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10. Genetic variation and fitness
Hardy Weinberg law
According to the Hardy Weinberg law gene
frequencies are constant. How can evolution occur?
Assume a gene with two alleles A and B that occur
with frequency p and q 1-p.
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
The frequency of heterozygotes is highest at p
q 1/2
None of these assumptions is fully met in
nature. Thus, gene frequencies permanently
change Therefore, evolution must occur!
What is the frequency after crossing?
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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
Equilibrium conditions
The change in p is the sum of forward and
backward mutations
Mutation rate m
The change in gene frequency is assumed to be
proportional to actual gene frequency multiplied
with the mutation rate.
At equilibrium dp/dt 0
Under constant forward and backward mutation
rates p and q will achieve equilibrium
frequencies. Otherwise they will permanently
change.
The change of gene frequency follows an
exponential function
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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. 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
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.
Constant immigration of individuals causes a
linear change in allele frequency
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Individuals are not equivalent
If individuals are not equivalent they have
different numbers of progenies. Selection sets in
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.
Five levels of natural selection
Zygotes
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
Ontogenetic selection
Compatability selection
Gametes
Children
Gametic selection
Viability selection

• Examples of such genes
• Transposons
• Cytoplasmatic genes

Mating success
Adults
Parents
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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
Diversifying selection
Stabilizing selection
Directional selection
Parent
Offspring
Phenotypic frequency
Phenotypic frequency
Phenotypic frequency
Phenotypic character value
Phenotypic character value
Phenotypic character value
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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.
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
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How do allele frequencies change after selection?
The mean fitness is defined as the average
fitness of all individuals of a population
relative to the fittest genotype.
The change of frequency of p is then
The general framework for studying allele
frequencies after selection.
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1. The dominant allele has the highest
fitness w11 w12 gt w22
2. Heterozygotes have the highest fitness
(heterosis effect) w11 lt w12 gt w22
w11 w12 1 w22 1 - s
w12 1 w11 1 - s , w22 1 - t
Rat poisoning with Warfarin in Wales shows how
fast advantageous alleles become dominant
The heterosis effect stabilizes even highly
disadvantageous alleles in a population
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3. Heterozygotes have the lowest fitness w11 gt
w12 lt w22
4. The recessive allele has the highest
fitness w11 w12 lt w22
w11 w22 1 w12 1 - s
w22 1 w12 1 - s , w11 1 - s
Heterozygote disadvantage leads to fast
elimination of the allele with initially lower
frequency.
Recessive allele frequency increases slowly. It
may take a long time for a rare recessive
advantageous allele to become established
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Reported values of selection coefficients
Survival difference
Endler (1986) compiled selection coefficient (s
1 w) for discrete polymorphic traits

• 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.

Reproductive difference
All values
Only statistically significant values
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Classical population genetics predicts a fast
elimination of disadvantageous alleles. Polymorphi
sm 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.
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
Sickle cell anaemia
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The fundamental theorem of natural selection
k groups with n members
Parents
k groups with n members
Children
The arithmetic mean and covariance of n elements
grouped into k classes is defined as
Now consider the average value of a morphological
or genetic character z that changes from parent
to child generation as Dz z-z.
The Price equation is the basic mathematical
description of evolution and selection
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The fundamental theorem of natural selection
If we take the change of w we get from zw
If w differs only slightly from w we get
Fishers fundamental theorem of natural selection
Sir Ronald Aylmer Fisher1890-1962
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.
Selection effect
Innovation effect
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
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Adaptive landscapes
Sewall Green Wright (1889-1988)
Species A
Species B
Global peak
Local peak
Mean fitness
Adaptive peak
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.
Adaptive landscapes
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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. By chance the
frequencies of red and yellow changed to 15 red
and 77 yellow ones. The new frequencies are
red 15/(1577) 0.16yellow 1-0.16 0.84
A3
A4
A5
During many generations changes in gene
frequencies can be viewed as a random walk
Time
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A random walk of allele occurrences
At low allele frequencies survival times are
approximately logarithmic functions of frequency
Survival times of alleles
The Foley equation of species extinction
probabilities applied to allele frequencies
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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.
The frequency of heterozygotes in a neutral
population is
For a mutation rate of u0 10-6 we get
Consider a population of effective population
size Ne. Let ue be the neutral mutation rate at a
given locus. Neutral mutations are those that
dont 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
At fairly high population sizes neutral theory
predicts high levels of polymorphism.
Neutral genetic drift explains the high degree of
polymorphism in natural populations.
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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. 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
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.
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Todays 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