Transcript Slajd 1

10. Genetic variation and fitness
Hardy Weinberg law
According to the Hardy Weinberg law
gene frequencies are constant.
Assume a gene with two alleles A and B that
occur with frequency p and q = 1-p.
A B
p
q
A p pp pq
Frequency
z
1
Assumptions of the Hardy Weinberg law
0.4
1. No mutations to generate new alleles
(no genetic variability)
0.2
( p  q)2  p 2  2 pq  q 2  1
After crossing
Frequency of B
pp
2pq
0.6
B q qp qq
AA
p2
qq
0.8
How can evolution occur?
0
0
0.2
0.4
0.6
0.8
1
Frequency p of allele A
The frequency of
heterozygotes is
highest at p = q = 1/2
AB
2pq
2pq / 2
BB
q2
q2
Sum
1
pq+q2
What is the frequency after crossing?
pq  q 2
q( p  q)

q
2
2
2
( p  2 pq  q ) ( p  q)
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!
Mutation rates
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  D  M  kD

M kD

N
N
Mutation rate 
The change in gene frequency is assumed
to be proportional to actual gene frequency
multiplied with the mutation rate.
dp
  p
dt
dq
 q
dt
Equilibrium conditions
The change in p is the sum of
forward and backward mutations
dp
   p  q    p  (1  p)
dt
At equilibrium dp/dt = 0
 p   q   (1  p)  p 

 
p  p0 e  t
q  q0 e  t
The change of gene frequency
follows an exponential function
Under constant forward and backward
mutation rates p and q will achieve
equilibrium frequencies.
Otherwise they will permanently change.
Immigration of alleles
Nonrandom mating
If mating is totally random a
population is said to be panmictic.
Assume a population has an allele A with
frequency p.
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
Nnew  N0 p  Ne p  Ni p*  ( N0  eN0 ) p  iN0 p *
pnew 
N 0 p  eN 0 p  iN 0 p * p(1  e)  ip *

N 0  eN 0  iN 0
1 e  i
p
dp
 pnew  p0  p0  p0  i( p0  p*)   i( p0  p*)
t
dt
Constant immigration of individuals causes a
linear change in allele frequency
A special type of nonrandom mating is
inbreeding.
Degree of relatedness z
Due to migration the next generation gets
individuals from outside by immigration and
looses individuals by emigration.
Inbreeding results in the
accumulations of homozygotes.
First
cousins
3/2
cousins
Second
cousins
Not
related
0
10
20
30
40
Percent offspring mortality
(< 21 years))
Inbreeding depression due to
homozygosity in Italian marriages
1903-1907.
Individuals are not equivalent
If individuals are not equivalent they
have different numbers of progenies.
Selection changes frequencies of genes.
Selection sets in
Five levels of natural selection
Zygotes
Compatability
selection
Ontogenetic
selection
Gametes
What is the unit of selection?
Children
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
Gametic
selection
Viability
selection
Mating
success
Parents
Examples of such genes
• Transposons
Adults
• Cytoplasmatic genes
Individuals are not equivalent
The ultimate outcome of selection are changes in gene frequencies due to differential mating
success.
Selection changes the frequency distribution of character states
Phenotypic character value
Parent
Offspring
Phenotypic character value
Stabilizing selection
Phenotypic frequency
Directional selection
Phenotypic frequency
Phenotypic frequency
Diversifying selection
Phenotypic character value
Selection changes the frequencies of alleles
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N
 rN
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
p
q
1
AA
AB,BA
BB
Before Selection
pp
2pq
qq
Relative fitness
w11
w12
w22
After selection
w11p2
2w12pq
w22q2
Initial allele frequencies
Crossing
Frequencies
1
w11p2+2w12pq+w22q2
A
B
Sum
p
q
1
AA
AB,BA
BB
Before Selection
pp
2pq
qq
Relative fitness
w11
w12
w22
After selection
w11p2
2w12pq
w22q2
Initial allele frequencies
Crossing
Frequencies
1
w11p2+2w12pq+w22q2
How do allele frequencies change after selection?
p(w11p  w12q)
p' 
w11p 2  2w12 pq  w 22q 2
The mean fitness is defined as the average
fitness of all individuals of a population
relative to the fittest genotype.
q(w12 p  w 22q)
q' 
w11p 2  2w12 pq  w 22q 2
w  w11p2  2w12pq  w 22q2
The change of frequency of p is then
p  p ' p 
p(w11p  w12q)
p
2
2
w11p  2w12 pq  w 22q
p(w11p  w12q)
dp

p
2
2
dt w11p  2w12 pq  w 22q
dp p( w11 p  w12q)
p( w11 p  w12q)  w p

p
dt
w
w
dp p( w11 p  w12q)  ( w11 p 2  2w12 pq  w22q 2 ) p

dt
w11 p 2  2w12 pq  w22q 2
dp pq[ p( w11  w12 )  q( w12  w22 )]

