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Atlas of Genetics and Cytogenetics
in Oncology and Haematology
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Selection
Robert Kalmes
Institut de Recherche sur la Biologie de l'Insecte, IRBI - CNRS - ESA 6035, Av. Monge, F-37200 Tours,
France (RK)
Published in Atlas Database: April 2002
Online updated version : http://AtlasGeneticsOncology.org/Educ/SelectionID30040ES.html
DOI: 10.4267/2042/37890
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 France Licence.
© 2002 Atlas of Genetics and Cytogenetics in Oncology and Haematology
I- Introduction
II- Modeling and selective values
III- Basic model
IV- Equation of the recurrence of allele frequencies
V- Change in the selective values
VI- Change in populations
VI- 1. Homozygote A1A1 is the most advantaged;
VI- 2. Homozygote A1A1 is the most disadvantaged;
VI- 3. Heterozygote A1A2 is the most advantaged;
VI- 4. Heterozygote A1A2 is the most disadvantaged;
VII- Conclusions
straightforward situation, selection during the haploid
phase will not be sufficient to preserve genetic
polymorphism. We will see that the situation is
different if selection occurs during the diploid phase;
and this is what we are going to look at.
I- Introduction
We are going to consider a panmictic population,
sufficiently large for the allele frequencies to be
unaffected by any factor other than selection. We will
also assume that the impact of selective factors remains
constant over the generations, and that there is no
overlapping of generations. In this population, let us
assume that gene A is present in 2 allele forms, A1 and
A2, of which the frequencies in generation n are p and
q respectively.
NB: in the context of selection involving only the
haploid phase, it can be shown that the allele that
confers the greatest advantage on the gametes carrying
it will establish itself in the population. In this
Atlas Genet Cytogenet Oncol Haematol. 2002; 6(3)
II- Modeling and selective values
Many studies have attempted to model the effects of
natural selection on changes in allele frequencies over
the generations. The basic parameter used to quantify
the effect of selection is known as the selective value
(or "adaptative value") of the phenotype (Darwinian
fitness), and it is conventionally represented as w.
In practice, the phenotype and genotype are linked by
the rules of genetic determinism, and the genotype is
directly linked to the selective value of the phenotype
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Selection
Kalmes R
that it determines. We shall also be discussing the
selective values of the various genotypes. So, in the
case of a diallele autosomal locus….
It should also be noted that it is possible to express
these values either as a difference from value 1, or in
the form w = 1-s. In this case, the parameter s is known
as the selective coefficient. In the example below, we
therefore have
w1 = 1, w2 = 1-s (with s = 0,1), w3 = 1 - t (with t = 0,25)
III- Basic model
In the simplest model that we are going to consider
here, these values represent all the constituents of the
selective value of each genotype for the prereproductive period: embryonic survival, larval or
juvenile survival …). This corresponds to the mean
number of descendants contributed to the next
generation by each of the genotypes.
Only the situation in which the selection operates
between fertilization and the moment when the product
of fertilization itself reaches the age of reproduction is
considered here; this component of selection is the
viability (v). According to this model, all the mature
individuals have the same reproductive potential, and
contribute the same mean number of descendants (f) to
the next generation. An individual with a survival
probability of vi therefore contributes vi.f descendants
to the next generation.
Many different components can contribute to the
selective value of an individual, but it is the global
effect that is taken into consideration by these models.
In the end, the selective value depends on the
probability of survival of the genotype concerned and
on its fecundity. The Table below shows how the
selective values can be estimated if we know the
number of descendants of each genotype.
In practice, it is often the ratio between these values
that is what matters for the change in allele frequencies.
In this case, what is used is the relative selective values,
calculated by relating the absolute values to the "best"
value of the genotype, hence here
w1 = 1, w2 = 0,9, w3 = 0,75.
Atlas Genet Cytogenet Oncol Haematol. 2002; 6(3)
We will consider a panmictic population, of infinite
size, with non-overlapping generations, and which is
not affected by any factors for evolutionary change
other than selection. It is assumed that the effect of the
selective factors remains constant over time (constant
selective values model), and that these factors only
affect the survival of individuals between the zygote
stage and the reproductive adult stage. This basic
model, therefore excludes selective differences that
could involve various possible crosses between
individuals of different genotypes.
It can be seen that if the three selective values are equal
to one another, in terms of their relative values w1 = w2
= w3, there is no selection differential, and the model
corresponds to the Hardy-Weinberg model.
In this population, let us assume that a gene A exists in
2 allele forms, A1 and A2, of which the frequencies in
generation n are p and q respectively. In the simplest
situation, it is only the probabilities of survival of the
genotypes that differ. In this case, how will the allele
frequencies evolve?
IV- Equation of the recurrence of
allele frequencies between two
successive generations
The Table below summarizes the steps in the
calculation, showing the values of the genotype
frequencies before and after selection.
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And hence we can deduce:
Still: W = w1p2 + 2w2pq + w3q2
W corresponds to the mean selective value of the
population. It is proportional to the mean number of
descendants contributed by a given individual to the
nith generation. This is the weighted mean of the
selective values of the different genotypes. This is an
important value that will crop up again.
