An Attempt of Population Genetics Considering Selection and Fitness

The HARDY-WEINBERG equilibrium in its simple form gives all alleles the same importance. This does not correspond with reality, and it is therefore necessary to include the fitness (W). Fitness and the selection coefficient that is proportional to the fitness are characterized by relative values. It can only be determined in retrospection, i.e. it is necessary to know the reproductive success of an individual in order to know something about its fitness.

The total (inclusive) fitness (W) of a population is made up from the fitness and the frequency of the single genotypes. If only two alleles with the frequencies p and q belonging to the same gene location are regarded, then A can be assigned the fitness value w₁ and a the value w₂. This means that

W = p²w₁₁ + 2 pq w₁₂ + q²w₂₂

The frequency of p after one generation shall be p₁. Without selection, nothing would change:

delta p = p₁ - p = 0

The same is true for q. If, nevertheless, the fitness values are taken into consideration, then the value of p₁ changes as follows:

p₁ = [p² w₁₁ + pqw₁₂] / [p² w₁₁ + 2 pqw₁₂ + q² w₁₂] = p₂ w₁₁ + pqw₁₂ / W

The change in the total fitness of the population (delta w = W₁ – W) after one generation is thus:

delta w = [p₁²w₁₁ + 2 p₁q₁ w₁₂ + q₁² w₂₂] - W = w₁₁ [p₁² - p²]+ 2 w₁₂ (p₁ q₁ - pq) + w₂₂(q₁₂ - q).

These equations are well suited for the extrapolation of the fate of single alleles in subsequent generations. The results can be depicted as exponential functions that show how long an unfavourable allele will survive in a given population or how fast a favourable one succeeds.

In order to use equations like the one above or to extend them according to need, it is necessary to have exact data of the population’s structure. Theoretical analyses do usually fail due to a lack of sufficient numbers. On the other hand, mathematical models are far too simple to cover the processes of natural populations completely. Mathematical attempts are nevertheless well suited in order to elucidate partial aspects of evolutionary research, in order to estimate how large the probability for the existence of a certain event is, and to learn something about the prerequisites required for the development of the evolution of complex biological systems.