dt
w11 p 2  2w12 pq  w22q 2
The general framework for studying allele
frequencies after selection.
1. The dominant allele has the highest
fitness
2. Heterozygotes have the highest
fitness (heterosis effect)
w11 = w12 > w22
w11 < w12 > w22
dp pq[ p ( w11  w12 )  q( w12  w22 )]

dt
w11 p 2  2w12 pq  w22q 2
w11 = w12 = 1
w22 = 1 - s
dp sp(1  p)2

dt 1  s(1  p) 2
1
1
f(p)
0.8
0.4
0.6
w11=w22=0.5
w11=w22=0 w11=w22=0.3
0.4
w11=w22=0.7
0.2
0.2
w22=0.9
5
10
15
20
Generation
Rat poisoning with
Warfarin in Wales
shows how fast
advantageous alleles
become dominant
25
0
100
80
individuals
0
w11=w22=0.9
0
z
0
Frequency of resistant
f(p)
0.6
w11 = 1 - s , w22 = 1 - t
dp p[1  p][sp  t(1  p)]

dt
1  sp2  t(1  p) 2
w22=0
w22=0.3
w22=0.5
w22=0.7
0.8
w12 = 1
60
5
10
15
20
25
Generation
Start of
Warfarin
poisoning
40
End of
Warfarin
poisoning
20
0
1975
1976
1977
Year
1978
The heterosis effect stabilizes even highly
disadvantageous alleles in a population
3. Heterozygotes have the lowest
fitness
4. The recessive allele has the highest
fitness
w11 > w12 < w22
w11 = w12 < w22
dp pq[ p ( w11  w12 )  q( w12  w22 )]

dt
w11 p 2  2w12 pq  w22q 2
w11 = w22 = 1
w12 = 1 - s
1
0.8
0.8
0.6
0.6
0.4
w12=0.9
0.2
w12=0.7
0
0
5
w12=0.5
10
15
20
1
w11=0.9
w11=0.7
f(p)
1
w12=0
w12=0.3
w12 = 1 - s , w11 = 1 - s
dp
spq 2

dt 1  sp(p  2q)
f(p)
f(p)
dp
spq(p  q)

dt 1  s(p 2  q 2 )
w22 = 1
w11=0.9
0.9
w11=0.7
0.4
0.2
w11=0
0
25
Generation
Heterozygote disadvantage leads to fast
elimination of the allele with initially
lower frequency.
0
5
w11=0.3
w11=0.5
10
20
15
Generation
25
w11=0.5
q0 = 0.01
w11=0.3
p0 = 0.99
0.8
0
10
20
30
Generation
Recessive allele frequency increases slowly.
It may take a long time for a rare recessive
advantageous allele to become established
40
Reported values of selection coefficients
Percentage z
16
14
Survival difference
12
N = 394
Endler (1986) compiled
selection coefficient
(s = 1 – w) for discrete
polymorphic traits
10
8
6
4
2
0
0.05
0.15
0.25
0.35
0.45
0.55
0.65
0.75
0.85
0.95
Selection coefficient
Percentage z
14
Reproductive difference
12
N = 172
10
8
Survival differences are:
• mostly small.
• Reproductive difference
are larger.
• The proportion of
significant differences in
reproductive success is
higher than for the
survival difference.
6
4
2
0
0.05
All values
0.15
0.25
0.35
0.45
0.55
0.65
Selection coefficient
Only statistically significant
values
0.75
0.85
0.95
• In many species only a
small proportion of the
population reproduces
successfully.
Classical population genetics predicts a fast elimination of disadvantageous alleles.
Polymorphism should be low.
Natural populations have a high degree of polymorphism
Balancing selection
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.
Heterozygote advantage
In heterozygote advantage, an individual who
is heterozygous at a particular gene locus
has a greater fitness than a homozygous
individual.
Cepaea
nemoralis
Shell colour and
habitat preference
of European
Helicidae
Shell
Nocturnal
Dark
Medium
Light
White
Polymorphic
9
8
0
0
0
Partly
nocturnal
5
15
1
0
0
Habitat
General
habitat
0
7
2
0
8
Exposed
0
14
10
1
10
Very
exposed
0
0
17
3
14
Sickle cell anaemia
The fundamental theorem of natural selection
k
k groups with n members
Parents
k groups with n’ members
E(x) 
x n
i
i 1
i
n
k
Children
The arithmetic mean and covariance of n
elements grouped into k classes is defined as
Cov(x, y) 
 n [x
i 1
i
i
 E(x)][yi  (E(y)]
n
 E(xy)  E(x)E(y)
Now consider the average value of a morphological or genetic character z that changes from
parent to child generation as z = z’-z.
Cov(w i , zi )  E(w i zi )  E(w i )E(zi )
E(w i zi )  E(w i zi ')  E(w i zi )
Cov(w i , zi )  E(w i zi )  E(w i zi )  wz  E(w i z i ')  E(w i z i )  E(w iz i' )  wz
k
n i'
n i' zi'
wi  ; z '  
n i'
ni
'
i 1 n '
n
z
i i
k
k
k
w i n i' zi'
ni
n i' zi' n ' k n i' zi'
'
E(w i zi )  


