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 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….
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.
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 Dp 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:
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:
VI1. 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:
VI2. 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:
VI3. 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:
VI4. 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:
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.
Contributor(s) 
Written  200204  Robert Kalmes 
Institut de Recherche sur la Biologie de l'Insecte, IRBI  CNRS  ESA 6035, Av. Monge, F37200 Tours, France 
© Atlas of Genetics and Cytogenetics in Oncology and Haematology  indexed on : Tue Nov 6 14:56:26 CET 2018 
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