside by immigration and

looses individuals by emigration.

Let i denote the immigration and e the emigrate rate. Both processes are assumed to be

proportional to actual density.

The total number of individuals before migration was N0. Ni individuals immigrated, Ne emigrated

0 0 0 0* ( ) * new e iN N p N p N p N eN p iN p

0 0 0 0 0( *) ( *)newp dpp p p p i p p i p pt dt

0 0 0

0 0 0

* (1 ) *1new

N p eN p iN p p e ippN eN iN e i

Constant immigration of individuals causes a linear change in allele frequency

Nonrandom matingIf 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

Deg

ree

of re

late

dnes

s z

Percent offspring mortality (< 21 years))

Individuals are not equivalent

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

Selection sets in

Zygotes

AdultsParents

Gametes Children

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

frequ

ency

Phe

noty

pic

frequ

ency

Phe

noty

pic

frequ

ency

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 changes the frequencies of alleles

A B SumInitial allele frequencies p q 1Crossing AA AB,BA BBFrequenciesBefore 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 a 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 NrNdt 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

A B SumInitial allele frequencies p q 1Crossing AA AB,BA BBFrequenciesBefore 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 pw p 2w pq w q

p(w p w q)dp pdt w p 2w pq w q

The change of frequency of p is then

The mean fitness is defined as the average fitness of all individuals of a population

relative to the fittest genotype.2 2

11 12 22w w p 2w pq w q

The general framework for studying allele frequencies after selection.

22212

211

22121211

22212

211

22212

2111211

12111211

2)]()([

2)2()(

)()(

qwpqwpwwwqwwppq

dtdp

qwpqwpwpqwpqwpwqwpwp

dtdp

wpwqwpwpp

wqwpwp

dtdp

1. The dominant allele has the highest fitness

w11 = w12 > w22

w11 = w12 = 1

w22 = 1 - s2

2

dp sp(1 p)dt 1 s(1 p)

2. 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)

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 25Generation

f(p)

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

w22=0.90

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25Generation

f(p) w11=w22=0 w11=w22=0.3

w11=w22=0.5

w11=w22=0.7

w11=w22=0.9

The heterosis effect stabilizes even highly disadvantageous alleles in a population

0

20

40

60

80

100

1975 1976 1977 1978Year

Freq

uenc

y of

resi

stan

t

zin

divi

dual

s Start of Warfarin poisoning

End of Warfarin poisoning

22212

211

22121211

2)]()([

qwpqwpwwwqwwppq

dtdp

3. Heterozygotes have the lowest fitness

w11 > w12 < w22

w11 = w22 = 1

w12 = 1 - s

w22 = 1

w12 = 1 - s , w11 = 1 - s

2 2

dp spq(p q)dt 1 s(p q )

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25Generation

f(p)

w12=0w12=0.3 w12=0.5

w12=0.7

w12=0.9

Heterozygote disadvantage leads to fast elimination of the allele with initially

lower frequency.

4. The recessive allele has the highest fitness

w11 = w12 < w22

2dp spqdt 1 sp(p 2q)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25Generation

f(p)

w11=0 w11=0.3 w11=0.5

w11=0.7

w11=0.9

Recessive allele frequency increases slowly.

It may take a long time for a rare recessive advantageous allele to become established

0.8

0.9

1

0 10 20 30 40Generation

f(p)

w11=0.3

w11=0.5

w11=0.7

w11=0.9

q0 = 0.01

p0 = 0.99

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Reported values of selection coefficients

02468

10121416

0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95

Selection coefficient

Per

cent

age

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

Per

cent

age

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 Heterozygote advantageBalancing 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.

Sickle cell anaemia

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

individual.

Shell Nocturnal Partly nocturnal

General 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 arithmetic mean and covariance of n elements grouped into k classes is defined as

k

i ii 1

k

i i ii 1

x nE(x)

n

n [x E(x)][y (E(y)]Cov(x, y) E(xy) E(x)E(y)

n

Parents

Children

k groups with n members

k groups with n’ members

Now consider the average value of a morphological or genetic character z that changes from parent to child generation as z = z’-z.

i i i i i iCov(w ,z ) E(w z ) E(w )E(z )

The fundamental theorem of natural selection

i i i i i i

'i i i i i i i i i i i i

E(w z ) E(w z ') E(w z )

Cov(w ,z ) E(w z ) E(w z ) wz E(w z ') E(w z ) E(w z ) wz

' ' 'ki i i

ii 1i

n n zw ;z 'n n '

''i

i i' ' ' ' ' 'k k k k' i i i i i i i i

i ii 1 i 1 i 1 i 1

n n zw n z n n z n zn 'E(w z ) wz '

n n n n n '

i i i iCov(w ,z ) E(w z ) wz ' wz w z

The Price equation is the basic mathematical description of evolution and selection

i i i iCov(w ,z ) E(w z ) w z

Sir Ronald Aylmer Fisher1890-1962

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

If we take the change of w we get from z=w

'i i i i i i i i i iCov(w ,w ) E(w w ) Var(w ) E(w w ) E(w w ) Var(w ) w w

ii

Var(w )Var(w ) w w ww

If w’ differs only slightly from w we get Fisher’s fundamental theorem of natural selection

The fundamental theorem of natural selection

i i i iCov(w ,z ) E(w z ) w z

Selection effect Innovation effect

iVar(w ) w w

Selection effect Change in fitnessThe Fisher Price equations are tautologies. They are simple restatements of the definitions of

mean and variance.

Nevertheless, they are the basic descriptions of evolutionary change

Because mean fitness and its variance cannot be negative, the fundamental theorem states that fitness always increases through time

Evolution has a direction

Sir Ronald Aylmer Fisher1890-1962

Adaptive landscapes

Sewall Green Wright (1889-1988)

2 211 12 22w w p 2w pq w q

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1p(A)

Mea

n fit

ness

x

p = 0.4unstable

equilibrium

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1p(A)

Mea

n fit

ness

x

p = 0.4stable

equilibrium

Adaptive landscapes

Adaptive peak

Species A

Mea

n fit

ness

Species B

Species occupy peaks in adaptive landscapes

To evolve they have to cross adaptive valleys

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.

Global peak

Local peak

Evolution without change in fitness

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 NVar

The Foley equation of species extinction probabilities applied to allele frequencies

0

200

400

600

800

1000

1200

1400

1 10 100 1000 10000 100000Initial number of allele A

Sur

viva

l tim

e

zAt 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 population of effective population size Ne.

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

Neutral mutations are those that don’t effect fitness.

The number of new mutations is 2Neue.

The number of neutral mutations that will be established in a population is therefore

(1/2Ne)*2Neue = ue

The frequency of heterozygotes in a neutral population is

e e

e e

4N uH4N 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 driftAssume a newly arisen neutral allele within a diploid population of effective size Ne.

The rate of genetic drift is therefore 1/2Ne.

Given a mutation rate of u of this allele u2Ne mutations will occur within the population.

The average number of neutral mutations is M = 4Neumeasuring M allows for an estimate of the effective population size Ne if u is constant.

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 = 0.03x-1.18

1

10

100

1000

10000

0.0001 0.001 0.01 0.1 1Neu

Gen

ome

size

(Mb)

z

Eucaryotes

Procaryotes

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