VIChange
in
populations
subjected to the effects of selection
V- Change in the selective values
between
two
successive
generations
We will now look at how p and q evolve, towards what
W tends, and what is the sign of ∆p in the 4
fundamental situations:
• Homozygote A1A1 is the most advantaged w1 > w2 >
w3
• Homozygote A1A1 is the most disadvantaged w1 <
w2 < w3
• Heterozygote A1 is the most advantaged w2 > (w1;
w3)
• Heterozygote A1 is the most disadvantaged w2 < (w1;
w3)
VI-1. Homozygote A1A1 is the most advantaged w1
> w2 > w3
∆p = pq/W [(w1 - w2) p + (w2 - w3)q]
Another important value for studying selection if the
change in allele frequencies between two successive
generations: ∆p = p’- p, where p is the frequency of
allele A1 in the nith generation.
The sign of ∆p tells us whether the frequency of allele
A1 has increased, decreased or remained the same. If it
has remained the same, then we are in a situation of
steady-state (or equilibrium). ∆p can be expressed as
follows:
After reducing to the same denominator
eliminating any common factors, this yields:
and
Note: w1 - w2 > 0 et w2 - w3 > 0 → ∆p > 0 regardless of
the values of p and q
→ establishment of the allele A1
Outcome of a simulation, where: w1 = 1, w2 = 0.9, w3 =
0.3:
However, 1 - p = q
and 1 - 2 p = q - p
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p = f (n) with the situation of the six different frequencies of p in the 0 generation.
The frequency of allele A1 always increases and tends towards 1.
The maximum value of W tends towards 1
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Selection
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∆p is always within the range [ 0 ; 1 ]
If the homozygote A1A1 is the most advantaged
genotype, allele A1 becomes established in the
population, and allele A2 is eliminated.
VI-2. Homozygote A1A1 is the most disadvantaged:
w1 < w2 < w3
∆p = pq/W [(w1 - w2) p + (w2 - w3)q]
Note: w1 - w2 < 0 et w2 - w3 < 0 → ∆p < 0, regardless of
Atlas Genet Cytogenet Oncol Haematol. 2002; 6(3)
the values of p and q
→ establishment of the allele A2
Outcome of a simulation, where: w1 = 0.6, w2 = 0.9, w3
= 1:
p = f (n) with the situation of the six different
frequencies of p in the 0 generation
If homozygote A1A1 is the most disadvantaged, the
frequency of allele A1 always falls and tends towards 0
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Selection
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Point p = 0 et q = 1 is an equilibrium point known as
"trivial", the value of W is maximum at the equilibrium
point.
→ Genetic polymorphism conserved / Stable
equilibrium
Outcome of a simulation, where: w1 = 0.9, w2 = 1, w3 =
0.95:
p = f (n) with the situation of the six different
frequencies of p in the 0 generation
∆p
is alwaysnegative over the range [ 0 ; 1]
VI-3. Heterozygote A1A2 is the most advantaged w2
> (w1; w3)
∆p = pq/W [(w1 - w2) p + (w2 - w3)q]
Note:
w1 - w2 < 0 → ∆p > 0 from 0 to equilibrium p
w2 - w3 > 0 → ∆p < 0 from equilibrium p to 1
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When the heterozygote genotype A1A2 is more
advantaged than either of the homozygotes, the
population tends towards a state of stable, polymorphic
equilibrium (and both the A1 and A2 alleles are
conserved).
p at equilibrium corresponds to (w3 - w2)/(w12w2+w3) = 0.33
Atlas Genet Cytogenet Oncol Haematol. 2002; 6(3)
The equilibrium frequency of allele A1(0.33)
corresponds to a maximum value of W.
W equilibrium = W1 p2 equilibrium + 2 W2 p q
equilibrium + W3 q2 equilibrium = 0.966
Obviously, it is for this value that ∆p is zero In the
range [ 0 ; 1 ].
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p = f (n) with the situation of the six different
frequencies of p in the 0 generation
When the heterozygote genotype is the most
disadvantaged of all the genotypes, the population
tends to fix either allele A1, or allele A2.
There is a specific point, the equilibrium point p, where
∆p is cancelled out in the range [ 0 ;1] .
VI-4. Heterozygote A1A2 is the most disadvantaged:
w2 < (w1; w3)
∆p = pq/W [(w1 - w2) p + (w2 - w3)q]
Note:
w1 - w2 > 0 → ∆p < 0 from 0 to p equilibrium
w2 - w3 <0 → ∆p > 0 from p equilibrium to 1
→ so either allele A1 or allele A2 will become
established: Unstable equilibrium
Outcome of a simulation, where: w1 = 0.9, w2 = 0.8, w3
= 1:
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Selection
Atlas Genet Cytogenet Oncol Haematol. 2002; 6(3)
Kalmes R
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Selection
Kalmes R
This is an unstable equilibrium point, which cannot
actually exist unless the population is infinite in size. In
a real population, random variations of p will be
observed.
This article should be referenced as such:
Kalmes R. Selection. Atlas
Haematol.2002;6(3):260-269
VII- Conclusions
In the model of selection with constant selective values,
the population always develops towards a situation in
which W is a maximum. This is a characteristic of the
"fundamental theory" of natural selection.
However, it is only exactly true in this model.
Despite this, in general, natural selection tends to
maximize the mean number of descendants of the
population. If there are different constraints (as, for
instance, in the model with variable selective values), it
may simply tend towards an optimum close to, but less
than the highest value of W.
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269
Genet
Cytogenet
Oncol