Commentaries or "Biological mathematics"

Let us model the following situation:

Let us assume that the population is big enough to breed and that the environment determines the population to be on the same quantitative level. The subsequent populations will be referred to as P0, P1, P2,... Pn,...

If in applet we perform computations on the input data given by the user, here we make our reasoning on the abstract data.

Our biological model may be described as the following mathematical model. Let us consider the sequence of numbers a0, a1, a2,... an,..., where a0 is a given initial value of the fraction of homozygotes (AA) in the diploidal population of individuals, an is the fraction of homozygotes (AA) in the n-th generation. The question is: What will be the subsequent numbers an in the sequence?. Does it converge and, if so, what is its limit?

The mechanism of generating the offspring population works as follows: we pick at random from the parent population two individuals: "the mother" and the "father", according to the distribution of genes in the parent population. In the case of the first population P0 it would mean that a homozygous parent (AA) is picked with the probability a0, a heterozygous parent (AB) is picked with the probability (1-a0). If a heterozygous parent is picked at random, we also draw at random the allel that goes to the offspring. Because we can chose one from two allels with the same probability, we assume that it is equal to 0.5

Let (AA) and (BB) mean picking at random the homozygous parent (AA) or the heterozygous parent (AB) respectively. Let [(AB)A] and [(AB)B] mean picking at random the heterozygous parent (AB) and transferring the allel A or B respectively to the offspring. [(AA)A][(AB)B] means: the homozygous (AA) " the father" transfers the allel A to the offspring and the heterozygous (AB) " the mother" transfers the allel B to the offspring.

In the case where a parent is homozygous (AA) with probability 1, the allel A is transferred to the offspring. This will be denoted by [(AA)].

Accordingly, the event that the child is homozygous (AA) may be described as follows:
    [(AA)][(AA)]+[(AA)][(AB)A]+[(AB)A][(AA)]+[(AB)A][(AB)A]

And the probability of the above event, Q(AA), is equal to:
    Q(AA) = (1+a)2*(1/4) (show details of calculations)

The event that the child is heterozygous (AB) may be described as follows:
    [(AA)][(AB)B]+[(AB)B][(AA)]+[(AB)A][(AB)B]+[(AB)B][(AB)A]

Probability of that event let us denote it by Q(AB) is equal:
    Q(AB) = (1-a2)*(1/2) (show details of calculations)

The event that the child is lethally homozygous (BB) may be described by:
    [(AB)B][(AB)B]

Probability of that event let us denote it by Q(BB) is equal:
    Q(BB) = [(1-a)*(1/2)]*[(1-a)*(1/2)] = (1-a)2*(1/4)

Let us check if the probabilities are well counted, they ought to sum up to one. Indeed:
    Q(AA) + Q(AB) + Q(BB) = 1 (show details of calculations)

Only individuals which grow up to the reproduction stage influence the evolution of the population. So we can pass over the homozygous individuals (BB) and take a look at the population of individuals with (AA) and (AB). So instead of the probabilities Q(AA), Q(AB) we will work with normed probabilities P(AA) and P(AB) where:
    P(AA) = Q(AA) / (Q(AA) + Q(AB)) = (1+a) / (3 -a) (show details of calculations)

So we obtain that:
    a1= P(AA) = (1+a) / (3 -a)

Also:
    P(AB) = Q(AB) / (Q(AA) + Q(AB)) = (2-2a) / (3 -a) (show details of calculations)

Because:
    P(AA) + P(AB) = 1,
our calculations are correct (show details of calculations)

Let us look at the sequence a0, a1, a2,... an,..., of the homozygous (AA) in the subsequent generations P0, P1, P2,... Pn,...
a0 - is given, assume that equal to number a ( 0<a<1 )
    from our calculation we know that:
    a1 = P(AA) = (1+a) / (3 -a)
    we can see that a0 < a1 (show details of calculations)

By similar reasoning we can deduce that the sequence a0, a1, a2,... an,..., is ascending.
We can also observe that  a1 is less than 1 (show details of calculations)

Similary we can see that all elements of the sequence (ai) are less than 1.
The same deduction give us that for generation P1 and P2 and fractions a1:
    a2 = (1 + a1) / (3 - a1)
    by calculations for a1 we obtain:
    a2 = 1 / (2 -a) (show details of calculations)

By similar deduction for a2, a3, and a4 we obtain:

Sequence (a2n) is an example of chain sequence.

What will happen if n tends to infinity? To what number will the fraction of the homozygous (AA) individuals in population converge?

We know that the sequence is ascending and bounded by 1, so we know that the sequence is convergent to the positive number not greater than 1. More over we know that the sequences (ai) and (a2i) converge to the same limit. We will count the limit of the sequence (a2i).
    Let us supose that the sequence converged to number z. We can than write

Last equality is easy to solve. We obtain:
    z = 1 (show details of calculations)

So the sequence (an) converge to 1, it's mean that the fraction of homozygous (AA) tends to 100%. During my talk with Mr. Sabath I learned that these calculations are correct and they confirm the observations made by biologists. If some allel is lethal in the homozygote and its occurrence do not give any advantage to the heterozygote, this allel is eliminated from the population. The situation is different when it gives. An example of this is the sickle-cell anemia. Allele of sickle-cell (let's call it a) anemia is lethal in homozygotes (aa) but heterozygotes(aA) are more immune from malaria than homozygotes (AA). As a result this allele is not removed from populations living in malaric areas.