Selection
Contributor(s)
Written  200204  Robert Kalmes 
Institut de Recherche sur la Biologie de lInsecte, IRBI  CNRS  ESA 6035, Av. Monge, F37200 Tours, France 
 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
 Homozygote A1A1 is the most advantaged
 Homozygote A1A1 is the most disadvantaged
 Heterozygote A1A2 is the most advantaged
 Heterozygote A1A2 is the most disadvantaged
 VII. Conclusions
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 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.
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 that it determines. We shall also be discussing the selective values of the various genotypes. So, in the case of a diallele autosomal locus?.
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
w_{1} = 1, w_{2} = 0,9, w_{3} = 0,75.
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 = 1s. In this case, the parameter s is known as the selective coefficient. In the example below, we therefore have
w_{1} = 1, w_{2} = 1s (with s = 0,1), w_{3} = 1 ? t (with t = 0,25)
III. Basic model
We will consider a panmictic population, of infinite size, with nonoverlapping 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 w_{1} = w_{2} = w_{3}, there is no selection differential, and the model corresponds to the HardyWeinberg 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.
Still: W = w_{1}p^{2} + 2w_{2}pq + w_{3}q^{2}
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.
V. Change in the selective values between two successive generations
Another important value for studying selection if the change in allele frequencies between two successive generations: Dp = p? p, where p is the frequency of allele A1 in the nith generation.
The sign of D 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 steadystate (or equilibrium). Dp can be expressed as follows:
However, 1 ? p = q
and 1 ? 2 p = q ? p
And hence we can deduce:
VI. Change in populations subjected to the effects of selection
We will now look at how p and q evolve, towards what W tends, and what is the sign of Dp in the 4 fundamental situations:
 Homozygote A1A1 is the most advantaged w_{1} > w_{2} > w_{3}
 Homozygote A1A1 is the most disadvantaged w_{1} < w_{2} < w_{3}
 Heterozygote A1 is the most advantaged w_{2} > (w_{1}; w_{3})
 Heterozygote A1 is the most disadvantaged w_{2} < (w_{1}; w_{3})
6.1 Homozygote A1A1 is the most advantaged w_{1} > w_{2} > w_{3}
Dp = pq/W [(w_{1}  w_{2}) p + (w_{2}  w_{3})q]
Note: w_{1}  w_{2} > 0 et w_{2}  w_{3} > 0 > Dp > 0 regardless of the values of p and q
> establishment of the allele A1
Outcome of a simulation, where: w_{1} = 1, w_{2} = 0.9, w_{3} = 0.3:
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
Dp 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.
6.2 Homozygote A1A1 is the most disadvantaged: w_{1} < w_{2} < w_{3}
Dp = pq/W [(w_{1}  w_{2}) p + (w_{2}  w_{3})q]
Note: w_{1}  w_{2} < 0 et w_{2}  w_{3} < 0 > Dp < 0, regardless of the values of p and q
> establishment of the allele A2
Outcome of a simulation, where:
w_{1} = 0.6, w_{2} = 0.9, w_{3} = 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.
Point p = 0 et q = 1 is an equilibrium point known as "trivial", the value of W is maximum at the equilibrium point.
Dp is always negative over the range [ 0 ; 1]
6.3 Heterozygote A1A2 is the most advantaged w_{2} > (w_{1}; w_{3})
Dp = pq/W [(w_{1}  w_{2}) p + (w_{2}  w_{3})q]
Note: w_{1}  w_{2} < 0 > Dp > 0 from 0 to equilibrium p
w_{2}  w_{3} > 0 > Dp < 0 from equilibrium p to 1
> Genetic polymorphism conserved / Stable equilibrium
Outcome of a simulation, where:
w_{1} = 0.9, w_{2} = 1, w_{3} = 0.95:
p = f (n) with the situation of the six different frequencies of p in the 0 generation
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
The equilibrium frequency of allele A1(0.33) corresponds to a maximum value of W.
W equilibrium = W1 p^{2} equilibrium + 2 W2 p q equilibrium + W3 q^{2} equilibrium = 0.966
Obviously, it is for this value that Dp is zero In the range [ 0 ; 1 ].
6.4 Heterozygote A1A2 is the most disadvantaged: w_{2} < (w_{1}; w_{3})
Dp = pq/W [(w_{1}  w_{2}) p + (w_{2}  w_{3})q]
Note: w_{1}  w_{2} > 0 > Dp < 0 from 0 to p equilibrium
w_{2}  w_{3} <0 > Dp > 0 from p equilibrium to 1
> so either allele A1 or allele A2 will become established: Unstable equilibrium
Outcome of a simulation, where:
w_{1} = 0.9, w_{2} = 0.8, w_{3} = 1:
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 Dp is cancelled out in the range [ 0 ;1] .
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.
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.
Citation
Kalmes R
Atlas of Genetics and Cytogenetics in Oncology and Haematology 20020401
Selection
Online version: http://atlasgeneticsoncology.org/teaching/30040/selection