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10. 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
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.
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 dp
p p p p i p p i p pt dt
0 0 0
0 0 0
* (1 ) *
1newN p eN p iN p p e ip
pN eN iN e i
Constant immigration of individuals causes a 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))
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 changes the frequencies 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 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 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
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 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
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2)2()(
)()(
qwpqwpwwwqwwppq
dtdp
qwpqwpwpqwpqwpwqwpwp
dtdp
wpwqwpwp
pw
qwpwpdtdp
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 25
Generation
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 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
0
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
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 25
Generation
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 spq
dt 1 sp(p 2q)
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
Generation
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 40
Generation
f(p
)
w11=0.3
w11=0.5
w11=0.7
w11=0.9
q0 = 0.01
p0 = 0.99
22212
211
22121211
2)]()([
qwpqwpwwwqwwppq
dtdp
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 Heterozygote advantage
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.
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 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 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 Dz = 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
nn z
w 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 w
w
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 fitness
The 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)
Me
an
fitn
ess
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)
Me
an
fitn
ess
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 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 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 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 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 1
Neu
Ge
no
me
siz
e (
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