 
 wz '
n
n
n
n
n
'
i 1
i 1
i 1
i 1
Cov(wi , zi )  E(wi zi )  wz '  wz  wz
The Price equation is the basic mathematical description of evolution and selection
The fundamental theorem of natural selection
Cov(wi , zi )  E(wi zi )  wz
If we take the change of w we
get from z=w
Cov(wi , wi )  E(wi wi )  Var(wi )  E(w i w i' )  E(w i w i )  Var(w i )  ww
If w’ differs only slightly from w we get Fisher’s
fundamental theorem of natural selection
Sir Ronald
Aylmer Fisher
1890-1962
Var(w i )
Var(w i )  w w  w 
w
The rate of increase in fitness of any organism at any time is equal to its genetic variance in
fitness at that time.
Cov(wi , zi )  E(wi zi )  wz
Selection effect
Innovation effect
Var(wi )  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
Adaptive landscapes
Mean fitness
x
w  w11p2  2w12pq  w 22q2
1
0.8
0.6
p = 0.4
unstable
equilibrium
0.4
0.2
Sewall Green Wright
(1889-1988)
Species A
0
0
0.2
0.4
0.6
0.8
1
Mean fitness
p(A)
Mean fitness
x
1
0.8
Adaptive peak
0.6
p = 0.4
stable
equilibrium
0.4
0.2
Species B
Global peak
Local peak
Species occupy peaks in adaptive landscapes
0
To evolve they have to cross adaptive valleys
0
0.2
0.4
0.6
0.8
p(A)
Adaptive landscapes
1
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.
Evolution without change in fitness
Genetic drift
A1
A2
Motoo Kimura
(1924-1994)
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.
A3
By chance the frequencies of red and yellow changed to
15 red and 77 yellow ones.
A4
The new frequencies are
red: 15/(15+77) = 0.16
yellow: 1-0.16 = 0.84
A5
Time
During many generations changes in gene
frequencies can be viewed as a random walk
A random walk of allele occurrences
9
i0 = 20
i80 = 12
7
z
1400
1200
6
Survival time
N
8
1000
5
4
3
800
600
400
200
2
0
1
1
10
100
1000
10000
100000
Initial number of allele A
0
0
20
40
60
80
Time
Survival times of alleles
TE 
2 ln(1/ p) 
ln(1/ p) 
ln(
N
)



Var(1/ p ) 
2 
The Foley equation of species extinction
probabilities applied to allele frequencies
At low allele frequencies survival
times are approximately logarithmic
functions of frequency
Effective population size
The frequency of heterozygotes in a
neutral population is
If we have N idividuals in a population not all
contribute genes to the next generation
(reproduce).
H
The effective population size is the mean
number of individuals of a population that
reproduce.
For a mutation rate of u0 = 10-6 we get
Consider a population of effective population
size Ne.
1
Let ue be the neutral mutation rate at a
given locus.
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
0.1
H
Neutral mutations are those that don’t effect
fitness.
4N e u e
4N e u e  1
0.01
u0 = 0.000001
0.001
0
20000
40000
60000
80000
Ne
At fairly high population sizes neutral
theory predicts high levels of
polymorphism.
Neutral genetic drift explains the high degree of polymorphism in natural populations.
Lynch and
Connery 2003
Genome complexity and genetic drift
Assume 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.
Mutations are
removed
10000
y = 0.03x-1.18
Mutations can be
fixed by genetic drift
107
Procaryotes
Unicellular
eucaryotes
106
Invertebrates
108
1000
Eucaryotes
Ne
Genome size (Mb)
z
The average number of neutral mutations is M = 4Neu
measuring M allows for an estimate of the effective population size Ne if u is constant.
100
105
10
10
Land plants
Vertebrata
4
Procaryotes
1
0.0001 0.001
0.01
0.1
1
Neu
In accordance with the Eigen equation
only small effective population sizes
allow for larger genome sizes.
-10-3
-10-4
-10-5
-10-6
-10-7
Negative
Selective effect of mutation
-10-8
Neutral
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.
Today’s reading
All about selection: http://en.wikipedia.org/wiki/Natural_selection
Polymorphism: http://en.wikipedia.org/wiki/Polymorphism_(biology)
Fundamental theorem of natural selection: http://stevefrank.org/reprints-pdf/92TREE-FTNS.pdf
and http://users.ox.ac.uk/~grafen/cv/fisher.pdf