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  p2  2 pq  q2  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!
Inbreeding
What is the probability for a children to get a certain allele from their grandparents?
Grandparents GM1
A,B
Parents
P(C)=0.25
GF1 GM2 GF2
C,D
E,F
G,H
M
F
P(C)=0
Ch
Childrens
GF1 is already inbred
A,B
P(C)=0.5
GF1 GM2 GF1
C,C
M
E,F
F
GF1 GM2 GF1
A,B
P(C)=0.25
E,F
M
F
C,D
P(C)=0.25
P(C)=0.25
The probability that Ch gets allele C is 0.25.
The mean probability to get an allele X from
one of the members of a lineage is called
the coefficient of inbreeding FX.
C,C
P(C)=0.5
Sewall Wright defined this
coefficient as
𝑛
Ch
C,D
Ch
P(C)=0.125
The probability that Ch gets allele C is 0.125.
GM1
GM1
P(C)=0.5
𝐹𝑋 =
1
2
𝑘+1
(𝐹𝐴 + 1)
𝑖=1
The probability that Ch gets allele C is 0.5.
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
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.
dq
 q
dt
dp
  p
dt
p  p0 e
 t
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 

 
Under constant forward and backward
mutation rates p and q will achieve
equilibrium frequencies.
q  q0 e  t
Otherwise they will permanently change.
The change of gene frequency
follows an exponential function
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.
Inbreeding results in the
accumulations of homozygotes.
Assortative mating describes a situation
where breeding occurs among individual
with similar genetic structure.
The opposite is called disassortative mating.
First
cousins
Domingue et al. 2014, PNAS
Quantile of genetic
similarity of pairs
Degree of relatedness z
A special type of nonrandom mating is
inbreeding.
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.
American pairs have
a slight (about 4.5% effect)
affinity to partners of similar
genetic predisposition
Quantile of cross sex genetic
similarity
Positive assortative mating increases the
degree of inbreeding
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 triggers the frequency of alleles
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N
 rN
dt
K
Absolute fitness is therefore equivalent to the reproduction rate of
a focal population
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+w22q
2
How do allele frequencies change after selection?
The change of frequency of p is then
p' 
p(w11p  w12 q)
w11p 2  2w12 pq  w 22 q 2
p  p ' p 
q' 
q(w12 p  w 22 q)
w11p 2  2w12 pq  w 22 q 2
p(w11p  w12q)
dp

p
dt w11p 2  2w12 pq  w 22q 2
p(w11p  w12 q)
p 
2
2
w11p  2w12 pq  w 22q
dp pq[ p( w11  w12 )  q( w12  w22 )]

dt
w11 p 2  2 w12 pq  w22 q 2
The general framework for studying allele frequencies after selection.
The basic equation of classical population genetics
The dominant allele has the highest fitness
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
Poison tolerance in rats
w22=0
w22=0.3
w22=0.5
w22=0.7
0.8
f(p)
100
0.6
0.4
0.2
w22=0.9
0
0
5
10
15
20
Generation
25
Frequency of resistant
individuals
1
z
dp sp(1  p)2

dt 1  s(1  p)2
Lactose tolerance in the
Neolithic
80
60
Start of
Warfarin
poisoning
40
End of
Warfarin
poisoning
20
0
1975
1976
1977
1978
Year
Rat poisoning with Warfarin
in Wales shows how fast
advantageous alleles
become dominant
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
dp p[1  p][sp  t(1  p)]

dt
1  sp 2  t(1  p) 2
In heterozygote advantage, an individual who
is heterozygous at a particular gene locus
has a greater fitness than a homozygous
individual.
1
f(p)
0.8
0.6
w11=w22=0.5
w11=w22=0 w11=w22=0.3
0.4
w11=w22=0.7
0.2
w11=w22=0.9
0
0
5
10
15
20
25
Generation
The heterosis effect stabilizes even highly
disadvantageous alleles in a population
Sickle cell anaemia
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 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.
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
The fundamental theorem of natural selection
a
b
c
d
e
f
g
h
i
j
k
Variance
Mean 
Difference in mean 

Gen. 1
0.48
0.44
0.82
0.28
0.59
0.88
0.05
0.59
0.16
0.86
0.22
0.10
0.46
Gen. 2
0.58
1.88
1.10
0.24
1.97
0.84
0.20
1.81
1.20
1.80
0.68
0.51
1.07
0.61
0.65
Fitness
Gen. 3
Gen. 4
Gen. 5
2.58
2.17
11.70
2.90
6.01
1.26
1000.00
2.73
3.60
11.28
3.15
3.00
7.38
1.98 100.00
1.67
10.61
2.81
4.59
3.11
2.51
3.03
4.06
10.00
3.41
4.98
14.13
0.57
7.51
4.22
1.00
1.13
2.24
4.23
0.22
6.67
6.38
1.45
4.93
20.31
0.10
2.12
4.20
8.22
0.10
1.05
2.08
4.02
2.23
8.74
33.08
s2
Allele
𝜎𝑤2 ∝ 𝑤∆𝑤
Selection effect
Sir Ronald
Aylmer Fisher
1890-1962
Gen. 6
15.72
6.43
31.95
30.23
5.25
1.93
12.26
6.43
0.84
18.55
0.30
125.92
21.32
10.00
13.10

279.18
Gen. 7
11.74
43.04
3.86
21.26
25.58
47.73
5.04
15.26
24.90
17.25
17.35
212.91
37.16
1000.00
15.84
588.74
Gen. 8
53.95
53.62
22.92
50.59
25.17
117.87
125.64
92.09
22.20
94.49
92.48
1693.89
203.74
166.58
33940.04
By definition variance and mean
fitness have positive values.
Change in fitness
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 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
Fitness
Species A
Species A
Species A
Global peak
Theodosius
Dobzhansky
(1900-1975)
Sewall G.
Wright
(1889-1988)
Species occupy peaks in adaptive
landscapes where altitude denotes fitness..
Local peak
Species A Species B Species C Species D
Species increase in fitness through time
Genetic composition / morphological structure
To evolve into new species they first have to
cross adaptive valleys
C
Fitness
AB
DE
F
Species
Genetic composition / morphological structure
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
Neutral evolution and 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
The frequency of heterozygotes in a
neutral population is
Effective population size
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.
4N e u e
4N e u e  1
For a mutation rate of u0 = 10-6 we get
1
Consider a diploid population of effective
population size Ne.
Neutral mutations are those that don’t
significantly effect fitness.
H
Let ue be the neutral mutation rate at a
given locus.
0.1
0.01
u0 = 0.000001
0.001
The number of new neutral mutations is
2Neue.
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 haploploid population of effective size Ne.
Given a mutation rate of u of this allele uNe mutations will occur within the population.
Eukaryotes
y = 0.0522x-0.548
Mutations can be
fixed by genetic drift
Procaryotes
Prokaryotes
Unicellular
Unicellular
eucaryotes
Eukaryotes
Invertebrates
Invertebrates
108
0.1
107
0.01
Ne
Genome size (MB)
1
Mutations are
removed
0.001
Prokaryotes
0.0001
1
10
100
Nu
106
105
Land
Landplants
plants
Vertebrata
Vertebrates
104
1000 10000
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.
Population size
Populations must not become to small
Bottleneck of very
low population size
Recovery
Extinction
Bottlenecks
increase the degree of inbreeding,
decrease the genetic variability,
increase the effect of genetic drift
Time
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_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